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OFFICE OF NAVAL RESEARCH if DEPARTMENT OF THE NAVY 



British Crown Copyright 
reserved. Reproduced with 
the permission ot the Control- 
ler of His Britannic Majesty’s 
Stationery Office. 


Bibliography 


Most of the papers in this volume contain references to further 
work, some of which may be included within this Compendium. Prop- 
erly qualified persons may obtain additional references o/material on 
application to the following establishments: 

1. Secretary, Undex Sub-Panel 

Naval Construction Research Establishment 
Rosvth 

Scotland 

2. Chief of Naval Research 
c / 0 Navy Research Section 
Library of Congress 
Washington 25, D. C. 

3. The Director 

David W. Taylor Model Basin 
Carderock, Maryland 

4. The Commander 

U.S. Naval Ordnance Laboratory 
White Oak, Maryland 


m 








PKSFACE 


( j During the recent war there arose on both sides of the Atlantic among research workers 

'' in the field of underwater explosions the feeling that some of the problems posed by the conditions 

oi undersea warfare had already presented themselves In the past and that various attempts had 
been made to solve them. Many of the records, however, had been lost or effectively hidden 
except for \vhat had crept into open publications and consequently a whole new literature had to 
be developed at co>' '('arable cost in both time and money, encompassing both old and new 
problems. A corol. ;■ of this experience has been the firm conviction that this new literature 
shmtld not suffer a similar fate. The idea of the joint publication of American and British 
research la the field of underwater explosions took form in the latter part of 1946 and the idea 
was further explored with the Bureau of Ordnance and the Bureau of Ships, United States Navy 
Eiepartment and with the British Admiralty. The Office of Naval Research, Navy Department, 
in its capacity of disseminator of scientific information undertook to sponsor the publication 
and 'has eventually seen the project through to its present form. 

The Compendium has three major purposes: first, to give a greater availability to many 
,j? papers which otherwise would exist in a very small number of copies, and to preserve and 

revive certain rare items, the scarcity of which was due to wartime shortages rather than to 
any deficiencies in the papers themselves; second, to present a representative summary of 
original source material and to display the scope of this mater al in a manner which might 
make it of more universal interest to schools and colleges as a branch of applied science; and 
third, to stimulate interest in this field for tte general benefit of the .sciences of Naval Archi- 
tecture and Naval Ordnance and to provide those working in these fields with ready reference 
material on many of the Important problems which they must face in their work. 


The scheme of the Compendium is as follows; All of the papers selected, which represent 
between idand 20 percent of the total quantity of material known to exist, have been divided 

into three volumes. The first volume is devoted to the primary unaes ater shoca wave, the 

second to the hydrodynamical effects falling under incompressible theory including the oscil- 
lations and behaviour of the gas globe formed by the explosion pr'^'iu.'ts, and the third to the 
effects of all of these phenomena on structures and to the measure ent and calculation of the 
rSsultihg damage. Three papers have been selected with the object of summarizing the knowledge 
over the field within the scope of the Compendium; these papers, which are placed in the first 
volume, servo to introduce the subject both in general terms, and also with some mathematical 
detail. 

The allocatton o! the original papers to the different volumes has, In a few cases, not been 
obvious and the editors, must assume full responsibility for any arbitrary assignments. A far 
greater responsibility of the editors has lain in the selection of the papers and in this, various 
considerations have had a voice. 



Many of the older papers have been included for their historical interest. Some papers 
have been used to provide suitable introductory or background material. Most of the other 
papers have been included intact and represent the opinions of the authors at the time of writing. 
A few of the papers have been reworked and ccitsist of new material incorporated Into the older 
original papers, or consist a summary of several progress reports which were too repetitive 
for economicalinclusion without condensation. Papers which have been rewritten are so marked 
yith the new date affixed. In general, selections have been made in an effort to give the best 
review of tfie entire subject in order to convey the most, and the best information within the 
space limitations imposed by the exige.'>>:ies of publication, and witliin the scope permitted by 
considerations of security. Both these features prevent this compilation from being exhaustive, 
and the latter feature prevents many succes.«(ful workers in this field from receiving recog- 
nition .here. 

The editors believe that this Compendium is a new venture in international co-operation 
and hope that this effort may prove useful in pointing the way for other similar joint enterprises 
which may be considered desirable. 

It i® our desire to acknowledge the continued Interest of Dr. A. T. Waterman, Deputy 
Chief, Oiiilce of Naval Research^ Navy Department, without whose help those volumes could not 


V 











have been produced, to thank Mr. Martin Jansson of the Techatcal Information Division, 

A *.•. «. a 4 ^^ .&u>k4M Mn<Ma^chtr4m«v cimH r«a'|*ofitl in tn6 

Ol H&vai i^e»€urcii> aau m» utipuDl^ St^«! Imi wro** v-«u ^,-s^a — ^ 

material tor reoroduction. We also acknowledge the guidance afforded by the Brltisn Uades 
Panel, particularly Dr. A. R. Bryant and Dr. E. N. Fox (a former member), the assistance of 
Mv. T. Ayes of the Department of Research Pr^ratiames & Planning and Miss E. Lord of the 
Department of Physical Research in the preparation of the British contribution, and to thank 
Dr. T. L. Brownyard of the Bureau of Ordimnce Navy Department for his help in some oi the 
correspondence and in some the problems of security clearance. 

G. K. Hartmann 

Chief, Explosive Research Department 
U.S. Naval Ordnance I,aboratory 

E. G. Hill 

Department of Physical Research 
Admiralty 


\ 

j 




( 





[S«k«lSS*5»n!«rarafi^^ 




TABLE OF CONTENTS 
VOLUME n 
THE GAS GLOBE 


5 • 


If-' ' 


#T^ 


Prefece v 

SUMMARY PAPERS 

REPORT OH THE THEORETICAL SHAPE OF THE PRESSURE-TIME CURVE 
AND ON THE GROWTH OF THE GAS-BUBBLE (B)* 

s. Butterworth, Admiralty Research Laboratory 1 

UNDERWATER EXPLOSIONS. TIME IhlTERVAL BETWEEN SUCCESSIVE 
EXPLOSIONS (B) 

H. F. Willis, H. M. Anti-Submarine Experimental Establishment 13 

THEORY OF THE PULSATIONS OF THE GAS BUBBLE PRODUCED BY AN 
UNDERWATER EXPLOHON (A)*' 

Conyers Herring, Bell Telephone Laboratories, Murray Hill, N. J. ........ 35 

THEORY OF PULSATIONS 

VERTICAL MOTION OF A SPHERICAL BUBBLE AND Tim PRESSURE 
SURROUNDING IT (B) 

G. I. Taylor, Cambridge University 131 

ON The changing form of a nearly spherical submarine 

BUBBLE (B) 

W, G. Penney and A. T. Price, Imperial College of Science and Technology, 

London 145 

THE BEHAVIOUR OF AN UNDERWATER EXPI..OSION BUBBLE (B) 

A. R. Bryant, Road Research Laboratoiy, London . 163 

RADIAL hlOnON OF WATER SURROUNDING A SPHERE OF GAS m 
RELATION TO PRESSURE WAVES (A) 

' E. H. Kennard, David. W. Taylor MmsI Basin .. 183 

The motion and shape of ’^HE hollow PRCHIUCED BY AN 
EXPLOSION IN A LIQUID (B) 

G. I. Taylor and R, M. Davies, Cambriijge University 227 

$TUDmS ON THE GAS BUBBLE RESULTING FROM UNDERWATER 
EXPLOSIONS: ON THE BEST LOCATION OF A MINE NEAR THE 
SEA BED (A) 

M. Shlffman and B. Friedman, Institute tor Mathematics and 

Mechanics, New York University 247 

THE BEHAVIOUR OF AN UNDERWATER EXPLOSION BUBBLE FURTHER 
APPROXIMATIONS (B) 

A. R. Bryant, Road Research Ijaboratory, London 321 

THEORY OF UNDERWATER EXPLOSION BUBBLES (A) 

S. Friedman, Institute for Mathematics and Mechanics, 

New York Uidversity 328 

♦ (A) American Contribution 
(B) B'ritish Contribution 

yii 










ij TABlj: OF CONTENTS (Continued) 

-■li 



THEORY OF MUGKATION 

MIGRATION OF UNDERWATER GAS GLOBES BUS TO GRAViTi AND 
NEIGHBORING SURFACES (A) 

E. H. Kennard, David W. Taylor Model Basin 

THE RATE OF RISK OF LARGE VOLUMES OF GAS IN WATER (B) 

G. I. Taylor and R. M. Davies, Cambridge University 

A SIMPUFIED THEORY OF THE EFFECT OF SURFACES ON THE 
MOTION OF AN EXPLOSON BUBBLE (B) 

,y. A. R. Bryant, Road Research Laboratory, London 

THE RATE OF RISE OF LARGE VOLUMES OF GAS IN WATER (B) 

H. N. V. Temperley and LI. G. Chambers, Admiralty Undex 

Works, Rosyth, Scd:l^d . . . . 

THE ATTRACTION OF AN UNDERWATER EXPLOSION BUBBLE TO A 
RIGID DISC (B) 

A. R. Bryant, Road Research Laboratory, London . 

THE EFFECT OF AN ADJACENT DEFOxVMXNG TARGET UPON THE 
BUBBLE DUE TO A SUBMARINE EXPLOSION (B) 

LI. G. Chambers, Admiralty Undex Works, Rosyth, Scotland 

PRESSURE PULSES 

THE PRESSURES PRODUCED BY EXPLOSIONS UNDERWATER OF 
SMALL CHARGES NEAP. A RIGID BOTTOM (B) 

G. Charlesworth, Road Research Laboratory, London 

SECONDARY PRESSURE PULSES DUE TO GAS GLOBE OSCILLATION 
IN UNDERWATER EXPLOSIONS. I. EXPERIMENTAL DATA (A) 

A. B. Arons, J, P. Sllfko, and A. Carter, Underwater Explosives 
Research Laboratory, Woods Hole Oceanographic Institution .... 



437 


445 

455 

465 

475 


SECONDARY PRESSURE PULSES DUE TO GAS GLOBE OSCILLATION 
IN UNDERWATER EXPLOSIONS, U. SELECTION OF ADIABATIC 
PARAMEifERS IN the: theory OF OSCILLATION (A) 

A. B. Arons, Stevens Institute of Technology, Hoboken, N. J. 481 



THE THEORETICAL SHAPE OF THE PRESSURE PULSE PRODUCED 
BY AN UNDERWATER EXPLOSION BUBBLE (B) 

A. R,. Bryant and Ll. G. Chambers^ Naval Construction Research 
Establishment, Rosyth, Scotland . . • 



THE OSCILIoATIOiN OF THE GAS-BUBBLE FORMED BY A DETONATOR 
EXPLODING UNDERWATER. DETAliLED COMPARISON OF THEORY 
WITH EXPERIMENT (B) 

H. N. V. Temperley, Admiralty Undex Works, Rosyth, Scotland 


487 


497 



(A) American Contribution 

(B) British Contribution 




Viii 


TABLE OF COKTSNTS (Continued) 


EXFESIIwSNTAL TECKHIQUES AM) MEASURESaE?iTS (C Ontinued ) 

PHOTOGRAPHIC MEASUREMENTS OF THE SIZE, SHAPE AND MOVEMENT 
OF THE BUBBLE PRODUCED BY CHARGES OF POLAR AMMON 

GELIGNITS DETONATED UNDERV/ATER AT « DEPTH OF 3 FEET (B) 

A. R< Bryatit, Road Research Laboratory, London 


A TECHNIQUE FOR MULTIFLASH PHOTOGRAPHY OF UNDERWATER 
EXPLOSION PHENOMENA (B) 


A. R. Bryant and K. J. Bobln, Road Research Laboratory, London 


PHOTOGRAPHIC MEASUREMENTS OF THE ATTRACTION OF AN UNDER- 
WATER EXPLOSION BUBBLE TO A BOX MODEL TARGET (B) 

A. R. Bryant, Road Research Laboratory, London 


MEASUREMENT OF BUBBLE PULSE PHENOMENA, IH. RADIUS AND 
PERIOD STUDIES (A) 

E. Swiit, Jr. and J. C. Decius, Underwater Explosives Research 
Laboratory, Woods Hole Oceanographic Institution ........... 


SURFACE PHENOMENA: DOMES AND PLUMES 

ANALYSIS OF THE SPRAY DOME (B) 

G. W. Walker, Mine Design Department, Admiralty 


A METHOD OF DETERMINING THE DEPTH OF UNDERWATER 
EXPLOaCONS (B) 

R, A. Shaw, Marine Aircraft Experimental Establishment, Sfcotland 


A FURTHER INVE^IGATION INTO THE DOME ANALYSIS METHOD OF 
DETERMINING THE DEPTH OF UNDERWATER EXPLOSIONS (B) 

R. A. Shaw, Marine Aircraft Experimental Establishment, Scotland . . 


THE RELATION BETWEEN THE APPEARANCE OF THE PLUMES AND 
THE GAS GLOBE BEHAVIOR IN UNDERWATER EXPLOSIONS (A) 

G. K. Hartmann, Naval Ordnance Laboratory 


THE DETERMINATION OF PEAK PRESSURE OF AN UNDERWATER 
EXPLOSION FROM A STUDY OF THE INITIAL DOME VELOCITY (A) 
D. A. Wilson, B. A. Cotter, and R. S. Price, Underwater Explosives 
Research Laboratory, Vt^oods Hole Oceanographic Institution 


SURFACE PHENOMENA; GRAVITY WAVES 

GRAVITY waves PRODUCED BY SURFACE AND UNDERWATER 
EXPLOSIONS (B) , 

W. G. Penney, Imperial College of Science and Technology, London 


SURFACE WAVES PRODUCED BY FIRING UNDERWATER 32-LB. 
CHARGES OF POLAR AMMON GELIGNITE AT VARIOUS DEPTHS (B) 
Q. Charlesworth, Road Research Laboratory, London 


* (A) American Contribution 
(3) British Contribution 






TABLE OF CONTENTS (Continued) 


SURFACE PHENOMENA: GRAVITY WAVES (Continued) 

SURFACE WAVES PRODUCED BY UNDERWATER EXPLOSIONS COMPARiSON 
OF THE THEORY OF W. G. PENNEY WITH EXPER^ENTAL RBiSULlS 
FOR A 32-LB. CHARGE (B) 

A. R. Bryant, Road Research Laboratory, London 

SURFACE WAVES FROM AN UNDERWATER EXPLOSION (A) 

J. Q. Kirkwood, Cornell University, N. Y., aisd 

R. J: Sssger, Naval Ordnance Laboratory 


* (A) American Contribution 
(B) British Contribution 


X 






REPORT ON THE THEORETICAL SHAPE OF THE PRESSURE-TIME 
CURVE AND ON THE GROWTH OF THE GAS-BUBBLE 


S. Butterworth 

Admiralty Research Laboratory 


British Contribution 


August 1923 




REPORT ON THE THEORETICAL SHAPE OF THE PRESSURE-TIME 
CURVE AND ON THE 6 R 0 WTH OF 


S. Buttefgerth 






fO 


mis 


Summary, 


lAjnb's tinary of under water explosions Is shown to give a pressure-time curve in very 
fair agreement with that ootalned hy experiment, provided that the two ass-constants are 
appropriately chosen. An attempt is made to ostlmate the effect or the compress IbM I ty of water 
by assuming that outside a sphere at radius 3.7 times that of the orlglhal charge the water has 
Its nornal compreasibullty and within this radius is Incompressible, the Incompressible e' ''v 
being Introduced to cover that region for which the prossures art too great tor tne usvai avuistic 
equations to apply. 

It Is found that, for an explosion In which a given quantity of energy Is released, the 
Initial gat pressure for this case must be of the order 360 tons per square Inch to account for 
the experimental prasiure-t Ine curve, whereas the Incompressible theory gives only 60 tons per 
square Inch for this pressure. The tneory thus Indicates that the maxlreum pressure may diminish 
with distance very rapidly in the Immediate vicinity of the charge. Lane's theory is extended 
to Include the effect of a constant external hydrostatic pressure, the result being that the 
bubble tends to oscillate In size the time of oscillation being of the order of one second for a 
&0 1b. charge. In deep water this msy result In a succession pulses which would, howeverr 
diminish rapidly In amplitude. , 


The only theory which has so far been published In regsro to the shape of tl» pressure-time 
curve due to an underwater explosion Is that due to Lamb*. Lamb assumes that the water may be 
r^arded as Incomprssslble and that the products of detonation during expansion follow the law 
n' * eonatant whore y remains Invariable throughout the expansion. He tshes tlio ease of 
sphorlcalsynthstry and worhs out the fonn of the prepfiure-tlmo curve for the values')"* 1 and 
7 ” 4/3. for ths latter case his result Is givaii In the following formulae. Let Kj, be the 
initial radius of the sphere containing the products of detonation and R the radius after tine t. 
Let Pg be the initial pressure In the sphere R^. Then !f p be the density of the surrounding 
incompressible fluid and/3*= R/R the relation betweerr^arx) t Is 


± - J. /£" W- ll* fs + a .'» ♦ «) 

A . «r . / fl n. ' r- -S 


15 / JP, 


( 1 ) 


Also if p Is the pressure at a time t at a poltit distant r from the centre of the bubble 

r 

R„ I R- 




(3- 1) 


(J) 


Formulae (l) and (2) can be used to calculate the pressure-time curve 'when S , p are known. 

It should be noted that (2) will hold only so long as r Is greater than R so that there are no 
negative values of p. 

Tnis formula may be compared with experimental results If we can assign a value to P . 

Two methods suggest themselves. First if w Is the energy released by the detonstlon of one'dram. 




of the . . 


Phil. Mag. January, 1923. 









2 


_ j _ 


of tM «xp)o$W«, then by tharnndynmics 

I’d ” {y ' l> «5 (3) 

8 Is the dsnslty of the explosive. 

For T.N.T. «* ” 3.88 x 10^® erpa per jrsm. 

8 • 1.2 

80 ttvst with 7 » 4/3 

Pj * 1.55 X 10^® aynes/se.cm. = 100 tons/f.quare Ineh. 

Mtsrnatlvaly equation (2) shoes that tiiaxIiMi pressure at distance r Is related to by 


so that If P^ Is detennlnad •xperlnentatty we can estlirate F^. Now for 180 lbs. T.N.T. at 
40 feet the plezo-electric muthod gives P^j, • 1.02 tons/squars Inch and since 9^ ■ 0,7 foot 

P^ » 58 tons/squara Inch. 

Taking the Tatter estinste the following Table is calculated frofb equations (1) end (2). 

I4MJ> 

Theoretical course of pressure-time curve 
for 100 lbs. T.V.T, at feet. 


/S 

e 

leOO 

1.02 

l.os 

1.1 

1.2 

1.3 

1.4 


m 

1.000 

0.961 

0,904 

0.826 

0.694 

0.592 

0.510 

10* t 

(seconds) • 

0 

0,45 

0,73 

1.06 

1.60 

2.10 

2.SS 

0 

a 

1.5 

1,6 

1.7 

i.e 

1.9 

2,0 

2.5 

®/'’l«X 

• 

0.4SM 

0,391 

0.346 

0*309 

0.277 

0.250 

0.16U 

10*t 


3,06 

3,w8 

4. 10 

4,70 

5.26 

6.65 

9.42 


These values are compared with those obtained by the plezo-electric method In Figure l In which A 
represents the theoretical curve and B the experimental curve. 

It is seen that apart from the finite time required to reach the moxlmiff! pressure there 
Is very fair agreement between the theoretical and experimental curves. 

As regards otlscr weights of charge, formula (2) (with (S • l) shewa that the maximum pressure 
at a given distance -varius us the linear dlraansion P^, that is as the cube root of the weights. 

This is in agroement with the experimental results at any rate so. long as the weight exceeds lOO lbs. 
The law of distance for maximum pressure shoulrt theoretically by r = constant. This las is 
found to hold experimentally throughout the range for which the plezo-olectric records have been 
obtained but observations with Hllllar crusher gauges at smell dlutancos show a tendency for F 
to be greater than this law wculd give. 

It r Is so large that the second turn In (2) may be neglected, the theoretical shape of 
the curve Is the Same at all distances. If we Include tho second term there Is a sonewhat more 
rapid fall of pressure near to the clxirge. The oxpor Imental evidirnce from the piezo-electric 
goiijcs is not sufficiently precise to test the constancy of shape. 




„ 3 - 


3 


In 9pUe of ihs agrsarant oatw««n tna inconpresslbla theory and observation as Illustrated 
In Figure 1 it Is Important to attempt to find the otfect of the finite compressibility of thn 
eater before ee can conclude that It Is safe to extrapolate by the Inverse distance law to obtain 
the preasurss In the Immediate nelgnboorhcod of the chargoa Lamb has stated that a complete 
solution including compresslbitlty appears hopeless* no know, however, that If we choose a distance 
sufficiently large the ordinary laws for propagation of sound will hold and that the comprossloll Ity 
of water diminishes for very high pressures. Tho following procedure should therefore give us 
rusui.S nuarei' t,ubh .nail li COiitpie.ft ita-uspresslwi * ity lo seSiSiiSda SuUpOSe vTiS gss bybbili 
surrounded by an incempresslble envelope so large that outside It the ordinary laws for the 
propagation of sound nay be taken U hold and let us attempt to find the form of the pressuie wave 
thrown off from this system. If a Is the radius of the boundary between the Incompressible and 
compressible region, then at a point r > a the velocity potential Is given by 


ip • F (t - )/r 

the equation of pressure Is 

« -p F’/r 


and the velocity Is 


U 


S4 

3T 


f; 

cr 


7^ 


( 5 ) 

( 6 ) 

(f) 


In these equations c Is the velocity of sound in water and p the norsal density. 

Within the region r < a since Incompressibility is assumed 

0 - X («)/f f ^ (t)/a (SB) 

P + ^pu* • -p X' (t)/r - (t)/a (4a) 

u ■ -X (t)/r* (7a) 

Th® function i/i Is possible since there Is no restriction to a pressure variation Independent 
of r, i/inust be cero In the outer region os p and u vanish at Infinity, 

Since p and u are continuous at r « a, 

F« = X' + 'P'’ '"e/c ♦ F • X- 

or 0 “ P - X • ' f (8) 

By (fi) and (8) since u^ Is negligible when r > a the pressure at a may be written 
p • pc 0/a* ( 0 ) 

SO that the function 0 gives the fonn of the pressure-tliM curve In the ou;ar nsdlun. 

To determine these functions we must form the energy equation from the assikm lew oi* 
expansion of the gas bubble, lot ths bubble have initial radius ■«. and radius R at time t so 
that by (7a) “ 



( 10 ) 


To foim the energy equation the work done by the bubble In expanding frets R to » must 
be equated to the energy stored In the neaium betweon R^ and a together with the energy that :ias 
escaped from the surface a. Tne energy escaping from a 


4 




= « 7T a 


« m 


u (p f ^ pu*) at 


0(* + ^‘) X « 




^P^ X at) 


The energy stores between R ana a is kinetic anc> Is to 

/-a 

Zvp r^u^or » 2 7/73 
- R 

The work done by the bubble 


dr .. - , v»' /I It 

72 ‘ ^ fu"a' 


4 77 P R ^ 

^ fRf"! 

PdV = 

-[-] I 


tne law of expansion oeinj pyh 'f ^ u constant. 

Epuatinp (ijl to (ll) and (12) 

' -tl 


2 P j i f ” 1^" ^ X^ 

\ I ^ ■[ T J J ‘ T * 


Xv’ 


This equation together with 

t//’ ♦ I 0 ♦ X' = 0 

obtained from (8) 

and S' + e 0 


(a) 


(12) 


( 1 ?) 


(u) 


(.15) 


(16) 


give three first order equations to deter, nine X, 'h R. When 0 is known the required pressure 
follows from (d), 

!f we chrose as our units of length and time, for length the initial radius R of the 
bubble and for time the ti.ne re.-.uired to traverse 2 R with the velocity of sound, equations (lu), 
(IS), (16), may be writt. 


t 4 ] 8 »-JX Sx--^S0 = 0 

. '* J * 

S 0 V SX * ^ 5t • 0 

a 

£K + X *■ 0 


I Which A a fl P /o 
o ~ 


(if) 


( 18 ) 


i 


n 


so that 


1 

f 



_ s _ 


5 


so mat 


«X 





SUi « -8X“20 St/a 

S R “ - X St/**^ 


(19) 


In this form If ws know X> <h 9 at a 9lven instant >«a can calculate S X> S </>, $ R and 
tlvarefora detsmiins X. </>• 9 at a future Instant. In particular when t*0, X*'/'”®. 9“t 
cs that wa can step out frets the initial conditions and calculata the curve of i/i. It my be 

remarked that If we put a equal to infinity In (iS), then (lu) and ('5) may be used to develop 
the Incompressible theory. The results obtained In this way agree nith those of LenO. 

Using the above seihod the results of Table II have been obtained. 

The values of A.n chosen In the three cases tabulated are such that the total enoray content 
of the gas buoble Is the same In all cases, so that the numbors illustrate the effect sterely of 
the alteration of the isode of delivery of a given total energy. 

They correspond respectively to-- 

Pg • 72, 2l«, 360 tons/square Inch 
with y • R/3. 2, 8/3 (see Footnote). 

The Incomprussiblc region in all cases Is taken to be Included within a sphere of radius 
3.7 Py The pressure-time curves are plotted In Figure 2 and It Is Imiisdiately seen that the 

effect of endowing the extnrnal region with the ordinary compressibility of water has been to 
alter very profoundly tne shape of the pressure -t line curve. 

The curve A fory* it/3 was that which on the Incompressible theory was found to fit the 
facts most closely. On this assumption It now - 


Table t i 


Fo otnote ; The large values of y Indicated are not physically 
imposslolc as at very high pressures. It Is probable that we are 
working on a very steep portion of tne adiabatic curve for the 
gas so tnat the value of y In the assumed law pyX • constant Is 
not the ratio of the specific heals of the gas but a value which 
fits tho slope of the lelevant portion of the adiabatic curve. 


1 


ii.4 


19.0 

5 


0,16 0.l6 1.00 
0.31 0»31 1.00 
0 . 4 ^ 

0.02 0.6l 


1.4.3 i.35 

1 .70 1«59 

1.93 t./a 


2.45 

2.14 

2.58 

2,21 

2.69 

2.26 

2.77 

2.30 

2.88 2.32 

2.96 

2.33 

5.03 

2.33 

3.09 

2.32 

3.14 

2.31 

5.18 

2.29 

3.22 

2.26 


5.25 

2.23 

3.30 

2,16 

3.33 

2.07 

3.34 

1.97 


0.52 0,52 1 

0.78 0.77 1 

1.03 1.01 1 

1.26 1.22 1 

1.77 1.70 1 

2.17 2.05 1 

2.4.7 2.29 1 

2,70 1 
a .88 2.58 1 

3.05 2,66 1 

3.15 2.70 1 

3.25 2.73 1 

5.33 2.74 1 

3.40 2.73 1 

3.45 2.71 1 

3*50 2.69 1 

3.54 2.65 1 

3.57 2.60 1 

3.59 2.55 2 

3.60 2.49 2 

3.61 2,43 2 

3.61 2.37 2 

3.60 2,23 2 

3.58 2.09 2 

3.54 1.94 2 

3.50 1,79 2 

3.45 1.64 2 


llf&blo Qontlmted 

















- 7 - 


7 


t 

"X 


R 

-X 


R 

-X 

■ " -/f 

R 

1 .S 

2,26 

1 .34 

2.00 

3.28 

1.62 

2.41 

3.39 

1.50 

2.54 

1.7 

2.29 

1 v 3 J 5 

2.06 

3-24 

1-50 

2.47 

5.33 

1.36 

2,59 

1.8 

2.32 

1.26 

2,11 

5.20 

1.38 

2.52 

3.27 

1.23 


1.9 

2.34 

1.21 

2.16 

3.16 

1.27 

2.57 

3.21 

1,10 

2.68 

2.0 

2.36 

1.16 

2.21 

3.12 

1.16 

2.62 

3.16 

0.99 

2.73 

2.2 

2.39 

1 .06 

2.31 

3.03 

0.94 

2.71 

3.06 

0.78 

2.82 

2.4 

2.40 

0.96 

2.40 

2.95 

0.76 

2.79 

2.95 

0.60 

2,89 

2.6 

2 . 4 X) 

0.86 

2.48 

2.89 

0,62 

2.87 

2.89 

0.47 

2,96 

2.8 

2 .W 

0.77 

2.56 

2.84 

0.50 

2.94 

2.85 

0,37 

3.03 

3»0 

2 .W 

0.69 

2.64 

2.81 

0.42 

3.01 

2.83 

0,31 

3.09 

3.2 

2.40 

0.60 

2.70 




2,83 

0.27 

3.15 

3 .t 

2.40 

0.54 

2.77 




2.64 

0.25 

3.21 

3.6 

2.40 

0,48 

2.83 




2.85 

0.23 

3.26 

1 


gives a pressure leas than half the observed pressure end Its rate of rise and fall are much slower 
than Is actuatly the case. Of the three It Is that curve corresponding to ■ 360 tons per square 

Inch which fits the observations most closely. 

In view of the enormous change that the partial introduction of conp/essIPlI Ity nas brought 
about It is seen that we cannot rely upon tne IncompressibVe theory either to estimate the 
properties of the gas or to extrapolato the experimental results to dnducs the presaores near the 
Charge. Thu Indications are that the Initial gas pressure may be much larger than 60 tons par 
square Inch, the value of y being correspoivJIngly increased so as to satisfy total energy 
considerations. If wo measure the areas of the pressure -.t I rre curves of figure 2 for the first 
10"^ second we obtain the following lesults:- 

Pj = 72, 2J6, 360 ton:i/square Inch 
[ pdt « 0.720, 0.938, 0.958 


While the plezo-electrie curve gives for the same Interval 0.932 Ib.-seeond-units. 

When phenomena Involving integral pressure are being studied the effect of the rata of 
energy del I wry will not bo so marked as In the case of those which depend on maxlnum pressure. 
The present Investigation shows, however, that mease renents of the pressure-wave ought to be 
made as near the charge as possible before we can estimate danege at small distances. 

Returning now to the case of Incompressibility It must be noted that the rate of growth 
of the gas bubble given by equation (l) will only hold If the pressure at Infinity Is zero. 
According to the formula the size of tne bubble continues to Increase Indefinitely. 

If we postulate a finite pressure at Infinity a modification occurs which limits the 
maximum size of the bubble and suggests certain Interesting phenomena. Let the pressure at 
Infinity be Q. Then the work done by the bubble during expansion from to R has to supply 
an amount of work equal to jTT 9 (R^ - R^^) In addition to Imparting kinetic energy to the fluid. 


Hence 



8 



i 

1 

1 


Hence by (l2)» (i.3)» (l6)» wo have when 3n = Ji anj a ** ® 


« 7T P R ^ 1 - ^ 1 V 1) (r5 - S > 2-np !? 

0 0 R 3 R 


Puttlog R “ ,3 R 


[at p R^= 1 ^ 3 P, J/ 


The Oubble Ci«ses to grow viban 


g/? 1 )/( /^ -/? ) 

or since 0 is small compared with P^, 0 Is large and approvIn»tely equal to (3 

Thus if Q ^ 2 at/nospheres (which will be the case for an explosion at a depth of about 

30 feet) and * 9000 atruospheres, 0 ^ 2.4. Thus if the IncomorKssfble theory held rlJldly 
the buboltJ woulo oscillate pcrpstually oatween the radii a«id 24 R^. In the actual case energy 
win be dissipated portly os a prossuro wave ano pirtly !n tnc fom of heat, but the tendency to 
oscillate Should not be entirely absent. Tne varied form of the surface effects as the bubble 
breaxs the surface n'^y ba accounted for by the variation in siie and internal pressuroi. The 
spout-like form of the surface upheaval which occasionally Is observed lo what we should expect 
from a hlgh-pr^issuro, small-volumo bubble, Tnc occasional spout like form of the surface 
uphiiaval Ic referred to by Hlllior, and he i^uotcs a case of two snots of 40 lbs, MO/60 amatol 
at a depth of 16 feet in wnich the upheaval reached a height of nearly 200 feet while In similar 

shots the upheaval only reached a height of 60 feet. The ccmpamtlve rareness of Its 

occurrence Is In accord with time' con.'Slderat tens whicn wo now proceed to investigation* 

By t-quatlon (20) 


* /2 Po . I 

p '(5- D* {1 

where Is written for 3 R^/Q sines Is large. 

The term Involving , 3 ^^ wi II only be of l.iiportanee when 3 Is large and then ^ * 3 * 1 • 

Hence 





tiib Integranfl In (2l) is Infinite when ;3 = i, /5 = /Sg and imaginary when <5 > 
If ws write for the Integral Ij + Ij + Ij where 







„ s .. . 8 

then Ij Is real and finite tnrouijhout the ranjs of Integration and inay therefore oe evaluated 
by any approxlnato iriothod. How for the case " 34 the expression In {} In equation (24) Is 
nearly constant. In fact It never differs by more than 34 from - 1.72 x 10"^ throughout Its 
while range. Hence to this apeuracy we may write Ij ■ - 0»57 x 10“-’ (0^ - l) when Bg ■ 34. 
Calculating Ij^, .j, 1^ from these formulae and using In (2l), we find that the time required 
for the bubble to reach Its maximum radius Is given by t • 3170 7 ,. In the case of 
100 lb. <har|e of T.H.T. with = 9000 atmosptvires “ 0.7 foot, t • O.J second, so that 
If the incompressible theory hold rigidly the bubble would r ‘ .to its original state in one 
second and a aeries of uxpl Vns at Intervals of one second would bo heard, *s a matter of 
fact two reports are oftsn nca. ..ut observers generally attribute the second report to the 
broaHlng of the surface b, the bubble. In Figure 3. curve A gives the charge of radius of 
bubble with time according tc equation (l) and curve 8 snows the modified course as calculated 
from the above data. Curve 8 Is one half of a periodic curve and shows that during the greater 
proportion of Its existence the bubble Is of large volume and low pressure thus accounting for 
the scarcity of spout-llko upheavals menttonrjd In the last section. 












UNDERWATER EXPLOSIONS 
TIME INTERVAL BETWEEN SUCCESSIVE EXPLOSION 


H. F. Willis 

H. M, Anti-Submarine Experimental Establishment 


British Contribution 


February 1941 




UNDERWATER EXPLOSIONS. 

TIME INTERVAL 8ETWEEN SUCCESS ItfE EXPLOSIONS 


13 


i - 



H. F. wAliia 
February 1941 




i^trodi^tion . 

I* 

It has frequently been observeq that an undereater explosion Is folliwed ufter a short tine 
by a aecona explosion. The tine interval may be of the order of a few seconds for large depth 
Charges, whereas for small detonators it Is about 20 mllHseconds, It is the purpose of this note 
to sutiAilt an explanation of this phenomenon, to calculate the magnitudes of the effects to be 
expected, and to compare these with available experimental data. 



Little Infottnatlnn appears to be available concerning large exploslorf, but a fair amount of 
wol'h has been done in connection witn smalt detonators. In these experiments, detonators have been 
fired at some distance from a 24" flat-response quart! receiver, and the resulting voltaaffiampl Ifiod 
and spread out on a C.R.O. 

Figure 1 shows a typical C.R.O. photograph. Figure 2 shows the same thing using greater 
srapllficatlon. Both photographs show evidence of a nurrt)er of explosions which become progressively 
weaker. The time Interval between them Is not constant but diminishes slowly with each explosion. 
The first explosion differs from the rest in that it shows an instantansous rise In pressure up to 
its peak value. In alt subsequent explosions the pressure Increase Is more gradual ano for the 
later explosions the pulses nave a nearly synmetrleal appoaiance. Between each pair of pulses 
there Is evidence of a small rarefaction which extends over the major part of the Intorvai. This 
Is particularly noticeable In tne case of the later explosions shown in Figure 2. 



In the, following calculations a natural explanation of all t.hese characteristics results 
from a consideration of what happens to the gaseous products of an explosion as they expand outwards 
fron the Instant of the explosion. The following assumptions are .nide!— 

(I) that the explosion takes place In an Infinitely short tir». Actially, for the type of 
detonator studied, the time Interval Is of the order of 2 microseconds. Subsequent 
consideration shows that there Is little change In the sUe of the bubble In this tlmo 
interval, so that little error Is Involved In treating the explosion as Instantaneous. 

(It) that the gaseous products do not dissolve In the water to any appreciable extent during 
the short period concerned. 

(ill) that the gaseous products at all times assume the form of a spherical bubble, and behave 
as a pernmnent gas. 

(Iv) tnat, to a first approximation, the water can be treated as an Incompressible fluid, 

The extent to which this Is Justified Is later considered In the light of tho results 
obtained. The assumption implies that there is no loss of energy by acoustic radiation. 

(v) that there Is no dissipation of energy by thermal conduction across the face of the bubble. 


Suggested explanation 


with thee* assumptions It Is now possible to see in a general way what happens after the 
explosion. At the instant of explosion a certain amount of gas is insts.-,tan«Odsly generateo at 


high 


1 

I 



14 


- 2 - 


hlyh pressure nnd ienpuruturs. This iimieuistely oejins to Tores outs^ras the wstsr in contact 
with it, n p.otioh which is caimjnlcateo to a U'saer extant to all parts of the surrounding fluid. 
The potential ensrjy initially nossessea by the gas bubble by virtue of Its pressure Is thus 
gradually connunlcatod to the water in the form of hlnctlc energy, fly reason of the inertia of 
the water, this motion ovarshoots the point at which the pressure In the bubble Is equal to the 
external pressure of the liquic. The bubble thereafter becomes rareded, and Its radial motion 
Is slcwied up and brought to rest. The external pressure n(svr compresses tne rarefied bubble, 

Spain the equilibrium confijuratlan Is overshot, and since by hypothesis there has been no loss 
of energy to the system by radiation or dissipation, it fallcss that the bubble comes to rest at 
the same pressure and volume as at the moiTient of tuO explcaioa. The physical asmrets of the 
explosion are therefore reproduced, and theoretically this process goes on repeatedly, with 
undimlnishsd amplitude. In practice, of coursa, energy Is lost by acoustic radiation and by 
dissipation, and this causes a progressive diminution of the amplitudes of the successive pulses. 

the fact that the C.R.o, records show a series of compress i onal pulses with only snail 
rarefactions Is readily explained. The peak pressure of the bubble Is of the order of bOOO 
atmospheres. The mexiiaim degree of rarui'actlon of the oubble ‘hat can occur Is equal to the 
external pressure of the sen {say 1 atmosphere). Thus although compressloas alternate with 
rarefactions tho magnitude of the latter ore small in comparison wltn the former. 

The fact that the C.R.O, traces show tne successive compressions as Isolated peaks, 
separated by long legions where tne amplitude appears to be zero, results from the high degree of 
asymmetry of the vibrational properties of a bubble when the pressure variations are excessively 
high. In such c,sses the vibrations no longer show the sinusoidal character of n bubble vibrating 
with small amplitude. Immudlately after the explosion the high pressure causes a very rapid 
expansion of the bubble. This, In conjunction with tne siaallness of the bubble, results In a 
rapid drop In pressure which soon becomes Immeasurably small In comparison with the peak value. 

The bubble, however, laoves relatively slowly when its size is greater, so that tho major part of 
tne time Interval between two successive compressions is taken up by the bubble In moving at 
pressures which are insignificant comparod with the peak value. 

Mathematical developrnent . 

The foregoing description of the expanding bubble Is now formulated mathenatlcally. 

The gaseous products of tne explosion forma bubble of radius r^^ and pressure p^, the 
radius and pressure at any subsequent lime t being r(t) and p(t), or simply r, p. The exterml 
pressure of the sea is P. It Is required in tne first instance to aetenmlnB the variation of p 
and r with t. This is most easily obtained "by writing down the engrgy equation for the system, 
which Is got as fjll ws:- 


dotential Energ y. 


This Is simply the work done by the bubble when It expands frem radius r to infinity, 

|d0 


I.e, 


v(r) 


(p -P) u w r* dr 


( 1 ) 


Kinetic Energy . 


This is the energy associated with the water by "eason of Its radial motion. By the 
conditions of continuity of the medium and spherical symmeto', the radial veiccity at a point 
distant y from tne centre of the bubble is r® dr 

? ' 3T 


The total K«&» of the nealum is theroforo 


to 

i . (*7T y^ dy). 

r 



( 2 ) 


K » 


2 


ITT pr’ 


- 3 - 


15 


The total enargy of the system is thurcfoie 


*■ 

S u 


+ K • Zvpr^ r~ ♦ (p - /’) 4 ■» Jr =■ 4 (a constant). 


The value of A Is determines from the fact that at the coiroiencenient of the motion »hen r a r^, 
r = 0 so that 


A » I (p -/’) » TT r^ dr 


and using this value, the energy equatlch bocomes 


E P - P J r^ dr 


This ei|uatlon Can oe Integrated when the nature of tho dependence of p on r Is hnown. 
It Is assumed that an adiabatic law Is obeyed: this of course Is implied by our previous 
assumption that no heat Is conducted across thu surface of tne bubble. 

The adisbatl. l-iw, pv’<^ a *onst., 'ivt-! 

pr^'y » 0. r.^ ^ f«) 


using this relation, equation (3) becomes 




This intejrates directly and jives 


i .L 

dt / 3 


» Pq 1 

Ty-i)p x’ / (l ♦ a) X - a x“ - X- ^ “ 


a • (y~ a P/p„ 


This result can now be written 


^ -r/r^ 

Po J, /'uT 


a) x-axP-x-^y* ■* 


This equation jives the relationship between r and t and specifies the radial motion of the bubble 
subsequent to the explosion. Tne pressure Inside the bubble at any instant Is furnished by 
equation (6) In conjunction with (<t). 

Pressu re vari a tiens inside the bubble.. 


In order to simplify further calculations, a definite value is now given toy. For the 
type of detonators studied experlmeniaiiy a value y» 1.3 Is quoted By the makers., The value 
y • U/J Is particularly convenient maihematlcally, and Is near enough to 1.3, and has tiiorofore 
bean taken. 


Equat Ion 












Equation (4) new becomes 




“ / 2 So 


yf (l + a) X *- a x*^ — i 


with a ■= /*/? P. 


The evaluation of this integral Is not posslOla by ordinary methods. Even graphical msthods 
run Into difficulties on account of the singularities of the Integrand. Hewever, a sufficiently 
accurate method Is avatlaWe on account of tne smallness of a. For the snail detonators, a has a 
value of about lb”“, and over the early part of the region of integration It therefore plays little 
part. The value of the Integrand, In fact, differs from ■>/ “P *“ ‘P® P®'"* 

X ■» 5. The same therefore appllos to the Integral, the integrand being always positive. But 

when X varies from x t to x • B the Internal pressure of the bubble deops from Its Initial value 

d to ^8 = ^ . This variation in pressure Is probably larger than one Is likely to be Interested 
° 5 * 625 

in, the pressures at subsequent points bolng of littio significance. Hence for practical purposes 
the pressure variations In the early stages correspond to 


/ X - t 




or, expressing this in terms of the Internal pressure p, 
frr •( 


/f M 

t*0 J 

[r-rl 


Time intervals betmin iuls os . 

Equation (P) cannot be used to determine the time interval T betwesn successive pulses. 

To determine this it Is necessary to revert to the general formula (f) ano to carry out the 
Integration over a complete cycle. In this formula the denominator of tne Integrand 

(1 f a) X - a X* - 1 

becomes zero for two. positive values of x. One of those values Is x =• 1. The other is a large 
number, which Is denoted by Xj. The physical significance of these zeros is that they correspond 
to the turning points or tne vibrations of the bubble. When x ° I tne bubble Is in its highest 
state of compression ano when x => Xj It is in Its highest state of rarefaction. Furihernore, 
the notion during compression corresponds completely with the motion during expansion, except for 
sign. Hence, if the integril of eqo'tlon ,T) Is evaluated between the limits x • 1 and x » x. 
the result will be t ■ T/2, Hence 


‘ Pq J j / (t f a) X - a x“ - 1 
where Xj Is the larger positive root of 

(l t a) X - a X** - 1 - 0, 

The approximate evaluation of ( 10 ) Is again possible by reason of the smallness of a. 
Putalls of this evaluation are given In appendix 2. The result Is 


IL 

V P„ 3 


wnsre 




_ 5 . 


17 


sin B id ■ UIA 


substltut Aj fora this becorD«s 


- i/Tl < f’ 
^ “ -pTr P 


It nay ba shown that the error Involved !n the derivation of this relation Is of the 
order i,o, about 1/ JO for the ootanat ,ii stucled. 

Equation (12) was (lariveO on thn asMimptlon that 7 = a>3. The corresponalnj formula whsn 
y has any value may be shown to be 


T = r (y- 1) 




Equation (I2a] can Oe expressed in aUernative and ^lore convenient forms ns follows* 
The equation of state for the gas In the Dudule givos 


m = qmntUy of gas In gm. molecules 
S • universal gas constant • 8 .;- 10 ergs. 


8^ • Initial gss tcmpfjrature i,.> , oolute scale, 

By virtue of (13) the term r 0 .*^^ can be eli ilnated from (l2a) with the result 


(y - i)' 


■pU: 


T can aUernatlvyly be e:fpresi- ' :-t f*? • :» of i.'.tlal potential energy of thu bubble 
For is tnc work bone by the buooK- in aiV’ is 


piv = p^v^'y 




4 n r 5 n 
3 (y - 1) ' “ o 


Eliminating r^ ^ between (l»| and (l2a) 


3 77*^ 


pl/2 ^ 1/3 ;,-b -6 


From equation (l 2 c) the following conclusions are orawn. 
t/ a 

(i) T Is proportional to »o Since, ’.or a given devonator, the successive pulses in 

practice Oecome weater, so that the potential energy at the stages of maximum compresslcii 
becomes less and less. It fellows that T should decrease slowly with each successive pulse. 
This a;^reo 3 with th 6 facts described earli*^r* 


In the case of the inlllcil p -lse mny be I intlfl‘^3 with the total enerjy of the explosion. 
T:e Interval between the first and socow p«’,sea u therefore proportional to the tube 
ujt of the heat value of the charge, irr.,pective of the nature of this charge. 


( 3 ) 


i8 


- 6 - 


i3l r varies inversely with the V6 power of the total oi<icrnal prossuro^ Shorter tlnwi 

inten*a1s are therefore to he cxpecte<J If a charje is fired at any approclahlo depth in 
the sea. 


P ressu^'e variations outside the bubble . 

Equation (») shows how the oressure inside tho bubble varies with tiraoi This variation 
Is not readily measureo, Whht Is frequently measured is tna form of ine prstssure puls© at a 

point outside the bubble. The form of this pulse is not the some as that of the Internal 
pro5-5ur».», and the nature of the d i f f'?rence Is new eensidered. 

At any instant* t* the expandinj bubble functions as a source whose strength S is equal to 
the rate at which watur is b^ing forcoo outwards, i.e. 

5 = A (ij r^) (15) 

«Jt ^ 


The velocity potential. X* ^ point distant a from the bubble Is^ by the usual formula. 


X 


s 

A 7T a 


1 

4 77 a 


0 /4 77 


r^) 



(16) 


No retarded potential is here involved as tne medium is assumed incompress ihle, and the velocity 
of sound is therefore infinite. The pressure at the point a is, therefore, by the usual 
hydrodynamic formula 


n “ o + i p v^ 
^ 3 t 2 


where v is the radial velocity at a 


e e 

a 





2 


( 17 ) 


The secaid term on the ri 9 ht hand side varies inversely with a**, and therefore rapidly pecomes 
neglljlole in comparison with the first term which varies inversely with a. Equation (17) therefore 
slmpi Iflas In practice to 


n . 


£ 

a 


at‘ 


(4) 


(18) 


This e.quatlon, tojether wit', the solution for r furnished oy equation (8), specifies the values of 
n for anytime t. Tnrs, howver. Is not j convenient solution, and a more useful eesult can t 
obtained In the followln;) way:- 

Equntion (1.8) Can be written 


r' + 2 r r^ = sJJ 

P 


(1«) 


Differentiation of the anciqy oquatioh (3) jives 

r^ r' ♦ i r f-7 a f (p _P ) 

2 P 


’'action 'jf ( 2 v; from. (l9) jives 


(p -/■) 




Substitatinj in (2l) the valu.; for r jiven by (.6), ana takinj y* s /3 jives 

n « - ^e-/'♦-S^(l+• 

I 


-flj^(l+a) x-a 




(25) 


( 21 ) 


(2 2) 


But 


1 


^ T 


19 


p >* -§ from (w) 


and ^ = 3 pQ a] 


inserting (these values In (2:2) 


f 3 

n « pQ ^ (i + «) - « a ) 


In the early 4taije of the motion, ivhlch is noM considered, the terms a ano a x'^ can oe ne^jlccteo in 
comparison elth unity, and then 

n - ^ (») 

This is a more useful formula than (is)* 

The peak value of the external pressure corrosponds to r ■ r^j Hnd is 

• Po (25) 

Equation (24) can now oe written as 


n ' 


so that, using equation (S) 




N l/ 2 ■ 5/2 


3 I n 


1/2 I J/2 


which shows how n varies with t In the early stages. 

The itiaxinum amplitude oT the rarefaction at a distant point Is Oeouciole from the sore 
accurate equation (23). From formula (lo) It apponrs that the itoxlmi*« expanolon of the bubble 
corresponds to * • x^, where is the larger positive root of 

(i •» a) X - a X* - 1 « 0 

and is approximately Xj • when a Is very snail. The corresponding value of 0 is given by 

equation (23) as 


Shis variation being superposed upon the existing pressure/*. Substituting in the above r - x, r 

• r * ° 

0 

n,. » 


) 


• - (from 25), 

The maxlrwii aniplltud.; of the rarefaction Is therefore a fraction of the initial positive 

pressure pulse. 

Comparison 









I 




20 


Cemjiarison ui tk e xperiment . 




Ex|)orln(nti maae with small 1 jm. detonators have enahlsd vho forejolng thecretlcel results 
to he testEd in various ways. 



Oetalled calculations of the turn of the external pressure pulses nave been made, and on 
the basis of the results the appoaranee of the first two pulses has seen sketched In figure 3, In 

these calculations, equation (27) was used In order to obtain tne form of the guises down to a 
pressure onplitude of 11^/100, a sepa'ate estimation of the maximum rarefaction Being made. 

Figure 1 shess a typical C.R.O. record of a detonator explosion. The receiver employed was 
a small 2i* quarts hydrophone calculated to have a principal resonance at aonut 400 he/s. and aimed 
to have a flat response at lower frequencies. Tho receiver was used In conjunction with an 
amplifier having a high Input Impedance (to - 20 megohms) and wide band frequency response 
(10 c/s to 100 Kc/s). The deflections on the C.R.O. records are proportional to the voltage 
variations across tne receiver and therefore, for s perfect receiver, proportional to the pressure 
variations n In the water. 

In a general may the appearance of the C.R.O. trace agrees with that calculated, as regards 
the sharpness and Isolated character of the pulses. There Is one difference, however; on the 
C.R.O. record the Initial pressure ris? Is followeo after a very short time by a small sharp negative 
pulse. This effect Is more apparent in Figure 2 where more amplification has been used, here It 
appears that tne negative peak Is only associated with ti,e first few explosions; It Is absent from 
later explosions where tho pressure variations appear to accord more nearly with theory. This 
affect may be oue to the failure of the receive! to respond rellao’y to the very high frequencies 
Which would be associated with the Initial explosloas. 



Equation (25) enables the peak pressure amplitude at any external point to be calculated. 

In Appendix I the I'ollowlng data are derived for tne detonators 

P_ * 370o atm. r. » 0.62 cms. 

f O 0 

Substituting thesu In equation (25), absolute values are got for the pressure amplitudes! It is 
calculated, for example, that tne pressure at a o I stance of lO feet rises to 7.5 atmospheres, 
Experimontally, a value of about 3 atmospheres Is obtained at the same renge. Accurate agreement Is 
here not to Oa expected, for tne calculated v.alue Is very materially dependent upon p^^ and r^, reliable 
values for which are not available. 

(c) The time IntorvrAl T between the first and second pulses . 

Equatlon"(l2b) has been used in order to get a theoretical value for T. Taking, for example, 
the case when a detonator Is fired at a depth of 15 feet below tne surface of the sca, the total 
pressure P at this depth Is found to be 

P •> 1,45 atm, - 1.45 10* dynes/cm^ 

In afloltlon, tho makors specify for their detonators 

‘ 3001'’ C " 3273'’ K. 

m ~ 0,0136 correspond i.ng tc 300 ccs. of gas (Appendix I). 

y “ 1.30. 

Using also the known values 

R = 8.3 10^ ergs/gm. molecule. 

I • 1 , 12 . 

formula 





- 9 - 


21 


tonsula (12o) now jWes 

T “ 18=« mil! Iseconds 

The corres ponding value obtalnea ay 3! reel nansureiiBnt Is 
T = i? fTiii 1 isteCcnds 

By the safne method the Interval T can be calculated tor the case of a depth charge* for 
this purpose tiquatlon \12c) Is more convenlents Consider for example the case of a 300 lb. charge 
of T.N.T. The heat value of T.K.T. Is given as lo»0 calorles/gm., from which the totaS energy 
of the entire cnarge Is found to oe 5.9 10^^ ergs. At a depth of (say) 30 feet P • t.9 atm. ■■ 

1.9 to* dynes/ctn^. Equation (I2c) then gives 

T = 1.2 seconds. 


An entparlmantal value 

T > 0.6 seconds 


has been quoted for a 300 1b. mine exploding at a depth of 30 feet. 

(d) Variation of T with pressure P. 

As already pointed out, the dependence of T upon P oxprossao by aquation (I2b) Implies that 
T Should vary witn the depth at whlcn a charge Is fired. Values of T for the detonators have been 

warxed out for various depths up to 300 feet Just as In the example worKed out above for IS feet. 
The resulting valuis aefino the continuous curve drawn in Figure 5. To test these values, direct 
measurements of T were carried out at sea down to a depth of 245 foot. These experimental Values 
are set 0:iS Ir. Table 1 and .re marked by the ilrcles l" njurn S. Thesa values He close to the 
theoretical curve, showing that equation (l2b) gives a reliable figure for the absolute value of T 
under external pressures? ranging from 1 to 8 atmosphares. 

Tnaonetleally T should vary inverseley wItnP®''*. To test the accuracy with which the 
index S/6 Is opeftitlve the ixporlmuntnl values for T and P have been plotted on a logarithmic scale 
In figure 6. « straight lino of slope -5/0 has been otawn tossing through these points, for 

purpose of comparison lines with slopes -2/3 ana -1 nave also been Indicated. it is evident that 
the axperiment.1l values discrimirvite quite sharply In favour of the index -5/6. 


(e) Time Intervals between successive pulses . 

AS was mentioned earlier. It Is found experimentally that the time Intervals oatween successive 
pulses get shorter and shorter, and this effect is in qualitative agreement wltn theory. It Is 
now possible to make quantitative comparison. 


Equation 1 2 2a) snows tnat tne time Interval between the first and second pulses is 


proportional to r p 


1/3 


where r and p refer to the initial values of r and p at the first explosion. 


...... .. g .5 0 -■'O ' - -- - 

For a given ma.as of gas, oy virtue of equation («). Is proportional to p^ (assuming 7* 4/3) 
so that T depends on tno initial pressure of tee gas according to the n lation 

r „ 1/12 


But from equations (4) and (2S) 

Po * 
rg 0 

Pq being the Internal andll^ the external pressure, ss that 


T <a n. 


1/9 


This Is strictly accurate only If the vibrations are undamped, JI^ being then the^same for the first 
and second pulses. For a damped train of pulses It Is not quite le.jltiniate to apply this formula 
to the pressure amplitude for the first pulse; It Is more accurate to regard the first half of the 


period 





22 10 - 

perio3 as belnij aetormlned oy the pressure 0^ of tho first pulse and tne second half as determined 

by the value n, for the second pulse. The time interval can than bo taaen to bo proportional to 

+ rij *). In practice little difference Is Involved, otherwise eguaiion (l2a) would requir 

correction for dampinj. The above result can be jenerallscd for any two consecutive pulses of a 

train arising from a single detonator. If 0^.0^ + 1 are the- amplitudes of the nth, (n ♦ l)th 

Pulses and T , the time interval between them, then 

n n ♦ * 


n ♦ 1 

V 


(n, 


1/9 




( 31 ) 


This relation hi^ been tested ajalnst experimental figures. Relative values of 
appropriate to successi"e pulses have been measured off from the C.R.O. records. This cannot be 
done satisfactorily rrcra one single photojrapn because n„ decreases too rapidly with n to allow of 
accurate measurement over a numoor of pulses. However, a series of photographs wer? taken In which 
the amplification was progressively increaseo. These enabled mean values to do given to the 
ratios n ^ 1 from various pairs of consecutive pulses. In this way it was founo' that the 
pressure amplitudes of the first five pulses were in tne ratio 


n,: 

Og: 

n^: 

n,: 

^5 

1 

45 : 

30 : 

6 ; 

2 : 

1 



From equation (jl) the corresponding time intervals are calcul.ated to be in the ratio 

^ 12 * ^21 * ^ 3 “ * 

= 1-45: 1.29: 1.12: 1.00 

The eorrosponding values got oy direct measurement from Figure 2 are 
1.67: 1.23: I.IO: 1.00 

Agreement is considered satisfactory in view of the fact that ecinplete reliance cannot be placed In 
the amplitudes of the pulses shown in Figure 2. The initial pulses are richer than the late ones 
In nigh frequency sound of the order of several hundred Kc/s., -and the receiver and amplifier do not 
necessarily repr.Juce tnem in true proportion to tne otners. This wouli affect the ealculntlcn of 
Tjjmpre than tne other intervals, 

Considerations o/ tie assumptions made. 


The foregoing results have oeen derived or. 
assumptions are now considered in tne light of tn^ 


■>is of certain assumptions. 
. , /btainoo 


These 


(a) 


The compressibility of the water . 


It is legitimate to treat tne water as an incompressible fluid only In so far as the radial 
velocity of the bubble is small compareo with that of sound in water. From equation (8) it may be 


shown that the maximum rsJlal velocity of the bubble occurs when • 
velocity Is 


j. 

16 / p 


_ 4 

■ 3 


, and that this maximum 


For the small detorators this amounts to 2.7 10 cras/second. The velocity of sound Is 15 10* cms/second, 
so that the radial velocity lo at most 1/6 that of sound. The assumption of incompress ibjllty Is 
therefore quite reasonable as a first approximation when it is remembered that this maximum velocity 
Is operative for only a relatively short part of tne expansion. 


(b) 


T he finite duration of the explosi on. 


It was assumed that the explosion took place instantaneously, whereas In fact for the 
detonators it lasts about 2 micro seconds. Assuming an instantaneous explosion, tne expansion which 
lakes place In the first 2/n secondils now considered. Equation (b), for small values of t, can 


be 


oe written 


« 11 - 


4. , i + 


Inserting actual values It Is found that, after 2 fi seconds 


There Is thus very little expansion In this time Interval, so that It raahes tlttlo difference uhether 
the explosion occurs at the beginning of this interval ijr Is distributed over it, 

(c) Th e ra diation o f acoustic ee ierjy. 

It was earlier ussumed that there was no radiation of sonic energy. Taxing the solution 
obtained on this assumption as a first approximation. It is now possible to calculate what energy 
Is actually radiated wnen account Is taken of the finite speed of sound, c, 

n2 

Tne Instantaneous flux of sonic energy per unit area per unit time at a range a Is -a • 

_ p 2 r * ^ 

Substituting foril from equation (2%) tn!s becomes - Q ' ^ ^ • integrating over the surface of a 

pc a r 

sphere of radius a« the rate of emission of energy from the bubble becomes 






The energy emitted in tho course of the flr^t pulse 1$ therefore 

9 9 * *4 

w = £j 

pc iT 

Jx « 1 

where has the same significance as in equation (lO). 

Putting “ * X In equation (8) and dlfferentletlng, jives 




and substituting this in (28) jIves 




2 7t/F -a— 4 
<^p 


In equation (29) the major part of the integral arises from values of x near to unity. This fact 
justifies the use cf equation (8) over the range of Integration x » i to x = x^ for equation (b) 
is valid only over the Initial part of this Interval (x =* I to x - 5), The error Introduced on 
this account Is extremely small, and for the same reason it is quite lajitimate, and more convenient 
to write the limits of integration in (29) as x ° 1 to x » cb. Then 


2 7t\/2 




This is now to be compered with tho total energy available at the rionent of the explosion. 
Taxing 7 - «/3, this latter by vl.'tue of equation (14) is *ir p^, so that the fraction cf the 


- 12 - 



I 

I 

i 

I 

I 


h 

V 
k 

V 

t- 

1 


c' 


3 , 

v; 


i' 






24 


Initial energy raolat?3 in the first (»ilse Is 

.W . JI_ i /S 

2v'2 e / p 

For the small detonators this works out to bo c.sj. 

The appreeiaole nvijnltuOa of this fljure Implies > ccnslieratie decree of fo'yin,- 
vlDratlons. This fact will affect to som.-. extent tna aciifapy off :: Mpreislons ct-'ni. 
p (t) anflll {t)i The deviatlcns arising will be cumulKti 'e aerf w'v;l htt ?Tipe<r jrsv‘- r 
validity of the fomijlao in th=,- earlier stajes of tne sx; ? ision. will bn 

ranje of vallolty of these formulae. Tne effect on the I irwij'i*.' fa-- ” -iiJ to ouite .■'«• 
with this the fact that the period of any simple narmonif .ibra-iry ust-m in little r‘. - 
quite heavy damping). 

Formula (32) shows that the dampinj increases wiM 0 ^,. tc '.I'i. aiiwlltuao o 
pulse should become rclstively loss. It Is possible, trt'efu -. cv.i r- is made I--, 
th& stiCond pulst. may te too weak to be noticed In compar i»n,. 




- 13 - 


25 


Af ȣKDIX I . 

liJ ii r i^ieures. 


The foltowinj Is the liltinutlon . urnUhed by the makers concernin) the detonators used. 
Th? f,!hourit of explasli^G osed li l n., par of wnicn Is lead azides The volume oT gas fobbed, at 
I !( 300 CCS. The inltirl lemperat ire ls 300o'’c ( • 32T3° k). The effective value 

■;f')' is 1.3. 

The Vfth,v o' p. Is hot a'lvii,->, but, ^nco the Ihitlal volume v^ is known, it may be deduced 
frwi the eriuatkh sute I13!. In this quatlon m is readily determined, for one gm, molecule 
-■ >'"■ a.f I" k.T." ecr.!',, les ii I'tros, so that the ,00 ccs. of gas resulting from the explosion 
C'.'r. Gspjiil to a v-.)ue 

f;, ■> -iSP- O.OUt* 

i 200 i 

Th« Vriijo of is n;t iui 'S fcC;tlri^lo.i to be about l cc* insorttng these valu'ds in (13) 

gives 

p.^ - 3100 X 10' - 5 ;„ie!,/cmV 

*>v r’sCC'iy'nry v:f Uili? • n Ihe accufscy with wh!ch v is knojw\, 

P ^ 

Thfy mechanical ‘-gy o the Jetonator explosion is. from liquation (lu), 

Hy feaso.i of eauatlon l / this can be w-lttcn - y * On suUtitutlng the values 
fjr m, and *y alnway quoted, .. ’Igor., of i200 Joules Is obtained for tne energy of the 
csetonsvur. 




Appendix II 



26 


- lU - 
APPENDIX II . 

Evaluation of the intceral for 
This integral U 


K (a) 


ax 


j •/ (1 + a) X - a x - l 


whsre Xj is the lar^. positive root at 

i < a' - a , " - 1 = C 

The integral defines a fjneticn of a. Tne asymptotic form of this function as a "• 0 is 
now calculatod. 

4 y Q 

When O'* 0 the value of x^ approaches a A numosr Xj is now chosen which is of a 

lowor order than Xj, Put still larje compared with unity. This can always Os done, ano such a 
number would be Xj ” o“ where q Is positive and less than 1/3. Than K (a) may be written in 
the form 


K (o) = 


dx 


1 / (1 ♦ a) X - o X* - 1 

I'hls is now compared with the function 




Kj (a) 


a- 1/3 


x** dx 


/ X- 


X - 1 


-sLgx. 


/ ( 1 f a) X - a x“ - 1 


/ X - a x^ 


(33) 


(3») 


It can be shewr. that, by virtue of the chclee of x^, the relative differences between corresponding 
terms of (33) and (3't) get less and lass as a-* o. Hence tne llmitinj forms of both K (a) and 
Kj fet) are the same. 

The integration of fho expross’lon for Xj (a) can oe carried out. “uttlng s\n^ 0 “ a x^ 
in the second integral, sh-/ result Is 


M “ l{ (Xj- l)^^* + I (iij- 1)3/2 ^ 5 (xj - l)°/3 } 

■nil 


♦ 2 =/< 


5 ''‘2 ■ *' ^ 5 '"a 

sin3/3 0 0 0, 


Kl/2 


'sin"‘ *2 * 

But since Xj >■> I it is nossiuie (o writs x^ for (x^ - 1) in the first torra. Then 
Xj (a) ■■ 2 { * I Xj3/2 ) ^^Jl j 




mil 


d ad - 


, -i ii X 3)1/2 

ySin ‘ *2 ' 


n^'30 ,0 


(35) 


Hut since %2 (n. Xj^)l^^« : s» that tne last integral on the right hand side simplifies to 


ft* ■'•1^ 


0^>^ae ‘ g("-Xj3) 


c. . 3 , 5 /» 


“uttlng 


I 


- 15 - 


27 


Putting this result In (36) 

jPT/J 

sln’''^(Ja6' n 1.12 


( 36 ) 


8ut If Xj l6 assumed to have the value a'^ then 


, 3/2 

Xj 


„5/6-3q/2 

a 


and since q < 1/3 this 
Thus tha first term of 
accordingly represents 


Is a positive power of a, whose value accord I ngly tends to zero asa-* o. 
(36) finally becaaies negligiole in comparison with tha second, which 
the limiting form of Kj (a). Hence 


1 (a) 


Kj (a) 


il «-5/6 


as a 0 , 


Whsna is net infinitesimally small, X |a} will differ from its limitinj) form. It has Doen 
astimatfed tMt the relctlviv error involved Is of tne order 


Table 1 


28 





TABU 1. 


Th eoretiral and observed values of T for 
va rious defiks at uihich detonators are fired . 


) 


Depth (in feet) at 
which detonator is 
fired 

Total external 
pressure P (in 
atmospheres) 

Calculated values 
of T (In nllU~ 
seconds) 

Observed values of T 
(In mill Iseconds) 

3 

U09 

23.5 

23.5, 24.1 

25 

1.75 

15.0 

15.2, 16.5, 14.5 

U5 

2.35 

12.4 

U.4, 11. 7 

65 

2.M 

10.3 

10.6, 9.2, 9.3 

85 

3.55 

3.75 

B.9 

105 

4.15 

7o76 

8.4. 7.2 

125 

4.75 

6.93 

6.9 

145 

S.36 

6.25 

6.4) 6.8 

165 

S.95 

5.73 

5.7 

185 

6.5S 

5.2B 

5.5, 5.7 

205 

7.1 5 

4.62 

5.0 

22S 

7.75 

4.60 

«.8, >.0 

245 

a.35 

4.33 

4.2 

1 . 







) 


) 





deton ATOa EXPLOSIONS 






















THEORY OF THE PULSATIONS OF THE GAS BUBBLE 
PRODUCED BY AN UNDERWATER EXPLOSION 


Conyers Herring 

Bell Telephone Laboratories, Murray Hill, N. 3 . 


American Contribution 


1949 


Most of the work on which this paper is based was done in 1941 for the Columbia University 
Division of National Defense Research, and appeared as a report in October 1941 with the NDRC 
number C4-sr20-010. In the present paper this report has been revised so as to omit material 
no longer of interest and to include additional material presented by the author at the 
Washington Meeting of the American Physical Society in the spring of 1947. 


35 


THE»Ry OF THE PULSATIONS OF THE GAS BUBBLE PRODUCED 
BY AN UNDERWATER EXPLOSION 
by 

Conyers Herring 
CONTENTS 

1. Introduction 1 

2. Utomlogy Rules 6 

3. Non-Compressive Theory with Spherical Synmetry 9 

4. Acoustic Radiation by a Spherically Syametrical 

Bubble 16 

5. Migration of a Shparical Bubble 22 

6. Departures from Spherical Shape 32 

7. The Energy Balance 35 

Appendix 1 : Pulsations of a Bubble in an Incojgqpresslble 

Fluid 39 

Appendix 2 t Pulsations of a Bubble in a Fluid of Finite 

Con^resslbility 44 

Appendix 3 : Calculations of the Rate of Radiation of 

Energy 54 

Appendix 4 : Effect of Gravity 59 

Appendix 5 : Effect of Proximity to a Free or to a 

Rigid Surface 64 

Appendix 6 : Pressure Distribution in Non-Compressive 

Radial Motion 79 

Appendix 7 ; Effect of the Inertia of the Gas on the 

Pressure Pulse 82 

Appendix 8 : Resume of Ramsauer's Experiments 84 




kTIQN 


* 0 * '*^ 1 * 


%in* ^mo*m 


{m, n) 
etc. 


ati*ejig'bli. of a slinple soupcs 

coeff’loients In expansion of velocity - 
potential 

padius of bubble 

tJ,me average of a, 

radius at which gas pressure equals p . 

«w 

maximum and minimum radius, and average 
of these. 

Incomplete beta function 

coefficients in expansion of velocity 
potential . 


C.y, etc* 


coefficients in expansion of velocity » 
potential 

velocity of sound in water 

velocity of sound in gas in bubble 

distance of receiver from bubble 

field strength in equivalent 
electrostatic problem 

charge of bubble in eciulvalenfc 
electrostatic probi’em 

defined by eq. (9), Appendix 8. 

potential energy of gas in bubble 

acceleration of gravity 

distance of center of bubble from 
plane free or rigid surface. 

thermal oorductlvlty 

length of a cylinder; any 
character! Stic length 

order of spherical harmonic 


M 



37 




momentum 

unit outward normal 

Logendrs polynomial 

pressure 

cofrfflclenta in expansion of 
pressure {Appendices 4 and 5) 

pressure of gas under static conditions 

p/'asstire of water at a great distance 
from bubble, at same depth as center 
of bubble. 

pressure due to image {Apnendlx 5) 

total energy released by explosion 

masses of explosive 

perturbed bubble radius (Appendices 
4 and 5) 


coefficients in spherical harmonic 
expansion of radius vector of 
bubble . 

distance from initial center of 
bubble . 

radius vector 

period of a pulsation 

period calculated from simplest 
theory 

period for small oscillations 
time 

time of maximum size of the bubble 

volume or molar volume 

velocity 

total energy of the pulsating motion 
(Appendix 1) 




(.Appendix 3) ; 
dlstfimo® (Appendix 5) 

^1/ajnax 

variable of integration (Appendix 1); 
oartsaian coordinate (Appendices 
4 and 5) 

defined by eq« (6), Appendix 3 
gamma function 

ratio of specific heats, or quantity 
playing same role in equation of 
state . 

Dirac delta function 
viscosity 

polar angle (Section 6 and 
Appendices A and 5); 
temperature (Section 7) 
radius of a cylinder 

\mspeclfled function 

density of water 

density of water at a great distance 
from bubble 

density of gas In bubble 

divergence of velocity 

element of volume 

unspecified function 

defined by eq. (14) or (15), 

Appendix ?. 

velocity - potential 

coefficients in expansion of 
velocity - potential 


39 


1 . INTHODUCTION 


During the last generation or two it has been noticed 
from time to time by different observers that when a charge of 
explosive is set off under water, the sound received by a hydro- 
phone some distance away consists of a number of sharp ’’pokes”, 
which decrease gradually in amplitude and become more and more 
closely spaced in time. The intervals between the pokes are 
greater the greater the charge of explosive and the smaller the 
depth of the charge below the surface; at shallow to moderate 
depths the intervals which have been observed range from one or 
a few hundredths of a second for detonating caps to around one 
second for charges of several hundred pounds of explosive. It 
•has been convincingly shown by V<illis^ that the periodic 
phenomenon involved is one of radial pulsations of the gas bubble 
produced by the explosion, x^urther investigations of this pulsa- 
tion phenomenon have shown that its interest is not limited to 
the field of the acoustics of explosive sound: it appears es- 

sential to a detailed understanding of the ’’plumes" of water 
thrown up over an underwater explosion; in some cases it may have 

\ an important effect on the damage inflicted by an underwater 

explosion on neighboring structures; it may provide a useful re- 
search tool in the experimental study of explosions and other 

1 

problems in hydrodynamics. Moreover, bubbles formed by explosions 
are not the only ones which jaay be made to undergo radial pulsa- 
tions of large amplitude: an example is the collapse of cavitation 


I ,/ 


See 


tViR article by Willis 


.in this Volume, 


40 


- 2 - 


bubbles after application of a positive pressure; another example 
is the observation by 14. Ewing and his associates at the Woods 
Hole Oceanographic Institution that multiple shocks are heard 
by a microphone placed near a submerged pipe with a diaphragm at 

2 

one end which is burst by water at high pressure inside the pipe. 

Although this article and the others in this volume are concerned 

primarily with explosion bubbles, and although additional factors 

such as surface tension may have to be invoked in applying the 

■> 

theory to tiny bubbles'^, it is worth bearing in mind that similar 
principles govern a variety of widely different phenomena. 

The essential features of the pulsation phenomenon arise 
from the fact that an explosion creates a cavity filled with high 
pressure gas, which pushes the water out radially against the 
opposing external hydrostatic pressure. The high velocity thus 
imparted to the water causes it to overshoot the equilibrium 
radius at which internal and external pressures are equal, and 
when the external pressure finally succeeds in bringing the ex- 
pansion to a halt a contraction sets in, which again overshoots 
and recompresses the gas to a high pressure. This sort of oscilla- 
tion may be repeated a number of times, until the original energy 
has become dissipated in one way or another. At each compression 


2 

Private communication from Dr. Ewing, 

.3 

See for example R. S. Silver, Engineering 154, 501 (1942). 


- 3 - 


41 

the high pressure developed gives rise to an acoustic impulse 
which can be heard at a aistanca. These features of the 
phenomenon will he discussed in detail in Sections 3 and h of the 
text, the forrasr dealing with the spherically symmetrical pulsa- 
tions of a bubble in an incompressible fluid, the latter with the 
effect of the finite compressibility of the water, which gives 
rise to acoustic radiation. Mathematical details are given in 
Appendix 1 and in Appendices 2 and 3, respectively. 

In actual experiments with explosion bubbles there are 
always asymmetrickl influences which prevent the motion of the 
water from conforming exactly with the spherically symmetrical 
theory Just mentioned. The most interesting features of the pulsa- 
tion phenomenon are, in fact, those which are associated with 
asymmetries in the motion, and many of the papers contained in this 
Volume are devoted to these features. If the only asymmetrical 
influence is the pressure gradient due to gravity, the bubble will 
rise; it turns out that the velocity of rise increases enormously 
during the contracted stages, and that appreciable departures 
from spherical shape may also occur in these stages. An additional 
"migration effect" can be caused by proximity of the bubble to a 
rigid body or to a free surface; roughly described, a rigid sur- 
face attracts a pulsating bubble while a free surface repels it; 
these effects may sometimes be more intense than that due to 
gravity. The importance of both kinds of asymmetry in the motion 
is obvious, since the position and characteristics of the bubble 
at the time of it's minimum volume will greatly influence the 


42 


“• 4 — 

damage which the high pressure aeeompanying this stage is 
capable of doing to neighboring structures; moreover, the fom 
of the plviraes sent up by the explosion will obviously depend on 
the way the bubble is moving v;hen it breaks the surface. A the- 
oretical discussion of the migration effects due to gravity and 
to neighboring surfaces is given in Section 5 » with mathematical 
details in Appendices 4 and 5 , respectively. The Mithematical 
methods used here, and those used in most of the other theoretical 
papers of this volume dealing with the migration effect, have not 
been elaborated far enough to provide a quantitative calculation 
of the extent of the departures of the bubble from spherical shape. 
However, some qualitative comments on these departures are given 
in Section 6. 

Experiments on the various aspects of the pulsation 
phenomenon have usually shown a quite satisfactory agreement with 
the predictions of the theory. A notable exception has to do 
with the apparent loss of energy between successive pulsations, 
a loss which occurs in the brief time when the radius of the 
bubble is near Its minimum. It has been established^ that some 
as yet unelucidated mechanism of dissipation does away with an 
araopnt of energy of the same order as the known energy loss from 
acoustic radiation. Some brief speculations on this topic are 
given in Section ?• 


^A. a. Arons. J. P. Slifko, and A. Carter, J. Acous. Soc. 

Am, 20,271 (l 94 ^i); A. B, Arons and U. K. Yennie, Rev. Mod. 
Phys. 20,519 ( 194 o), also Volume I of this Compendium. 


5 


43 

^ : No Introduetlcn to th« subject of bubble pulsations 

can be e<^plete without nantion of several early papers, theoret- 
ical ones by Lamb and by Rayleigh^, and as experisental one by 
Ramsauer^, which preceded the work of Butterv;orth and Willis 
presented elsetdiere in thi;« Volu)sa« Rayleigh was •eMfiasM with the 
collapse of cavitation bubbles, while Laab and Ramsauer studied 
only the expanding phase of the motion, and make no meution of 
( the phenomenon of successive pulsations* however, their work 

contains the essential ideas of the theory of Section 3 and has 
other interesting features as well. A brief account of Raasauer's 
work is given in Appendix S. 


Lamb, Phil. Hag. 45,257U923)} Loru Rayleigh, Phil. Mag. 
34,94 (1917). 

^C. Ramsauer, Ann. d. Physlk 72,265 (1923). 


44 


•w 6 

2, HOMOLOGY RULES 


Before undertaking a detailed study of bubble motion 
it is worth while to mention a few very general conclusions 
which can be drawn directly from the basic equations governing 
tbs motion. The motion is completely deterruined by: 

(a) the three Hugoniot equations at the advancing front 
of the shock wave generated by the explosion^; 

(b) the Euler equations and the equation of continuity for 
the water inside the shock front; 

(c) similar equations for the motion of the explosion 
products inside the bubble; 

(d) the equations of state of the water and the explosion 
products. 

In the stages before the detonation is complete, equations of the 
Hugoniot type at the advancing detonation front must also be in- 
cluded. Now if it is legitimate to assume 

(i) that there is no body-force term in the Euler equa- 
tions - i.e., that gravity may be neglected - and 

(ii) that the pressure in the explosion products is a 
unique function of the density, independent of the 
rate of change of density, 

then all the equations (a) to (d) will be unchanged if all dis- 
tances and times are changed by the same constant factor, pres- 
sures being left unchanged. This means that if (i) and (ii) can 

"^See Volume I of this Compendium. 


- 7 - 



45 

be assumed „ the pressure and velocity distributions produced 
by different amounts of the same explosive are identical if 
referred to units of distance and time which are proportional 
to the linear dimensions of the charges. Mathematically ex- 
pressed, if a mass of a given explosive produces the pressure 
and velocity distributions 


P - 77”{r,t) 


V - y (r,t) 


(t to be measured from the start of the explosion), then a 
mass qg of the same substance produces the distributions 


7T 


1/3 1/3 


P2) 


(‘^2) 


-r 


(q )l/3 


( 1 ) 


Strictly speaking, it must also be specified that the initial 
shape of the explosive be the same in the two cases, and that the 
behavior of surrounding obstacles or surfaces, if any are present, 
can also be scaled. This rule has long been known and applied to 
characteristics of shock waves; it is equally applicable to bubble 
pulsations, provided (i) and (ii) hold. 

Another homology rule which is approximately valid for 
bubble pulsations results from the fact, which will be demonstrated 
in the next section, that over most of the cycle the motion ap- 
proximate fairly closely to the motion which would be executed 
in an incompressible fluid by a bubble with no gas 5.nside it at all. 




•• 8 — 


46 

Of the equations enumerated above » only those enumerated under 
(b) are needed to determine the latter motion. In the absence 
of gravity these eqxjations are invariant under any scale changes 
for pressures, lengths and times which satisfy P^/Pg * 
(L2^/L2)V(tiAj5) . In this approximation explosion bubbles from 
all sizes of ohargs ab all depths execute homologous motions. 
Taylor has shown that if gravity is not neglected, this homology 
rule does not completely disappear, but reduces to a homology 
over one degree of freedom Instead of two. 

Departures from the scaling law ( 1 ) can be produced by 
irreversible processes taking place inside the bubble, and de- 
partures from both types of scaling law may be expected if 
turbulence becomes serious in the water. Failures of scaling, 
if properly ihterpreted, may thus be of considerable significance 
in detecting the presence of Irreversible phenomena. 


•o mm am m 9m m Ml mm M M w w ^ wm m M 

g lift tfe‘* 

See the article by Taylor^ this Volume. " 


• V e IM. 


47 





3. !T0y-0QM?RE33IVS THE0H3T WITH SPHBRI&AL SKMMBTRg 

Let UB aanaider no~ t.be simple at possible model of 
the motion of the bubble, l)y assuming the water incompressible 
and the motion epherioally symmetrical. This model, simple 
though it is, turns out to be capable of giving a fairly 
satisfactory aeoount of the radius- time curve under most 
conditions. It is not surprising that it does so, in view 
of the following faotsj 

(a) The shock wave advsynoes so much faster than the 
boundary of the bubble that, before an appreciable 
part of the first pulsation period has elapsed, the 
motion of the water has bsoome fairly clearly 
separated into a shock wave region and a "bubble 
region", between which the water is relatively 
quiescent, as shown in Pig. 1 (b).* This empirical 
fact is accounted for by theoretical calculations 
on the form of the shock wave.^ 

(b) In the "bubble region" the pressure is never 
large enough to change the density of water by more 
than a few percent, 

(c) With the possible exception of times very ©lose 

to the minimum of the contraction, the pressure in the 


q 

"^See the articles by Kirkwood and Brinkley and by Temperley 
and draig In Volume I of this Cfompendium.. 

'■^Refer to the end of this article for all referenced figures. 


gas bubl>le (and therefore that in the water) ohauigeB 
hy oniy a very eiuaxX fraction of its vaXue in the 
time required for an acoustic wave to be propagated 
through a di stance of the order of the radius of the 
bubble. This ensures that the flow in the bubble 
region will keep in phase with the motion of the 
boundary of the bubble. 

(d) In many oases, though not all, the unilateral 
migration of the bubble in its first period, due to 
gravity or other influences, is of fairly small 
ma^itude compared with the maximum radius of the 
bubble, and the bubble remains fairly accurately spherical 
in shape. As will be shown later, this is the case 
when the else of the explosive charge is auffioiently 
small In comparison with the external hydrostatic 
pressure. 

The more refined theories to be considered later In this article 
will explain the validity and limitations of the simple 
theory in a more quantitative way. 

We shall therefore assume for the present that the water 
moves radially outward or Inward in such manner that the 
amovints of water oroeelng any two oonoentric spheres in a 
given time are always the same; If the motion is spherically 
aymmetrloal this means that the variation of velocity v with 
radius r Is given by 


49 




da 

dt 


( 2 ) 

where a(t) Is the radius of the bubbles Under cmr assumptions 
the total energy, which Is constant, equals the kinetic energy 
of the water, plua the potential energy of the compressed gas, 
plus the potential energy due to expansion against the pressure 
p of the water far from the bubble. The Icinetlc energy la 




V3) 


The work done against p Is, to within an additive constant 




4// a'" 




(4) 


Denoting the potential energy of the gas for the present merely 
by G, the energy equation is 


ZTTfB: 


s 


(dt) 


4 ^ 0 ^“ 

g-“ 


P 




0(a) 


W 


( 6 ) 


where W is constant in time. This equation can be solved for 
^ and integrated to get the motion. The term G(a) complicates 

dii 

the Integration considerably: fortunately however It can be 
shown that for the fli’st pulse two the calculated period Is 
very little altered by setting 6 s 0. The reason for this is 
that G la appreciable only in stages of the motion when the 
bubble Is small, and those stages occupy only a small fraction of 
a period. A rough estimate of the error Introduced by Ignoring 
G can be obtained from Appendix 1. For the later pulsations 
the amplitude of the motion is much loss, and eventually the 


50 


pulaafclona can be rogardad aa small oacllla'bions in the neigh-- 
bonhood of the radius at which tho gas presaur-’ equa2.8 the 
apri’c-Mndlng hydroatatie pr-sasxtrs^ For these sniRll oscillations 
G la eaaentlalj bxvb the calculation of the period is simplified 
by the assumption of small omplltudea* 

In Appendix 1 it is ahotvn that when G m 0 In (5}> the 
period of the motion Is 

, 1,136/° ^ (6) 

This formula has been derived by Willis^® and others. Of oours®, 
the comparison of this formula with observation is not complete 
unless a definite value can be inserted for W. Prom the 
discussion given above It is clear that W will be appreciably 
leas than the total energy released by the explosion. 

Theoretical calculations reported In Volume I of this Compendium 
agree with observation^^ In assigning to the first pulftatlon 
of the bubble a value of W of the order of half the chemical 
energy Q released by the explosive, the remaining energy being 
distributed In comparable proportions between energy carried 
off to appreciable distances by the shock wave and energy 
dissipated as heat by Irreversible processes at the shock 
front In the very early stages of Its motion. A convenient 

^®See the article by Wills In this Volume. 

^^3ee the article ty Arons and Yennle In Volume I of 
this Compendium, or Rev. Mod. Phys. 20, 519 (1948). 





51 


altsirnatlve to darlving W from Q Is to derive It from the 
maximum radius ot the bubble, If msasurementB of this 
quantity are available. If is neglactad^^, 

W » (AiT'/ 3) fimax^ becomaa 

a 1.829 ^ * pj^ (7) 


The corresponding expression for the period of small oscillations 
about the equilibrium radius la, according to Appendix 1, 


T 

a 


2 71' 

7rr 


p 


( 8 ) 


where Y la the ratio of the specific heats of the gas. For 
Y' s 1,5, (8} becomes 

3.18 /o ^ p a_ 

Willls^5 haa shown that (6) does in fact describe the 
periods ol oscillation of the bubbles produced by charges of 
vartoue sizes at various depths. With a suitable choice of 


12 

The rather Indirect experimental determination of 
reported In reference 11 gives a value of the order of 0.2W at the 
maximum of the first pulsation, presumably at 500 feet depth; however, 
this value Is open to suspicion since It Is 50% higher than the value 
obtained in a similar way for the^ second pulsation, whereas it should 
be smaller. Thus the correct G(an,ax) might be as small as O.IW at 
this depth; at the surface, where p^ is 16 times smaller, a 3 will 

l» ro^hlj 16 tl... lareer. and S(a„p/, ,U1 b. ..all.r bv 

X • "X 

a factor 16 " ^ i , iT we use Arons’ value of 1.25 for the 

effective ratio of specific heats 7' . 


13 


'See the article by Willis In this Volume. 


52 


W/ft , oonetant for a glvgn type of explosive, (6) oan be made 
to give results correct to within a fraction of a percent, an 
accuracy eomparabla to the degree of reproducibility of 
experiments. However, the value of W which one must use in 
(6) to get the beat fit. to a gi.ven range of experimental data 
will of course differ a little from the correct value of the 
energy of the motior;., because of the sllglit effect of gas 
pressure on the period. More refined comparisons of theory 
and experiment than those given by Willis have been made 
subsequently, and are reported elsewhere In this Volume. 

k rough Idea of the error involved in neglecting gas 

pressure may be conveyed by quoting the results whlTh one obtains 

by substituting into the theoretical calculations of period 

l4 

made by Shiffman and Friedman the empirical value 1,25 
obtained by Arons^^ for the adiabatic exponent in the equation 

7 

pV •=; const, and the value obtained by Arons for the energy of 
the first pulsation. This gives a period which is lower than 
that given by (6) by less than a percent, if W is Interpreted 
as including the total energy of the gas relative to infinite 
adiabatic expansion; the ratio of T to a is greater than 
that given by (7) by an amount which varies from about 4^ at 
sea level to about 9% at 500 feet depth. 

Fig, 2 shows a comparison of an observed radius- time 
curve with the curve calculated in Appendix 1 by Integration 
of (5) with G = 0 . The observed points were taken by Ewing 
and Grary from a motion picture taken by Sdgerton In 1941 of a 

14 

See the article by Friedman in this Volume. 

15 

A, Bo Aronr^, J. Acous. Soc. Am. 20, 27? (1948). 


53 






■bubble produced by a detonating cap a foot below the surface; 
they refer to the first oeolllatlon. The constant W has been 
chosen to aafe© the theoretical maximum radius agree with the 
observed one; the times of maximum size have also 'been made to 
ooinolde. The agreement is reasonably good; however, the 
Goaparison is not capable of Indicating Just how accurate the 
simple theory is, because of the deviations due to the presence 
of the free surface, deviations which will be discussed at 
length In Section 5 • 

In the present approximation the shape of the theoretical 
curve Is Independent of the size and depth of the explosion, 
there being only the single adjustable parameter Thus the 

curve of Pig. 2 Is applicable to all explosions, if the 
horizontal and vertical scales are expanded or contracted In 
proportion to 

As the successive pulsations of a given bub'ole decrease 
in amplitude because of acoustic radiation and ither dissipative 
processes, the periods will shorten and approach ths value (8). 

If the bubble can be made to remain reasonably spherical until 
the amplitude of oscillation has become small, measurements of 
or Tg , or both, may provide a check on the equation of 
state of the cxaloslon products. 

It is interesting to speculate that the use of 
propellent charges, which give no shock wave, might result 
in values of TT/Q, approaching unity, with correspondingly more 
violent bubble pulsations. 


54 


4, AGOUSTIQ RADIATION A 3PH5RI GALLY a viMSTRIG-AL BUBBLE 


An exact oalculation of the motion of a pulsating gas 
bubble In a oompresaible fluid would be very diffioult; Lamb^^ 
has derived a partial differential equation governing the 
variation of velocity with radlue and time for the epherioally 
symmetrical caafij but this equation la complicated and would 
be very difficult to solve, fortunately, however, the effect 
of the finite compressibility of water on the motion of the 
bubble produced by an explosion amotints, except In the very 
early stages represented by Fig. 1 (a), only to a small to 
moderate correction to the simple theory of the preceding 
section. It la therefore possible to worte out the details of 
the motion, ae affected by tne finite compresalblllty, by an 
iterative process of successive approximations, taking the 
non-oompre salve motion as the zeroth approximation. On© way 
of doing this la carried through in Apnendlx 2. This method 
involves transforming the equations of motion of the boundary 
of the bubble into a form in which the correction for 
oompre s Sibil It y le represented by a radial integral whose 
integrand decreases very rapidly with Increafel -i-uance 
from the center of the bubble; In thla form bull little error 
is made if the non-oompresslve approximation to the Integrand 
le used. The result la a differantial equation — Eq. (19) 
or (20) of Appsndix 2 — for the time variation of the radius 

16 


H, Lamb, Phil. Mag. 45, 257 (1923). 


55 


a of the bubble. This equation la almllar to that which would 
be obtained by dlfferoiitlating: Eq. (5) above with respeat to 
time* but contain 8 several additional toms, of which the most 
Important Is one proportional to a^(da/dt) j|dp(a)/d^/ , 
where o is the velocity of sound in water. As this expression 
is Intrinsically negative, one might be tempted to Interpret It 
as the Instantaneous rate of loss of energy by acoustic radiation. 
This is not quite correct, however, althougli it gives the 
correct total loss of energy over the whole interval when a 1s 
small. For the fact that the term Just written is proportional 
to the time rate of change of fcho left of (5} does not Imply 
that it is proportional, with a suitable constant time lag, 
to the rate of reception of acoustic energy at a distance. 

The equations of Appendix 2 are therefore directly 
applicable only to the calculation of the total energy of a 
pulse and to the calculation of how the time variation of a 
is affected by compressibility. It la easy to show from these 
equations that the effect of the finite compreselbiUty of 
water on the curve of a against; t ^s negligible when a is 
more than a few times its miulmum value; details are given in 
Appendix 3. Wear the minimum the effect may theoretically be 
either email or large, depending on the equation of state of 
the gas in the bubble. Rsoent experiments have shown that 

17 

A. Arons, J. P. Sllfko, and A, darter, J. Acous. 

Soo. Am. 20 , 271 (19^); A, B, Arons and D. R. Yennle, Rev. 

Mod. Phys. 20, 519 (1948), also Volume i of this Oompendlum. 


b6 

with oouanon axplosivea the total aooustlo Iobb of energy 

during the first oontraoted stage is never more than a 

fraction {''•’25/6) of the total energy W of the pulsation j so 

that the oorreetlon to the saotton la small to moderate. It 

might therefore be supposed that the radlua-time curve could 

be fairly accurately calculated by the methods of Appendix 2. 

However, this does not seem to be the oaea; the energy lose 

between the first pulsation and the second la found experimentally^^ 

to be mueh g 2 <eater than the observed acoustic energy in the ' ^ 

first bubble pulse, indicating that some dissipative mechanism 

is acting which has not been taken into account in the present 

theory. As an adequate discussion of this question of energy 

balance requires a knowledge of the effects to be expected 

from migration and asymmetry of the bubble, further remarks 

on this point will be postponed until Section 7. 

Let us now consider the way in which the pressure in 

the acoustic pulse radiated by the bubble varies with time. 

This 18 most conveniently computed in the way suggested by 
X8 

Willis : if a value r^^ of r can be found which is small 

enough so that the relative change of pressure in time r^/o 
is small, and which is at the same time large enough so 
that the linear approximation of acoustic theory is valid 

for r>.r^ , then the acoustic impulse received at any large ) 

distance r should have an amplitude given approximately by 

18 

See the article by Willis in this Volume* 


57 


p(r,t) - - (r^/r) jp(rj^,t-r/c) - p^J (9) 

The quantity In braotceta can be computed by using non-oompreasive 
theory for the motion In side r^. The well-known equation which 
relates pressure to velocity for Irrotattonal motion of an 
in© oapres Bible fluid is 


P - P = r - f. (10) 

^ at 2 

where ^ is the velocity potential and v ■ - is the 

velocity. This expression should give a fairly reliable value 


for the pressure at all distances from the bubble. Inside r^^ 
we may substitute the non-oompresslve approximations to v and 
vis.; 


v 


_ a^ d» 
“r2il * 


5^ 


a 


a2 da 
r dt 


» 


J 


It is then easily shown (Appendix 6) that (10) reduces to 



The maximum of (11) will occur at the time when a is a minimum, 
if the gas behaves at all like a perfeat gas. It is easy to see 
that the pressure given by (11) at the minimum of the first 
oontraotion, when is of the order of twice the initial 

radius of the explosive, cannot be more than a fraotlon of the 
pressure produced aii the same point when the original shook 
wave orossee it, 57 (9) this implies that the peak pressuz^e 
in the bubble pulse is at most a fraction of that in the shock 


58 


19 

wave; experiments give a ratio of about one fifth . 

The uXtuatlon Is quite different, however, with regard 
to the impulaa of pressure. Assume for slmpllolty that r 
is several times the second term on the right of (10) 

will then at all times be negligible compared with the first, 
and we have 




( p - ) at 




(19) 


This will be valid regardlaae of wfhether or not r is small 
enough for the non -compressive approximation to be valid at r 
at times near the minimum of^ on traction; It is only neoeeaary 
that the non-coninre sslve approximation be valid at t-j^ and tg. 

has Its maximum value, as shown in Appendix 6, when 


2 

ITow a da/dt 


a = 2 ®raax» i*®* * when the bubble is still fairly large. If 
we take t-, and t2 to be the times when a has this value In the 
contracting and expanding phases respectively, (12) will give 
the maximum Impulse of pressure d:ue to the non-oompreeslve motion. 
Exprjssing ^ in-terms of more oonvenlent quantities, the 

at 

maximum Impulse (12) can be reduced to 


1/6 2 

maximum Impulse of pressure = 2 am^x 

r 

r 

This turns out to be of the order of six or eight times the 
Impulse measured in the shoctc wave by the tlms the pressure 
^^A.B. Arons and D.R* Yennle, Rev. Mod. Phye. 20,519 (19^8), 
also Volume I, this Qompendlum. 



) 


I 



has drappeiS ta l/e^ a 
ba expeoted ta be 20"* 

0 i UUt> I.JM-. aoi*. 

predictlan Is In app 
In 19^1. whe 


dana ( sonia af tha a , 
from small explasl; 
same warkers suggestad ana*. *. 
af the gas In the bubble during ^ 
athars regarded It ms a spurious 
tioal analysis made at that time, 

indicated that the former hypo the . i 

subsequent exaerlnients have sha ., 4 ±t!,--.ubly 

symmetrical candltlans the pressur: • le curve far the bubble 
pulse has a smooth bell- shaped farm. 

It must be emphasized that all the formulas of the present 
section and Appendices 2, 3i and 6 have been derived only far a 
pulsation in which the matian is snherloally symmetrical, 
although generalization ta other oases Is not difficult. In most 
actual explosions gravity and proximity to surface, bottom, or 
objeots will Introduce asymmetrical influences. As will be 
shown in the next two sections, these Influences have an effect 
on the motion, which, though it may be slight when the bubble 
is large, is greatly enhanced in the contracted stages. Taylar"^^ 
and others have shown that these effects can greatly modify the 
acoustic pulse given out in the contracted stages. In certain 
cases, however, it is possible to balance the asymmetrical influ- 
ence of surface and bottom against that of gravity, and in such 
cases the theories dlscuesed in the present section should bey4pplloaute, 

^ ----- 


See the article by Taylar in this Volume. 


60 


5 , 14IGRATIQN OK A SPHKKICAL BOBBLE 

The motion pictures from which Fig, 2 was c on struct tv:. 
showed the migration effect mentioned in the introductions tUt 
center of the gas bubble was observed to rise slightly up to : e 
time of maximum size and later to sink rapidly, with periodic 
fluctuations, through a distance of several inches. Only afl ;* 
the pulsations had practically ceased did the bubble rise ag< 1 -u 
The explanation of this phenomenon, interestingly enough, ap, 
peared automatically as a by-product of an attempt to undergo nd 
the deviations of the points of Figure 2 from the theoreticf.'. 
curve, in a way which will now be described. 

It will be noticed that the theoretical curve is us 
broad, and that there seems to be a slight asymmetry in the 
experimental curve. Now the theory from which the curve wa 
constructed has disregarded a number of factors, of which 
most important are; 

(1) The gas pressure (responsible for the term G in 5)) 
has been disregarded. 

(ii) The water has been assumed incompressible, so 
that the damping effect of acoustic radiation 
has been ignored. 

(iii) The bubble has been assumed spherically 
symmetrical, 

(iv) The explosion has been assunied to take place ir, 
an infinite body of water. In the photographs 


61 


represented in Fig, 2 he cap wae only 12 inches 
from the surface, and ;as so close to a slanting 
at@el plate beneath that the bubble almost 
touched the plate a the time of its maximum size. 

The correction to the simple t‘ ory necessitated by (i) is 
estimated in Appendix 1, and t- t necessitated by (11) in Section 
A and Appendices 2 and 3* 1‘h< ^ i quantitative calculations sho>f 

that, as one would expect, tb^ corrections from factors (i) and 
(11) are negligible when a i near and tend anyway to 

broaden the curve in time, nr. compared with the simple theory. 

The correction from (ill) cs’-, lardly be appreciable in the 
neighborhood of the maximum ' dlus, since the bubble is observed 
to be very nicely spherical a this stage. There remains the 
factor (iv). The effect on ta motion of proximity to a free 

surface can easily be predl ■ )d qualitatively. The water between 

the bubble and the surface ' n be more easily given a radial 
acceleration when the surfi c is near than when it is distant. 
Consequently the stream li'^v. t tend to bend toward the surface, 
with the result that for a ivsn a and , the kinetic energy 
of the water is less than V s value (3) which applies in the 
absence of a free surface The potential energy is of course the 

same function of a and < as in the absence of a surface. 

Thus the effect of the c* ' ace is like decreasing the inertia of 
a simple oscillator withr- changing the spring constant, and so 
it decreases th. s «^iod, ..n Appendix 5 this effect Is worked 


\ 


62 


out quantitatively by the method of images, and it is shown that 
when the center of the bubble is a distance h below the surface, 
the period ?(h) is related to the period T(oo) in the absence 
of a surface (hut for the same ) bv 

Tih) - T(oo) U - ♦ o(^) {14) 


where a is the time average of a over a complete period. 
Near the maximum, the radius-time curve is contracted in the 

time direction by the factor compax-ed 


with the curve for h oo« 


The theory of Appendix 5 also predicts that when the 
explosion takes place at distance h from a plane rigid surface 
the period will be longer than in the absence of such a surface: 
qxiant i ta t ively 


T(h) - T(oo) U + ♦ O(j^) (15) 

The qualitative explanation follows the pattern outlined above 
for a free surface. The stream lines avoid the rigid surface, 
so that for given a. gf" » the kinetic energy is greater than in 
the absence of the surface. Thus the effective irsrtia of the 
water is increased and the period lengthened. The formula (15) is 


) 





-1 


an asymptotic one which should be a good approximation when h 
is two or three times • A clue as to how T varies for 

smaller values of n ia provided by considering the case h ■ 0, 
The motion of the water for this case must be the same as the 
motion to one side of an imaginary plane drawn through the center 
of the bubble produced by the explosion in free water of double 
the charge used. Since T is proportional to the cube root 


of the charge, 


T(0) » T(oo) 


When free and rigid surfaces are simultaneously present 
in the neighborhood of the bubble, the effect on the period, 
though calculable, is not simply the sum of the effects due to 
the various surfaces separately. For example, it is shown in 
Appendix 5 that when the explosion takes place halfway between 
the surface and a horizontal rigid bottom, and at distance h 
from either, the effects of surface and bottom do not cancel 
each other; instead we have 


I (■> 


a log 2 

T . 1 (00) (1 jip- ) 


Similar calculations can be made when a free and a rigid surface 
intersect at an angle. In general, when both kinds of surface 
are present the effect of the free surface tends to predominate. 


64 


A quantitative check of this theory by means of the 
curvature of the a vs. t curve of Fig. 2 at its maximum seams 
out of the question, since the distances Involved are not vary 
accurately kno%m, and since flow of water around the sides of 
the steel plate and even bending of the plate may have been 

quite appreciable. The factor (1 - which would describe 

the narrowing of the curve near the maximum due to the surface 
effect in the abaence of a steel plate, has the value 0.^6. 

This figtire is only slightly less than the ratio of the curva- 
tures of the observed and theoretical curves in Fig. 2; apparent- 
ly therefore the ateel plate had a suzi>rislngly slight effect, 
or else the measurements of radius ware falsified by bending of 
the mirror with which the photographs wars taken, or by distortion 
of the surfacs. A similar situation exists with regard to the 
period. The experimental errors Just mentioned could not falsify 
the observed period, but they could affect the calculated period, 
which is proportional to a ^^ . The period calculated from (7) 
la T^(ao) - .0297 sec., assuming normal atmospheric pressure and 
fresh water (p “ 1,00). If only the free surface were present, 

(14) would modify this to T^(h) .026^. The observed period Is 

T » .026^, and the difference between this and T^{h) could be amply 
accounted for by the effect of gas pressure alone, without radiation 
damping or any effect of the sttrel plate. 

The complete theory of the motion in the presence of a 
free or rigid surface, as worked out in Appendix 5» predicts that 


65 


the bubble, besides being sucked back and forth periodically by 
its large, should be continually repelled from a free surface and 
attracted to a rigid one. In the case of the bubble of Fig. 2, 
the repulsion from the free surface was actually great enough to 
make the bubble sink. As the mathematical details of the theory 
are rather involved, only a rather loose picture of the mechanisms 
involved will be given in the present section; moreover, it will 
be advantageous to discuss first the simpler theory of the migra- 
tion due to gravity alone. This theoi*y, which is worked out in 
detail in Appendix 4, can be described by saying that the bubble 
has associated with it a vertical momentum eqxial to its velocity 
of rise times (2n/3)pa^, and that the time rate of change of this 
momentum equals the buoyant force (4tr/3)pa^. This gives 

velocity of rise due to gravity - £s ^ dt ^ 

Note that the velocity of rise becomes enormously accelerated 
during the contraction, when a becomes small while the integral 
remains large. Although Appendix 4 assumes gravity to be a small 
perturbation, so that its conclusions are rigorously valid only 
when the velocity of rise is small. Taylor has shown that the 
effect of this rise on the motion can be calculated, to a fair 
approximation, by assuming that the bubble is constrained to re- 
main spherical, so that (17) becomes valid at all times. 


21 

See the article by Taylor in this Volume. 


66 


The migration due to a neighboring free or rigid aurfaee 
can be explained with similar concepts. The image of the bubble 
in the surface affects conditions in the neighborhood of the real 
bubble in two ways: it gives to the %iater there a velocity norosal 

to the surface, and it creates a pressure gradient in this direction. 
As is shown at the close of Appendix the migration can be cal- 
culated, correctly to the second order in \/h, by a slmpl* momentuis 
argument. To begin with, the pressure gradient due to the Itaage 
will impel the bubble in the direction of lower pressure, Just as 
the buoyancy of the bubble does when the pressure gradient <.s due 
to gravity. The normal momentum due to this effect can therefore 
be obtained by Integrating the product of the image pressure 
gradient by the volume of the bubble, and the corresponding normal 
velocity can be computed. The sum of this velocity and the velociv/ 
field of the image, evaluated at the position of the center of the 
bubble, tu.Tis out to give the correct value of the velocity of 
migration normal to the surface, to order 1/h , as determined by 
the more rigorous calculation given earlier in Appendix 5. This 
velocity is 




for a bubble at a distance h from a free surface; if the free sur- 
face is replaced by a rigid one, the sign of dh/dt is reversed. 


.... 1 






67 


It- will be noticed that (13) consists of a periodic 
term and a monotonic one, the latter corresponding to repulsion 
from a free surface and attraction to a rigid one. This can be 
understood qiialitatively from the fact that the pressure gradient 
due to the image is most effective, in impax*ting normal momentiim 
to the bubble, when the bubble is large. At this time the image 
pressure gradient is toward a free surface or away from a rigid 
surface. The predominant motion is in the direction opposite to 
this pressure gradient. Just as in the case of the rise due to 
gravity, the velocity of migration due to the surface contains a 
term proportional to the reciprocal of the volume of the bubble, 
which becomes extremely large in the contracted stages. 

In making comparisons of theoretical and observed migra- 
tion rates, one must usually deal with cases where both gravity and 
boundary surfaces are present. To the order of approximation used 
in Appendices 4 and 5» the velocities (17) and (13) can merely be 
added in such cases. The relative importance of the migration 
effects ( 17 ) and (l3) as compared with the rest of the motion can 
be measured by a ratio such as (dh/dt )/(da/dt) . For (13) this 
ratio at any stage is essentially a function of and is 

practically independent of or the size of the explosion. For 
( 17 ), however, the corresponding ratio is proportional to 
pga^ny/P^ , so that the gravity rise is more important the larger 
the explosion or the smaller p^^^ . 

For the experiments of Ramsauer mentioned in the intro- 
duction and described in Appendix 3, the combined effects of 




6S 


surface and bottom would never be more than a fraction of (17). 

If we set t ■ 2 T in (1?) and use the simple theory of Appendix 1 
fox' the variation of a with t, the right of ( 13 ) becomes 


gT 


(velocity of rise at 


\si6 may, therefore, expect distance risen by this time to 
be very roughly 0.15 gT" " 0.5 " '■ - . if we set a. 



"00 


max 


150 cm. corresponding to a typical one of Raiftsauer’s measure- 
ments, we find the distance risen to be only a little under the 
figure 10^ of a ^^^^ which he reported. By way of contrast, the 
distance which the Edgerton bubble would have risen by the 
time of the first maximum, due to gravity alone, is only a little 
over a millimeter, or less than of 

The slight asymmetXT' of the experimental points in 
Fig. 2 about the maximum is hard to account fur theoretically* 

It is shown in Appendix 2 that the chief effect of the finite 
compressibility of the water is to produce a radiation of energy, 
and in Appendix 3 that only a negligible radiation of energy 
occurs during the time when a is greater than say . 

The calculations of Appendices 4 and 5 show that, to the first 
approximation, neither gravity nor proxirai^ / to the surface can 
produce an asymmetry, 'oreover, the surface waves produced by 
the expansion of the bubble could not travel an appreciable 
distance before the contraction sets in. A retardation of the 





69 

contractior- could of course be produced by the gradual burning 
of some constituent of the cap which did not explode, but an 
improbably large amount of such material would be required to 
produce an appreciable affect while the bubble is large. During 
the later stages of the contraction the measurements of the 
radius may have been systematically too large, since the bubble 
grew an opaque "beard”. Near the maximum, however, the only 
explanation of the asymmetry would seem to be the distortion of 
the pictures by disturbance of the water surface or bending of 
the mirror, as already mentioned. 


70 


6, DEPARTURES FKQM SPHERICAL SHAPE 

In the analysis of Appendices 4 and 5; which has Just 

been discussed, the effects of gravity and of nearby surfaces are 

treated as small perturbations on the motion. As is shown in 

these appendices, the perturbed motion is in first approximation 

merely a superposition of radial and translatory motions, without 

any change in the spherical shape of the bubble. Moreover, 

22 

Taylor has shown that the observed motion of bubbles can be 
approximately calculated, even when the migration is large, by a 
theory in which the bubble is constrained to be spherical at all 
times. However, it is clear that in higher approximations depar- 
tures from sphericity will occur, and photographs taken during 
the later war years^^ do in fact show that in the contracted 
stages the bubble becomes flattened in a plane, normal to its 
direction of migration, and sometimes develops a mushroom-like 
shape. As this flattening has a significant effect on the rate 
of migration and on the intensity of the pressure pulse, some 
theoretical study has been devoted to it^^. The present section 

02 

See the article by Taylor in this volume. 

^^See the article "Motion and Shape of the Hollow Produced by 
an Explosion in Liquid" by Taylor and Davies in this volume. 

^^See for example the article of Penney and Price in this 
volvune . 


71 

will be devoted to aoae qualitative comments on the physical 
causes of this flattening and of finer-scale departures from 
sphericifcv » 

An easy way to see why a rapidly moving bubble must be°° 
come flattened is to consider the non->uniformity in the distribu- 
tion of pressure over the surface of a bubble which is constrained 
to remain spherical* In Taylor’s approximation, where the bubble, 
though constrained to be spherical, is free to expand or con- 
tract and to move up and down, it is clear that the average pres- 
sure over the surface of the bubble will equal the gas pressure 
and that the first moment of the pressure will be zero. It is not 
hard to show further that the second moment of the pressure, i»e., 
the coefficient of the second order spherical harmonic in the 
expansion of the pressure at the surface of the bubble, must in 
this approximation be the same as for a uniformly moving rigid 
sphere whose size and translational velocity are the same as the 
instantaneous values for the bubble. This can be verified from 

Taylor’s explicit expression for the pressure in which the 
2 2 

only terms in cos 9 or sin 9 are simply proportional to the 
square of the velocity of migration, independent of radial velocity 
and acceleration. Now it is well known that in the steady motion 
of a rigid sphere through an incompressible frictionless fluid the 
pressure is higher at the front and rear stagnation points than 

^^See the article by Taylor in this volume* 


72 

at the sides where the fluid moves tangentially to the sphere. 

It is therefore to be expeeted that an explosion bubble will 
flatten if Taylor’s constraint of sphericity is replaced by the 
boundary condition that the pressure be constant over the boundary. 
Small-scale distortions of the surface of the bubble 
tend to become greatly exaggerated during the contracted stages. 
This is illustrated by the fact that near the minimum radius 
bubbles usually present a ^decidedly prickly appearance , The 
phenomenon can be understood mathematically in terms of the treat- 
ment given by Penney and Price* for the motion of a bubblfa- d«»part- 
ing slightly from spherical shape. Qualitatively, the cause 
seems to be that illustrated in Fig, 3. Here the top curve may 
be taken to represent a small portion of the surface of the bubble, 
as distorted by some accidental ripples. This surface is of course 
a contour of constant pressure. As we go away, from the surface 
into the water (down in the figure) the contours of constant 
pressure must become smoother, as shown. This means that the 
pressure gradient must be numerically greater at the ’’troughs" 
of the ripples in the figure than at the crests. If the pressure 
is higher in the bubble than a short distance outside it, as is 
the case in the contracted stages, the troughs will be accelerated 
downwai^i much more than the crests, and the amplitude of the 
riples will increase • I X on other hand the pressure gradient 
is in the opposite direction, as it is when the bubble is large, 
the troughs will be accelerated upwasd relatively to the crests, 
and the ripples will be leveled out. 

^ tm m ^ tm m mm m ^ mm mm m ^ m mm 

^^See the article by Penney and Price in this volume. 


73 


?. THE BafERgy BALAPTC-B! 

It has already been mentioned In Section 4 above 
that the ensrjsy lost between successive pulsations seams to 
be considerably greater than the acoustic energy radiated. 

Thia l8 Illustrated in Pig. 4, which shows, on a smaller scale, 
the Buooesalve pulsations of the same bubble as Fig. 2 . The 
energies of the first two pulsations are roughly proportional 
to the cubes of the maximum radii, hence are in about the 
ratio ls0«31 • As a check, the kinetic energies shortly 
before and shortly after the first minimum appear to be In 
about the same ratio. Although no acoustic measurements of 
the bubble pulse were made for this explosion, it seems quite 
certain from other cases in which such measurements have been 
taken that nowhere near 10% of the energy of the first 
oscillation could have been radiated acoustically. Several 
procasser may be coneldered in an effort to find a cause 
for this additional dissipatlont 

(1) Turbulenoe. It has been noticed that for some 
cases, where the bubble is migrating rapidly the non-acoustlo 
dissipation of energy le of the same order of magnitude as 
that which would be expeated for the dissipation in the 
turbulent wake of a uniformly moving solid sphere of the same 
size and* veloolty as the bubble at its minimum, In a time 
of the same order as the duration of the contracted stage. 

Since the Reynolds number for such oases may be of the order 

27 

See for example A, B. Arons and D. R. X^ennle, Bsv. 
Mod. Phys. 20, 519 (1948), also Volume I of this Compendium, 



of 10*^, it is certainly worth while to examine seriously the 
possibility of turbulent dlsaipation. However, there are two 
considerations which make it seem unlikely that this ia the 
principal cause of the energy loss. In the first place« there 
Is to the author's knowledge no evidence that the disappearance 
of energy Is significantly less serious for bubbles vrtiich do 
not migrate than for bubbles which migrate rapidly; since one 
expects turbulence to be much less serious for a stationary bubble, 
a large energy disappearance for such oases would probably have 
to be attributed to other mechanisms. In the second place, it 
is questionable whether turbulence could develop quickly enough 
to produce a steady-state rata of dissipation in the very short 
time covered the contracted stage. For there are no solid 
boundaries to help start the turbulence, and a calculation of 
the stresses due to viscosity in the velocity field of Taylor’s 
theory shows these stresses to be negligible in comparison with 
the hydrostatic pressure. 

(11) Cavitation. It is Just possible that the pressure 
a short distance above a rising bubble may be low enough to 
produce a cavitation. However, even if this should occur for 
a rising bubble it would not explain the energy loss of a 
stationary bubble. 

(ill) Transfer of beat from the compressed gas to the 
water. Loss of energy through themal conduction can easily be 
shown to be negligible if the motion of the gas in the bubble 
is non -turbulent. If an appreciable portion of the energy of 
the gas la to be lost by conduction, the drop in temperature be- 
tween center and boundary of the bubble must be distributed over 

an appreciable fraction of the radius a. In such a case the 


15 


flow of heat outward In one period has the order of magnitude 

a'^. KASdt (19) 


ft a^. KAOdt 
Jo a 


~ Wr a=!s «.— » a 1 M ^ 4 «w4 4* vr ^ ^ o <T.Ck ^ AW) ^ ‘f'-Vl A 

wn©r'e a. xs uriw Ttut^xaiaJL C ^nv-t JC w x w x « j w*-'*- »»«•-. - 

difference in temperature between the center of the bubble 
and the water at the boundary. The ratio of (19) to Q or W 
becomes smaller the larger the explosion. However, it Is 
easily verified that (19) is entirely negligible as compar-ad 
with the total energy Q or W, even for the smallest explosions. 

It seems unlikely that there can be any appreciable departure 
from adiabatic conditions, even when convection inside the 
bubble is taken Into account. 

(Iv) A lag in the achievement of thermal equilibrium 
in the gas as it Is comnressad. If such a lag is present the 
work done by the water in compressing the gas will be greater 
than the work given up by the gas in re-expanding, and the dif- 
ference will be manifested as an irreversible heating of the 
gas. It is not unlikely that such an effect exists , especially 
since soot particles and possibly water droplets may be present 
in the gas. However, If this Is the predominant cause of the 
energy loss, the unbalance in the energy equation ought to depend 
strongly on the scale of the explosion. For the dissipation 
due to the lag in equilibrium should become very small when the 
duration of the contracted stage becomes either very short or 
very long compared with the time required to establish thermal 
equilibrium In the gas. To the best of the author's knowledge, 
no such dependence on scale has been noticed; however, there 
is not a very wide range of charge alaes for which accurate . 



78 

measurements sf bubble sizes and acoustic pressures have been 
made . 


1 



77 


APPENDIX 1 PULSAT-IOWS OF A BOBBLE IN AK 
IN CdMP^~SSiBLE P^UID 


V/e have to integrate the energy equation (5) of 


Section 5. Thie equation la 


2 rp P« ■t' 0 ( a) e w 

Consider first the ease v/here the potential energy 0 of 
the gas is neglected. Solving (1) for dt and intograting 


we have for the expanding phase 


tm - = 


^ da 


2 Pq<> 

rrp" '• I e ® 


where tj,j. la the time of maximum size. (It la convenient 
to choose the naximxim as origin because the approximations 
of this simplified theory are most nearly fulfilled near 
the maximtaa. ) Putting 


%41X 


\/ 4T Poo 

V “3" nr 


vire have from (2) 
t 


y V 


/ <^y 

y 'vnrryT 


The most convenient way to evaluate the integral for values 
of the lower limit different from zero is to transform It 
Into an incomplete beta function. If we set 


vje hav© 


z s § “ i VT"- y’ 
y^s 4 z (1-z) 

da s 

'■ 'h - r 


BO that 


where s 


dx is the ineoraplets beta 


.lx”- - (1-x) 

function. How In the range from 0 to 1/2 the factor 

r'A 


(l-z) varies quite slowly, and it can be quite satis- 
factorily approximated over the range of integration by 
a parabola. By this means the curve shown in Fig. 2 
was constructed. The radius-time curve is thus 


tja ~ t s 



sjr* ? 






(■?) 


where z is given by (5) and (3) and Bg is to be read from 

Pig. 5 . 


The period Is obtained In the present approximation 
by setting 2 - 0 in (7) and multiplying by two. v;e have 

/k c\ /k c\ i r’f’wll 

2Bi 


-piy- ' 


SO the period is 

Tjj r 1.135 W ^ (0) 

Let us now consider the other extreme, that of 
email oscillations about the equillbritjm radius aj , if 
the pressure-volume relation for the gas is 



79 


( 



( ) 


whore T is the ratio of the specific heats, we may set 

0 = ^ 7 av = (f yV’ (10) 

If ive sat X m a^, the kinetic energy term of (i) Is 



( 11 ) 


If we expand t>ie potential energy terms in powers of 
(x - Xj), where x, s a, , v;e find 

QA V ^ 

Potential energy - constant + TTzJ^Poo */ (x-x^)+*** (12) 


From (11) and (12) the limiting period for snail oscillations 
comes out as 


= 


I 0irp/2S 


2 -!r 


V 24T^poiX ‘^/25 Yz tT 


a 


(13) 


No detailed calculation of the period for Intex*- 
medlato cases will be attempted here. The order of mag- 
nitude of the deviation from (0) caused by gas pressure 
con, hov/ever, be estimated as follov/s. .‘^Ince in (13) the 
radius a, is the mean of the maximum and minimum radii for 
the small oscillations, it is tempting to rewrite (0) in 

terns of the moan of the two radii a_„_ aiid 0. If this is 

max 

done by putting 

1 44 

%ean “ 5 * wax » 3 ^ *raax P «» " ^ » 

and if in (13) wo take - 1.4 to get a nunerlcal value, w© 
find from (0) and (13) respectively 


oO 


->L 


2 1*0S9 ®jaax 

g 3,66 ^p;f^%esn 
Tg . 3.07f*^p^^a, 


(14) 
(14a) 

(15) 


Tlvus^ the coelTlcient of the mean radius In the oxprsaeion 
for the period varies only slichtly from infinitely large 
to Infinitely small amplitudes. A numerical calculation 
made by Professor Kennard elves the result that for ~iT's 1,4, 
®n:ax ® ®min • 2.0v , 

>.30 s 1.93 


T = 3. 


nax 


Thus, for largo arAplitudos (14) cXvqb a better approximation 
to T than (14a), while for small amplitudes the reverse is 
the case. 

A calculation of the fona of the a vs. t curve 
talcing account of gas pressure could be made as fcllov/s. 

For small a and reasonably large amplitiides, the second 
tena of (1) could be neglected or treated ns a small 
correction? the Integral for t could then be reduced to 
an incomplete beta function with indices involving 
For large values of a, tho term G(a) could be regarded as 
a small correction, and another tractable integi’a! obtained. 
It is hardly worth v/hlle to carry out the calculations for 
small a, however^ since when the amplitude of motion is 
large the a vg. t curve in this region v/ill be influenced 
by dissipation (see Section e 4 and 7) . The calculations 

for largo a will be briefly sketched here, in order to show 
that tho effect of gas prossura on the outer parts of the 


.3 


81 


motion le alight. Th« correct refinement of (2) ia 


*in - t 


/^ax 

■i 




da 


*1/ V/ G /. \ _ B G(a) . S(amx) 

K 2Tf 2Tf If*'- 


(16) 


'/)'$ may treat the laat tt»o terms under the radical as small 
compared with the reaialning ones and so replace the Integrand 
by the first two terms of its expansion in powers of these* 
Setting W - Q(a >nay ) s .^Pco^iax we have, on making the 
same substitutions as before, 


tm - t !3 




2 (r-1) j-sp^ 

7 

v/hore /®aiax * first term in the brackete would 

give the motion as previously calculated. Tho second can be 
evaluatewi with sufficient accuracy for values of y near 1 by 
setting 


-1 


1-y" 


s C<-1)^ 21L£!zil d-y"^) 
2 




dy (17) 


and then introducing the variable s defined by (5). !Thd 
details will be omitted here, since we slmll only need the 
order of magnitude of the ratio of the second term of (17) to 
the first when y is near 1* This ratio is asymptotically 
y^ /8, which is small for the value of y, ('^0.4) 
which characterizes the first pulsation* 

It should be remembered that ^ beoome's large almost 
Instantaneously after ^he detonation, so that the Influence of 
gas pressure does not Increase the time taken by the first ex>* 
pansion In the way that it increases the time of the ensuing 
contraction* 



APPENDIX 2 


PULSA!TIONS OF A BUBBLE IN A FLUID 

6¥¥tffw compMssi^^^ “ 


la this discussion v;e shall start from tha 
equation of motion of a viscous fluid, although It will 
turn out that the effect of viscosity on the pulsation 
of a spherically symmetrical bubble is negligible. The 
Stokes-Navler equation Is 

^ - 4- * - Vp (1) 

3 1 3 ^ r ^ 

v/here the symbols have the meanings explained on the 
notation sheet. Por fspherically syranotrlcal motion we 
have for the radial components of the various terms 




^'4 


V ) 


a 

d r 


^ ^ 


V 

and so (1) beeomos 


3~v 2 

i 

dr r 


d V . 2v 
dr r * 


■aT+ar'*''' y? ar l^Sr+ -f“ 

Let US Integrate this equation on r, at a given Instant of 
time, from the surface of the bubble to Infinity. The order 
of mafTiitud© of the effect of the viscosity tem is not 
likely to be much Influenced by assuming that ^ is 
constant for all stages of compression; this simplifying 


aseumption will therefore be made, since this term will 


83 


( 



( 





turn out to be negllcl-ble anyway. It will be convenient 
to separate the intecrated equation into terras charac- 
teristic of incoiapresslble motion end texnus involving 

B (r^v) s fi“» say. Accordingly, we note that 

r"" 3r 

J (r‘*V)d(-Vr) s r |V| ^ 


d’\ 




r«a 


The intecrated equation (2) now bdeornes 


fi d a _ a da 






r»ia 


dCT' , 
r ^ dr 




da 

rSt‘ y 


4- 




»r ^ r 


rsa 



rso® 


Jl 


or finally. 


d*a , 3 ^ 





oo 


y/if 



(3) 


This equation should be valid even for rat..,.-r early stages 
of the motion, when tho shock v/ave front has advanced only 
a fev/ times the initial bubble radius. This is because the 
starting equation (1) is valid richt throxigh the shock 


S4 


wave front. In the Interpretation of the right side of (3) 
w© should rerneraber that, strictly spoalting, p Is not quite 
the sfurie function of ^ in th© shook wave front that it is 
elsewliere, because heat is being goneratod in this region 
by irreversible processes. In practice wo may ignore this 
complication, however, as the extra pressure due to this 
heating is negligible. If this is done, the right aide 
of (3) will be a known function of the pressure at r a n„ 
l.".oreover, we have 


cr(a) 




rssB 



(4) 


v/hera c the velocity of sound In water. Thus, 

if ;re assme that the pressure Is uniform throughout the 
bubble, so that p(a) is a knov/n fmiction corresponding 
to on adiabatic law, (3) will become an ordinary differential 

^ OO 

equation for a(t), except for the r dr . iVo 

shell now trj to transform this term. 

An inspection of (3) shows that the only tenas 
which can be responsible for a dissipation of energy are 
th© last two on the left, since all the others are un- 
changed when the sign of ^ is changed. The first of 
these must account for the energy radiated away in the 
form of sovind. A hint on how to evaluate this term is 
provided by considering the special case in ?/hioh the 
amplitude of the motion is small enough so that the 
ordinary acoustical theory can bo used. In such a case 
will obey the wave equation and^ since only outgoing 




1j 

1 


■ i! 

< 

3 


" i 


85 


waves satisfy our boundary conditions, we may writ© 

r<r<r,t) * f(t ) j 
e 

where c is the velocity of soxmd in water. We then .have 

-oo ^ oe 

^ <1? s “O / dr s ea(r(a) 

'a /a 


(5) 


( 6 ) 


For the larce amplitudes with which we are chiefly con- 
c-amed the acoustical theory on which (5) is baaed is of 
course invalid. It v/ill be shown below, however, that 
for larce ainplitudes of motion the integral on the left 
of (6) can bo represented by the same term, cacCa) , 
plus other terms of lesser Importance, 

Let us start by taking the divergence of the 
Euler equation (i.e,, of equation (1) v/ithout the viscosity 
terms) and then differentiate with respect to time. The 
result Is 


If-) 


(7) 


1 

X 



^'low ^ ^ - V. V ^ 

For simplicity it will bo assvuned that c is net appreciably 
changed under the compressions to be encountered, so that 
it can be treated as a constant in the differentiation. 

This approximation is only bad at the start of the first 
pulse, a stage which we have already excluded from the 
present treatment because at this stage the pressure cannot 
be assuvaed constant throughout the gas bubble. For constant 




ni 

SJ 


I I; 

I ^ 

I I 


\ i: 

■I 


i 

i 

' '' P 






86 

0 , (7) bocomes 

“ . ^ ■* 


A 


at 


X - e 7 <r = 7 




(e) 


1.9., In the present case of spherical symmetry 

3”1[r<r) , 

t — ^ " ® ' a ^ 

Let us multiply the equation (9) by an arbitrary function 


(9) 


t •(- XS:i^-l| and Integrate on t from -«» to 0 and 


on r 


from r^ to «o, where r^< r, . (The function 8 will play the 
role of the "Dlrab delta function” used in quanttsn 
mechanics. ) 

On integrating by parts and taking 0 * and F as 
vanishing at t » *oc> or at r a oo » we obtain 

X 4^) ■ /f-^J ^ 


But 



^-i£l 


r<r 


3 0 


Srj 


dt 


TmV, 



tnO 

p^e 

P^dtdr (10) 


i; 


and 




in the third integral on the left we can use s e 
80 that (10) becomes 



a*- 


-”*tHO 


tsO 

dt -./T, 

— *rsr_ y„ yioe 


P S dtdr 


(11) 


J* Oo 


) 


It ■' 


87 


How lot us spaclfy that S(x) be dlfferont rrom zero only 

^ 6f© 

in a very narrow range about x s 0, and that / f(x)dx s 1» 

*o 

As we make |(x) narrower and narx°ower the last two terms on 
the left of (11) will eventually vanish, while the rest of 
the equation approaches 




s OJ F 

tmO 

rsr, 


(r, -ot,t)dfc 


( 12 ) 



Integrating (12) on r^ gives now, for t s 0 

Oo ^ 

® ca<r(a)'^ j jF (r^ -ot,t) dtdr^ (13) 

a 

The double integral on the right represents the correction 
to (6) necessitated by the fact that the amplitude of motion 
is large* It can be approximately evaluated by Inserting 
for P a funetlon obtained from the non-compresslve approxi- 
mation* We have fr«xn (8) and (9) 


where 


How 


P 8 a 

^ ^ r*- 

§ = j^o*-Vlogf - 


(14) 


c~71oB f a = 


_1, 

€ 


VP 


2 — T9 V - _ — i» 

IPP v»Vv 


f = 


-2v .MZ “ 5 V 






so we have 


(16) 


Since V In tii® lion-oompreaslve appj?oxlwatlon varies inversely 
as r’, $ will dl® off very rapidly as v/e qo away from the 
bubble, and the chief contribution to the double integral 
in (IS) will com® frois values of (r^- et) very close to a. 
Therefore, there can hardly be a very large error intro- 
duced by using the- nea-compresalvo approximation to v in 
tho evaltiatlon of this integral* Also, we may note that 
if -^^c, most of the variation in P during the Intogration 
(m t is due to the change In the argument (r^- et), and 
very little to the change In the argument t. Thus, we 
may set 


P(r, - ct,t) P(r, - ct,a)+t P(r^ -ct,0) 


(16) 


and gauge the adequacy of the approximation by the smallness 


of the effect of the second term* We have 

'O /^*V*”** 




yOStj^ 09 

f f 


J yp(r^ - et,t)dtdr,«s-!' J J P(r,0) ^ dr^y P(r,0) 


dr , 
rs" dr 

C ( 


-oe 


si/ (r-a) P (r,0) dr - /(r-a)^P(r,0)dr (17) 


Using (14) the first of these Integrals becomes just 
~^(a,0) . The second can be evaluated using (14) and (15) 
with the non-corapreasive approximation to v. The details of 
this will not be given here and we shall merely state the 
result. It is that the effect on the motion of the second 
term of (17) is rather lass than that of tho first during 
the first contraction of the bubble. Both terms are largest 
for the stages near that of minimum radius, although tho 




flrat 7«nis&e« at this spadiua. The effoct of the second 
tern may be larfjer than that of the first just after the 
«»ploSiOn takes place, but the present analysis is pre- 
vented by other reasons from applying to these stages. The 
larger th@ radius of the bubble, the smaller the effect of 
the second tem compared s?ith the firsts a numerical cal- 
culation gives a value of about .05 for the ratio of the 
second term to the first when 3™ a | . Since the first 
term itself is not very Important, we shall drop the 
second term and write {17) as 

j * ct,t) dtdr^ I f (a,0) 

✓ a ,f—oo ■ 




We are now ready to solve (S) by using (18), 

(1'3), and (4). The last term on the left of ( 3 ) ohn now 
be seen to be entirely negligible, since the principal part 
of the correction to the non-eoaprosslv© approximation is 
the term ca<r(a) in (15) s we have in fact. In c.g.s. units, 
y^'^10 while oa ~lo^to 10 ^ , We thus write (3) 

• 1 ^)“' - -h 4e 

yr»B 

Mow the left side of (19) is 




90 


while on fche right we have 



p(a) -p^ 

VI 6a 



where ^ la the density at infinity. The second term Is 
negligible for all stages of the motion except those 
Immediately follovsrlng the explosion, so we may write (19) 
In Integrated form as 



The equation (20) la the extension of the energy aquation (6) 

of Seotlon 5 » to which It reduces as c — , 

4 ^ 


The factor ^1 » “e/ left of (20) has 

merely the effect of making the curve of e against t 

asymraetrlool U*e.» of making the contrnctlon slower than the 

expansion), without however producing any dissipation of 

energy. Since rtf' c for all those stages of the motion 
dt 

which we are considering here, this factor Is rather unim- 
portant. The dissipation of energy arises from the tem 
dp 


In ^ on the right. For we have 


/a 


( 21 ) 


%ax 


/'^^“max 


Since ^ 0 , this term Is negative and becomes ever 
greater in magnitude in the course of time. The effect of 
this dissipation on the motion la dlscuBsod briefly In 


) 






Appendix 3, where It is shown that only a negligible 
amount of energy is radiated when the bubble is large, 
bub that at the flrat oontraotlon an appraelable 
radiation of energy may take place. 

Before concluding, it should be mentioned that 
the preceding analysis disposes of an objection which may 
be raised against the simple theory of Appendix 1. The 
objection is that for a certain range of depths the 
pressure Jump at the shook wave remains greater than p<m< 
for a large part of the period of the first pulsation. 

It Is therefore nob immediately obvious that the motion 
of the bubble v/ill be the same in the actual case as in 
the non- compressive approximation, for which the pressure- 
radius curve of Figure 1 would be replaced by a monotonic 
curve with asymptote poo • 


92 


JIPPBTOIX 3 Qg THE SAW .0?. 

ftAMATloll OF wm(F£ 

In tililB AppoiiSiA tlis Tate <5* j*&diatioii of 6HS*'Cy 
fyom th® pulsatlne bubble will be calculated on the assump- 
tlcm that the rate of energy loss is so smal- that it has 
a negllciblo effect on the course of the motion. Tills 
assumption will turn out to be somewhat rough near the first 
ffliniaum or two, but the ealculationa here will at least 
indicate the order of macnltude of the energy loss. Our 
starting point will be equation (20) of Appendix 2. As 
explained there, it is a fairly goo<^ approximation to 
neglect the ^/o on both sides, so we write the equation 
AS / 

j (p(^) " ^ ® 

^ ®max '^Ajnax 

The left of (1) differs only by an additive constant from 
the left of equation (6) of Section. 3 of the texti it thus 
represents the kinetic and potential energy which would be 
present for the motion of a bubble in an incompressible / ^ 




93 


fluid with the seme a and ^ , There oannot ba much 
error involved In assuming that the amount of energy radiated 
during the contracted stage is equal to the total change in the 
right of fl} during this stage. 

The right of (1) will be evaluated by insertine 

the value of ^ \idilch would obtain if there were no 

dissipation. We shall consider three extreme cases s 

(i) Small vibrations about the equilibrium radius e;|^. 

This case provides a check on the starting 
formula (1)^ which should yield the same result 
as the usual acoustical theory for a simple 
source. 

(ii) Near the maximum radius for vibrations of 
any amplitude. 

(iil) Hear the mlnlmtim radius, for vibrations of 
large amplitude. 

In all three cases we shall assume for simplicity that the 

pressure-'Volume relation is 

4-r jrjr 

pa s Poo a, or s Pjaax ®min 


(1) From (1) and (2) the energy dissipated in one 


f S 

m 


P . 5 ' » 

m o 
1 1 


cycle la 
- 4T 




If a*dt 


which in the limit of very small oscillations is 




90 /daY 


where Tg is the period. At the same time the total energy is 




• ■ ii •• 


so that the fraction of this energy lost per cycle Is 


3?oof, 




JP-o 

= 2T-^ 121 


by (13) of Appendix 1. Jn the usual acoustical theory 
the energy radiated by a simple souroa in one period is 


"?T Jp A** 
T c Tf 


where A = 4TTa, 


is the "strength” of the source* 


The ratio of this to (3) is easily verified, by using (13) 
of Appendix 1, to be the same as (4) . For s 1 atmosphere, 
a 1*4, 0 s 1»5 X 10® cm/see, (4) comes out to be .086. 

(11) In the neighborhood of the maximum radius let 

where the time t«0 is taken as the time of maximum radius. 

The negative of the right of (1) is then asymptotically 




(2^t) 


- IsTj^/a 


„ V- ^*ittex’' 
*max i 


The kinetic energy is 


2 ’T? erf 


and the ratio of (6) to (7) is / 

-.aa# 




t 

®aax 


z' . 

/ a, \ ^ 

C ®maay 




95 


( 





by (14) of Appendix 1. 'llie quantity (3) represents the 


fractional chanee in 


(^T 


produced by the dissipation. 


as compared with the ideal motion of Appendix 1» This 

change is negllGible in the neighborhood of the maximiim 

radius, since for IT's 1.4, 1 atmosphere, 

c s 1»5 X 10® cm/sec, ^ reduces to 

.4 . ®^max ® 

3.4 X 10 

(ill) For large axiplltudes of motion we may Ignore poo 
in the neighborhood of the minimum radius, and thus obtain 
from aquation (1) of Appendix 1 

^ r Vw « o' 

V 2 rf a3 


35 



Where 0 s W 
Using this and (S), the right of (1) becomes 


ISTyp^ax ®mto V W 

Trr^ 







e-sir 


dx 


(9) 

( 10 ) 


( 11 ) 


where x = % — • «Ve want to find out how much energy la 
•tain 

radiated away during the entire portion of a cycle when 
the radius of the bubble is small (we have seen that the 
r .dlation is negligible at other times). To do this let 
us take the limits of the Integral in (11) to be 1 and Oo 
(the exact value of the upper limit is unimportant) and 
multiply by two. The resulting Integral can be expressed in 
terms of gamma functions, and dividing (11) by W we have 
finally 


96 


Fraction of 
total energy « 
radiated 


tjIsF • r(s/gi fj 


”na:x min 


If wo set 


P 

w - 47T min max 

w . -Trrrr * 


(IS) takes the more convenient form 


Fraction - 
radiated - ® i 


r(3/2)rl^) 


M'S ^1) r* /'-Sx-rA-S 
' (sr-sy 


Tlio coefficient of 


tT 

“Ttir 


c has the values 
Coefficient 


It must ba remembsred, of course, that (13) will not be 

exactly correct even for a spherically symmetrical motion, 

because we have ignore'^ the modification of the motion 

produced by the radiation and by other dleslpatlve mechanisms. 

When aBymraetrlcal influences are present, the radiated energy 

2B 

may be very much less, as Taylor has shown, because the 


bubble will not shrink to so small a radius. 



( 


APPENDIX 4 


EFFECT OF GRAVITY 


O 


Whenever the radius of the pulsating bubble Is 
small enough eoraparod with the hel^^t of the column of 
water necessary to produce pressui’o pc» « ws may expect 
the effect of gravity on the ptilsatlons to be small. Ae- 
eordingly let us expand the. velocity potential f and the 
shape of the bubble in powora of g. clhooaing the initial 
position of the center of the bubble as origin of coordl- 
nateSy w© may denote the radial distance to a point on 
the boundary of the bubble by R(0 )» where is the 

angle with the vertical. He thus sot 

^ (?,t) s 

P (ihb) sPo-f-gP,-#- ..... (2; 

R(6,t) £ a(t)-4-gRj(5,t)-^'»..». 15) 

Tailng p«> to be the pressure at infinity in the plane z»0, 
the equation of motion is 

^ (K7^) s (4) 

and the first order part of this is 

ail -iii- ^ii- = (6) 

To Integrate this wo must use throe additional relations: 

(i) Since “ 0 » the expansion of in 

spherical harmonics must have the form 

^ , ^ !<=) 


1 m 

■.'■I 1; 




98 


pi?ovlded ther© are no obataelea o:? fr-aa aui>facoa 

nearby. 

(ii) The boundary condition Tfdiich p, must aatiafy 
Is deteziBined by the fact that p(R) is a fixed 
function of the voliana V of the bubble. Indepen- 
dent of g. Thus, to the first order in g 

p (R,t) s p^ (a,t) + gR,jj|£i^^ gp, (a,t) 


s 

from which 






± 23 . 


dr 


r»a t 




(lil) Since 


dR 


z ^ 


(i 


£) 

dr / 

/ r» 


) : i 


r»R 


'■} ! 


i 


we have 


dRi 


«'V V a'’ ^ 


* + +- 


a' 

The procedure is now to substitute (6) end (7) In (6), and 
to use (8) to express A^, B, , in terms of the coef- 

ficients in the spherical harmonic expansion of R^ « If we 
let _ 

M 


PO 


R 


. - 2 ^t) (CO30) 


( 8 ) 


( 9 ) 




tho differential equation whieh results for be 

hotnoceneoue when i>l, BO that the solution which satisflea 

(?) gJf) 

' » 0, " * 0 at tsG (start ef motion) 

will bo simply 0. The same will be true when J^s 0» 

since the second term on tho right of (7) is proportional 

(o) a 

to Rj . Thus we have only the equation for x s 1 to ec«i« 

aider: this equation is 


'r*a ~ 

or, using (8) and the relation 


a. 

g 


a 3 da dR?^ 2 

2 - a 


This integrates to 

dSp ^ - 2 

" "P" 


The equation 


J 

a dt 


Rs a-^-gR,*oos0 

moans that to the first order in g the bubble is displaced 
upward by the amount without change of site or shape* 

If one is interested only in the relation (12) « a nmch 
simpler derivation can be given* Consider the momentum 
of the water contained in a large vertical cylinder of 
radius length L with center at the origin. 

This momentum is 

1? » ^Jjf^ ^ " f Jf f 

vy „ 


bubble 


cylinder 




iOO 

where is fch® outward normal in each case. As 
the second surface Integral £oes to zero. If we take the 
origin of coordinates at the Instantaneous oenter of the 
bubble, the evaluation of the first surface Integral Is 
simpllflodi we have » 0 at the given instant, so that 
to the first order in g 



by (0). How the time rat© of change of is equal to the 
total gravity force acting on the water in the cylinder, 
plus the total pressure force on the ends, plus the momen- 
tusi entering across the surface. In the limit the 

last contribution vanishes, and the first two give 


11m -^2L 
L-^oo dt 


4r 

T" 


S 


(14) 


Equating (13) to the time integral of (14) gives (12). 

At the maximum of the first expanslctfi the value 
can be obtained by substituting the value of ^ 
from Appendix 1 into (12) : disregarding gas pressure, the 
result turns out to be 




0.62 gT^j 


( 15 ) 


The value of gR^^ at this stage is obtained, as to order of 
magnitude at least, by multiplying half of (16) by To/2» 






101 


The rise of tho bubble during fche first expansion is thus 
proportional to and the ratio of the distance risen to 
proportional to T. ^-xs rise is therefor© negligible 
for mall eaqplosions, but considerable for large ones. 

A conspicuous feature of (12) is that the rise 
is onoraously accelerated during the contracting stage, 
because of the a in the denminator. This is an illus- 
tration of the general Instability of the contracting 
stage. Even for rather snail explosions, the velocity 
of rise will probably become so groat during the con- 
traction that the approximation of treating the gravity 
correction as small will break down. It is certain, 
however* that a vary rapid vertical acceleration will 
take place, which will tend to create turbulence much 
sooner than it vrould be created in the absence of gravity. 


APPENDIX 6 


EFFECT OP PROXIMITY TO A FREE 



Lot the center- of tlie q&s bubble at the atart of 
the motion be a distance h beneath the free surface of the 
v/atcr, or alternatively a distance h from a plane rigid 
stirface. The vaster v;lll be t rented as lnc(»npres8iblef and 
gravity will be neglected, since it is easy to show that 
the first-order effects of the surface and of gravity are 
additive. V«e shall be interested in two effects due to 
the presence of the surface, namely, the change in period 
and the rising or sinking of the bubble. The complete 
theory, which is similar to that of Appendix 4, is rather 
compllcatedi however, the calculation of the effect on 
the period can be made quite simply, end this ?/ill be 
given first. 

The boujidary condition which must be satisflod 
at a free surface is that the pressure p must be constant 
over the surface. This means that the velocity potential 4* 
must satisfy 

(’f)" =0 

on the surface. Since ^ is of order l/r, where r is the 
distance from the center of the bubble, the ratio of the 
second tern of (1) to the first will be of order l/h as 
h — It la not hard to show that this ratio Is quite 
small when h is even a few times so that it v/111 be 

good enough for our purposes toieplace (1) by the condition 


103 




I f 

E V 


f - 


on tho free surface 


Altomatlvoly* v/e must have 


• V ^ s 


on the rigid surface 


where n is the unit normal to the surface. It will be 
convenient to picture as the electrostatic potential 
due to a charge (and, if necessary, a set of multipoles) 
located at the 5.nitial center of the bubble, in the presence 
of (a) a grounded conducting plane, or (b) a plane boundary 
of a medi'um of dielectric constant aero. The total ” charge” 


of the bubble is 


e - - 1 
" TT 


f a 0 d,3 s gp- 

bubble 

surface 


where V Is the volume of tho bubble. As is v/ell known, the 
field due to the charges induced on the plane in case (a) is 
the same as that produced by an oppositely charged mirror 
image of the charged bubble: the linos of force (stream 
lines in the hydrodynamlcal problem) accordingly look as 
shown in Figure 6 (a). Similarly in case (b) the field 
produced by the polarized dielectric is the same as that 
produced by an image with the same sign of charge, and 
the lines of force are as shown In Figure 6 (b) . Now the 
kinetic energy of the hydrodynamlcal problem la 


ar, 


2 

entire 

fluid 


S' 

I 

n 


104 



which differs only by a consten.t factor from the sl©ct-ro“ 
static onerey 

-hjff 4 ’’ 

of the electrical problem. The chan^^e in the latter when 
the bubble Is brouj^t from infinity to a distance h from 
the plane is equal to the work done in this operation. Zf 
h 1s a few times larger then the radius a of the bubble, the 
field E acting on the bubble, due to its image, when it is 
a distance x from the plane, can be taken as — e'*/4x‘*, with 
th© upper sign In case (a), the lower in case (b). The 
wox4c done, therefore, is 





) 





105 


* B 


Any redlatrlbution of charge (norroal velocity) on the 

etirf&oe of the bubble^ end any alterable in the shape of 

the bubble, due to Ita proximity to the gurfase, will give 

a contribution to this work which is of higher order In l/b, 

and negligible if h la several times a. The work (4) is to 

be compared with the aieetrogtatle energy ©“^/"a of the 

bubble when it le. spherical and at an Infinite distance 

from the plane* Thus, to the first order In l/b. we have 

for given V, ^ , 
dt 


!,;■ * 


Kinetic energy in _ . 

presence of surface “ 


a . / Kinetic energy ) 

•gS") 3C I in absence of / (6) 
1 surface ^ 


where the upper aign is to be used for a free surface, the 
lower sign for a rigid surface* 

The motion of the bubble in the presence of the 
stxrface is accordingly given to the first order by multi- 
plying the first tern of (1) of Appendix 1 by the factor 
(1 q: ’■^)* Bnd Integrating the resulting equation. This 
equation may be written simply 


<ir= m) 


(R is used rather than a for the radius of the bubble, since it 
will be convenient to use a(t) for the radius in the absence 
of pertiarblng Influences.) The fom of the function f(R) 
caruiot depend on h, since both the total ©ner^, and uhe 
form of the potential energy function, are independent of h* 


i06 


It «e set 


9 • • « 


R(t) a Rj(t)4-ia^Rjt)-#- 

the first order part of (6) is just 

? t ^ ^ “ 2 ^ |g| 

3 ^ 


(7) 


s 


2 R 


d\ 

• 


(S) 


Since R, * 0 at the start of the motion# which we take at 
t«0# the solution of (6) Is 

a dt (9) 

^0 

H« B,/h^ is simply th® time interval by which the 
curve of R against t lags behind the curve of a against t 
for a fixed radius, llie value of this quantity after one 
complete cycle represents the amovint by which the period 
is shortened because of the surface. Thus# we have to 
tiie first order In 1/ii 


n, - 


-f- 1 da 


I 


T (h) s T (oo) 


UO) 


where “a Is the time average of a# and where the upper sign 
Is for a free surface, the loiver for a rigid one. 

When there are two or more plane surfaces near 
the bubble, their effects on the motion are not simply addi- 
tive, even in the first order. In some cases the combined 
effect can be calculated rather easily by the image methods 
of electrostatics already used- For example, consider a 


) 






bubble halfway between the surface and a horizontal rigid 
bottoa. The field between these two planes in the equivalent 
sleotrostatle probletn is that of the original charge a and 
an infinite set of images, ae shown in Pig. 7* The two 
nearest Images give equal and opposite contributions to 
the potential at the bubble, and thus cancel; however, the 
next two images both have charge -e , so give a negative 
contribution. The potential at the bubble due to the infinite 
set of images comes out to be ~ log 2 , 


which differs 


i08 

only by th® factor log 2 ® Q*S9 from the valu® which 

m would have If the bubble were a distance h from the 
surface In infinitely deep water* Thus for this case 

T a 5 (-«) a - ) 


Iff on the other hand* the bubble were betv/een two parallel 
Infinite rigid surfaces, its kinetic energy when espanding 
or contracting would be infinite} for this case T«o» in 
the nonocoinpressive approximation. 

In the diaeusaion so far, no account has been 
taken of any displacement of the center of the bubble, 
or of any deviation from spherical shape, due to the proxi- 
mity of the swface. Following the ideas used in Appen- 
dix 4, let us set 

<j> (?,t) a i +- I -h (11) 

P (r*t) - (12) 

R (d,t) s ^2*. (15) 


where H is the distance from the origin (taken at the 
initial position of the center of the bubble) to the 
boundary of the bubble in a direction mp’;lng angle & 
v/lth th® a axis, vAilc’. wo shall draw fran the origin toward 
the surface and normal to the latter* The equation of motion 
is 



(14) 


/ 


109 


and thd fix's t end saoond order porta of this are 

Aa In Appendix 4, v/e use three additional relations to 
Interjrate these: 

(1) Since ® expansion of cj>^ in 

spherical Uamonlcs must have the form 

t C • + V(-^ ^ B^r) cos e 


~t ("tJ" + ^a. ( cos ® ) + 


where the toxtsis In a”, b”, c”, 

n n n 


represent 


the field produced by the ima^^e of the bubble In 
the surface . The potential ^ ” at the orij 3 in due 
to this image Is ^ correct to the second 

order in l/h* where e is given by (3) and where 
the upper sign is for t I'rae surface^ the lower 
for a rigid onoc Similarly, the gradient of the 
Image potential at the origin is, to the second 
order in l/h 

K 3z / +TV^ 



lie 


and. In q&OlQTbI, the derivative of at the 

“k**X 

origin is of order h“ . Thus, we have 


a. 


da 


1 + m • 1 


''l * 


Q' 

\*e/ 


w 


Ag s 0, Dg ■ I ^ , Cg S 0 


( 19 ) 


(li) To order l/h*', the pressure at the boundaiy 
of the bubble Is 

P(R#t) = p^(e,t) -h (R,A + 4^ 


or 


-t 


;u 

a Ps, 


fc , P, (a,t) A *h 


2 h'^ a r' 


h-" dr 


Also, to the first order In lA» 

p(R»t) - (a,t) *f dTa”** ^ 

dp 

= 

(») 

where la the spherieal average of R,o Since 
at the boundary of the bubble 

^ ^ da 5 Po 




(81) 


ar 


(20) and (21) give 




( 82 ) 


Into this v/e may Insert from (14) which la. 


Ill 


i - 
ii 


since • 


^ 

r dt 


^ a-de 

* “ 2 r'^ dt 


We shall not be interested in the spherically 
syronietrical part of p, ; if we anticipate the result 

jo 

to be proved below that is apherlcally aynEietrl~ 
cal and remember that p(R,t) is constant over the 
sTirface of the bubble, we have from (20) 


pja, B ,t) s - rJ 


"apJ 


spherically 

syiranetrical 

terms 


(Hi) In the equation ^ S - 

insert _ r*I 

^ (^) 


let us 


^ ('‘8) ^ 
\ P^CcoaS) 

a 

v^e obtain, as in Appendix 4, 




dR^i'^ s - S da „ 
-a^ “ alF« 


- .r) 


- 




To obtain the differential equations for the n) / 
we first Insert (18) into (26) and solve these equations for 
A/# ®/» expression (17) for can 


110 


be evaluated in terms of the $ a, and ^ 


If this is now inserted into (15) and the richt of (IS) ex- 
pressed in teime of and of a and its derivatives, by 


(22) and (23), a set of second order differential aquations 
for the R fAt) result. All the equations will be horao- 
eeneous except that for since at t»0, 

this means that 

6 

r; -0 for X>Q 


Por/*0 v/e obtain the equation 

Co) 


a da d*R 




'S£ “dt^ 


dR ; ^ ^ a 

(3 


d^a 


3 

d a 

a 


a ) „(« 
/ 


- + 




It can be verified that (8) la an InteGral of (29), so that 
the solution (0) obtained by the simpler method Is checked. 
Another check on (29) is obtained by notlnc that the left 

• ^ , correspondlns 


'^da'f , a^ da dfa ( 


side reduces to aero if we set R, • 4c » 


to a shift In the starting time. 


The same procedure can be used to get the dlf» 


ferential equations for the using (19), (27), (17), 




(16), and (84). For />1 the resulting equation is homo- 
geneous, so 

for 




R' '* 0 


For U v/o have on collecting the terms 


a 




113 



We conclude that as far as the second order in 
l/h, the effect of the surface is to change the period of 
the raotlon (first order) and to shift the center of the 
bubble a distance toward the surface, 'fhis latter 

displacement consists of a periodic part and a raonotonlc 
part* Tlie periodic part represents a sucking of the bubble 
back and forth by its image: the velocity at the origin 
doe to the image is — e/4h*' s ^ a j the first term 

of (3S) Is three times this because the bubble acquires a 
dipole mcmont Just sufficient to keep the pressure constant 
over its surface. Tlie monotonlc second term of (32) is 
negligible compared to the first in the limit of small 
velocities, although not for the velocities encountered 
in ssploslons. This term, like (19) of Appendix 4, con- 
tains in the denominator, and thus illustrates the in- 
stability of deviations from spherical sysunetry during the 
contraction, Tlio ratio of the raonotonlc part of the velocity 
, due to the surface, to the gravity torm g of 



The Integrals oui be evaluated Tor t«»T» by the methods 
of Appendix 1, in the approximation tdilch neglects gas 
pressure. The results are 




1.14 '/-£T 
' Pao 



80 that the ratio is 

) suy^’ace term j 
I gravity term f 


0.20 




®max 

"S~ 


For small explosions In shallow water or close to a rigid 
surface this ratio can become appreciably greater than one, 
so that the bubble can, for example, sink instead of rise. 
It must be reraembered of course that when h is as small 
as two or three times 8^,^ the rate of drift given hy (38) 
becomes so large during the contraction that It can by no 
means bo regarded as a small correction to the motion. 
However, (32) should give the drift fairly well over the 
greater part of the first period, so the qualitative con- 
clusion that a bubble will rise or sink in a given experi- 
ment is probably safe. 


(34) 


(SB) 


( 36 ) 


ii§ 

In oonoluesloa. It will bs shown that the 
aatlsfylng (31) oan hs represented as a b\m of the 
dlsplactsssnt due to the pressure gradient produced by the 
Image, and the masa motion produced by the image. The 
preaaure gradient due to the image is 


^ ?o + 4 (m)"j ^ WhJ) <3T) 

Equating the time rate of change of the normal momentum 
(compare (13) and (14) of Appendix 4) to the buoyant effect 
of this pressure gradient would give a value for the 
coordinate of the bubble in the z-direction, satisfying 


d d»pT s . 4ir 

dt[ 3 ITj 3 




(38) 


'PS 


In combining this with (37) we may eliminate p(a) by the 
differentiated energy equation (5) of the text— i.e», by 
Sq. (19) of Appendix 2 with c set equal to infinity — viz., 

, 2 - 

- - J_ 7 LQ,&i 

dt5 


P(a) - ^ 


d- 




The result is, to order l/hf. 




(39) 


Now as was stated above in the paragraph following Eq. 
(53)*. the velocity field due to the image would* in the 
absence of the bubble itself, cause a velocity of notion 
ds^dt in the s-ooordinate of the water at the position of 


116 


the bubble center, where 


d*. 


— ’ - d. j;!- d- 0(1,) , 

4h2 dt 


n" 


whenosp to or-ler l/h , 


d_r 3 dzv! , ± -h 5,%f 1 


(40) 


Adding (59) and (40) gives 


^ p fe <*p ‘ M* 0 ^ j' 


domparieon of (31) and (41) ahowa that to this accuracy (ap+^v^ 
aatlaflea the same differential equation as Rg '/h • 


(41) 




APPENDIX 6 PRESSURE DISTRIBUTION IN 




The Euler equation for radial flow is 

"Or ^ 

Intetjratinc this from r to Infinity and writing 
S z f V 63 * BO that V s “ ^ .^.. 

^ Jt 9 ' 

we find| assvuning ^ constant 

F a (© ^ 

For non-oompresalve motion outside a bubble of radius a 


v/e have 


so that 


a da 
^ s 7X 


a** da 


r ^ 


Substituting (4) and (5) into (3) gives 


P - P- * •6^ ^+- 2#£(iT- (10 


Setting r <■ a gives 


pCa) -Po«a 

and solving (7) for d a and Inserting in (6) gives finally 

P(r) - P„ s I |p(a) - p_-t ( 1 - 


118 


It is obvious that the maximum value of (8) will 
occur vdien a Is less than the radius at v/hlch ^ Is a 
maxljffiujn, but it la not Immediately obvious that (0) is a 
maximum when a is a minimum, eilnce in this region increas- 
ing a decreases p(a) but increases . Hov/ever, if ^e 
assiane p(a) to follow an adiabatic law pa » constant and 
neglect the tern in equation (1) of Appendix 1, we have 


w = 0 (=mln) . ^ (10, 


If (9) aiid (10) are assumed it can bo verified that (B) 
Increases with deoreaalnc a all the way down to 
whenever Tf> 1, as is always the case for perfect gases. 

If r Is largo enough so that the last term of (5) 
can be neglected we have 


(P - 


k 


^da] 


( 11 ) 


The maximum of (11) will come when t, and t are two times 


for which. 


lA da 


3F 


is a maximum, in the expanding and 
contracting stages respectively. Prom equation (1) of 
Appendix 1 wo have, neglecting Q, 




( 12 ) 


This is a maximum when 


119 


W 


8 P« i 8 P, 

^ — amax ‘ S 


1 • d • f wlxoxx a N 8 a ^timr 
This maxltuum value of (18) Is 


( 13 ) 


2 ‘Va “% p 

5 ajaa-t (2 -8)38 




Insertion of this Into (11) gives 

maxlimaa Impulse . ^'4 V" 


T 


a«Jex 


of pressure 


: 2 




p, 


od 


(14) 


f I 


ISO 


APPMSIX 7 SFFWCT OF THE INERTIA OF THE a AS 

ON THE PR B33URE Plj LSE 

!t seems unlllcely that oscillations of the in 
the bubble, l.a., non-uniformity of the pressure In the 
bubble due to the Inertia of the gas, can cause any 
appreciable wiggles In the pressure pulse emitted during 
the contracted stages. The reasons are as follows: 

(i) Kie pressure gradient at the outer edge of the . / 

gas ahere is 



where (®g is the density of the gas. The 
equation of motion of the gas can apparently 


be satisfied by a velocity distribution 
approximating a uniform contraction or ex- 
pansion ooniblneOL with a pressure distribution 
which is roixghly parabolic in r with the slope 
{!) at the boundary. For such a distribution 
it is easy to calculate the fractional deviation 
of p (a) from the pressure p which the gas 
would have under static conditions at the same 
volume; the result is 


p(a) ~ P 



5F 



At the radius, for example. 


( 2 ) 



d^a 


P.(»l 


(eq. (19), Appendix 2) 






Initial stages of the es^losion. Thus a motion is 

possible for v^ioh gas inertia la negligible, al<» 

thou^ it may not be the motion nhioh actually occurs* 

(ii) We mi^t suppose the gas to oscillate in one of its 

normal modes, with an amplitude that increases as 

the radius gets smaller* Now the lowest normal 

29 

mode of a sphere of gas has the period (for small 

amplitudes) l*40a/o * where is the velocity 

g S 

of aovind in the gas. Hear the minimum radius 
the gas is hot, so that c is high, and the 

o 

period of the gas oscillation is rather small even 

on the scale of the a vs. t curve at this stage. 

We may, therefore, expect the efficiency of the 
water in exciting suoh an oscillation to be small* 

(ill) If the amplitude of such a normal mods of vibration 
is to increase greatly as the bubble contracts, the 
normal mode cannot lose much of its energy to the 
water in a period. But if little energy 1s lost in 
this wa^ the effect of the gas oscillations on the 
sound leceived at a dlstfaice in the water Will be 




122 


30 

APPENDIX 8 RgaUMS OF RMiSAUBR*3 SXPSRIMSINT 3 , | 

/ 

In the experlBenta of Raneauar aentloned In the intro- 
»auot.lon, the ohjeotlve was to obtain measurements of the radius-time 
curve for the first expansion of the bubble. Rameauerr worked with 
chaises of one or two klograme of guncotton at depths up to 9 
meters, at a place where the total depth of water was 12 meters. 

The charge was suspended near a aubaerged iron framework i^hloh 
carried a number of rigid electi'odes. Electrolytic currents 
flowed to each of these electrodes from a single eleotrode 
some dlstanod away. When the expanding gas bubble from the explo- 
sion reached any particular sleotrods, that electrode became iso- 
lated from the water and the current in the circuit to which it 
was connected ceased. Measurement of the time of cessation of 
the current in the circuit of each eleotrode thus gave a point 
on the radius- time curve of the bubble. Only the expanding 
phase of the motion was studied, and the difficulty of supporting 
electrodes rigidly close to the explosion prevented measurements 
from being taken at less than about a quarter of the maximum 
radius. 

Ramsauer found the variation of anjjjj^ with depth and 

charge of explosive to agree nicely with the prediction W r 

i#SU®max simple theory, with W - 0.41Q. This Is In 

fair agreement with values of W/Q found in recent work for ' f 

other high explosives. He also attempted to check the energy 

equation (5) by using his measured values of da and some calculated 

dt 

30 

■^C. Ramsauer, ton. d. Physik 72, 265 (1923) 




123 

values of the gas energy G(a); however, hi a meaeurements of 

velocity are not very accurate, and the assumptions from which 

he obtsinsd the equation of state of the gas may he oouclder- 

ahly in error. Plotting the left side of (5) as a function 

of a ha drew the conclusion that about four»flfths of the 

quantity (Q=?) represented acoustical energy radiated in the 

stages a^ag^ while one»fifth zwpresented dissipation due 

to viscosity and thermal conduction in the later stages of 

the expansion. Very little weight can he given to this, 

however, because of the uncertainty in both the kinetic and 

potential energies. Bemsauer also concluded that the tempera- 

0 

ture of the gas at the end of the expansion was 160 0., so 
that none of the water formed by the explosion could condense. 
This conclusion of course rests on the validity of Ramsauer's 
assumptions rsv^ardlng the equation of state of the explosion 
products. 

Ramsauer noticed that at maximum size the bubble ex' 
tended about 10^ farther, measured from the Initial center of 
the explosive, in the horizontal direction than in the down- 
ward direction. This rising of the gas bubble is to be 
expected, of course, and is in approximate quantitative 
agreement with the theory of Section 5 and appendix 4, another 
incidental effect was observed which- if real Is a little puszling; 
%ax ^ slightly decreased by placing an equal volume 

of air in the tube which contained the guncotton. 




shook wave 

















-e o 


-e O 


SURFACE 


BUBBLE' 


BOTTOM 


-e O 


-e O 

Parr, of the Infinite series of Images required for 
the calculation of the flow about a bubble placed 
midway between a free surface and a rigid bottom. 


VERTICAL MOTION OF A SPHERICAL BUBBLE AND THE 
PRESSURE SURROUNDING IT 


G. I. Taylor 
Cambridge University 


British Contribution 


August 1942 





m 




i 

n 







I 131 


VERTICAL MOTION OF A SPHERICAL BUBBLE AND THE 
PRESSURE SURROUND INC IT 

0« I. Tasylaf 

Awgmt 1942 

Summar y, 

T.ha upward motion of the spherical hollow which Is formed In water by an exptosion is 
dletusseu on the assumption that the spherical fom is preserved. Ourinp the expanding stage 
the buoyancy of the hollow gives a large amount of vertical momentum to the surrounding water. 
During the contracting stage this momentum Is concentrated round a rapidly diminishing volune. 
An Intense concentration of vertical momentum is thus prcducad in the neighbourhood of the 
vertical line through the charge. The pressure at points shove ths charge and near this litte 
rises to considerable values owing to two causes:- 

(a) When the hollow Is near Ita greatest contraction at the end of its first oscillation 
Its centre is nearar to points vertically abcMO the seat of the explosion than It was 
at the time of the explosion (but further from points on the level of the explosion). 

(b) The pressure due to the combination of vertical motion and nxpanston on the second 
expansion Is In certain circumstances Urge. 

For comparison with observation, the maximum displacements to be expected In a 
rectangular plate 6 feat x u foot x o.i73 inchos attacked by S.65 1b, of T,h,T. placed 
(a) Ik feet below the plate; (b) is feet away horizontally are calculated. They are found to 
be k ,3 and 1.5 Inches respectively. The observeu displacements were *.35 and 1.15 inches. 

Though complete dynamical similarity is not possible when charges of varying, sizes are 
exploded under water, yet in certain circumstances dynamical similarity should be very nearly 
attained. it is shown, hwiever, that there are large ranges of depth end charge weight which 
could not be explored even approximately by model experiments unless the experimental apparatus 
were so constructed that the atmospherical pressure above the water surface could be reduced. 


The radial motion of the water near the seat of an explosion has been studied by many 
writers, Recently Conyers Harrlngx has analysed the small change In shape which the expanding 
spherical hollow suffers owing to the varying hydrostatic pressure in the water due to gravity. 

He showed that during the period when this effect Is ssbII It merely displaces the sphere vertically 
without changing Its shape. This vortical motion may be regarded as being due to the vertical 
momentum given to the fluid surrounding the spherical hollow by the floating power of the hollow. 

The momentum asscclated with a spherical hollow of radius a moving with velocity — 

{jir pa?] H , z being the depth of Its centre and p the density of water. The rate at which 
upward morontum Is cotmiunlcated by the fioating power of the hollow Is ^npa^g, so that 


a^g 


dz 


d 


(a^ SI 


i* I 


a^dt 


(t) 


Starting with a very small at t 0, the time of the explosion, it will be seen that during 
the early stages while a Is Increasing rapidly - dz/dt is less than 2gt, !.e. the hollow naves 
upwards more slowly than It would if a were constant, on the other hand, during the period while 
the- hollow Is large /a’dt acquires a considerable itognltude so that when the bubble contracts. 


I 


Af\0 


132 


- 2 - 


and a b«con»e smsll. f a^dt becomss iruch larger than a^t so that - <Sz/dt becomes large compared 
with 2gt. Thus the centre of the bubble will be displaced upwards at a rate which rapidly 
increases as a diminishes, in fact the analysis by wnlch Conyers Herring ahcwod that the hollos 
Is displaced upwards rapidly ceases to apply. Herring describes this rapid increase In 
displacement as an 'Illustration of the general instability of the contracting stage*. 

Though Instability vsould probably prevent tho hollow from remaining approximately spherical 
during the contracting stage, yet photographs of the b;£ible from the explosion of a detonator by 
Edgerton^ have given measurable radii over rather mors than 2 complete pulsations, so that the 
hollow appears to remain sufficiently coherent for this time to pulsate as though It were spherical. 

This experimental result encourages one to pursue the subject further and to calculate 
what the motion would be if the bubble were constrained to be spherical by some kind of constraint 
which absorbs no energy and no Inertia. (i) is still applicable. The other equation which must 
be satisfied Is that which ensures constancy of energy. The kinetic energy of the water 
surrounding a sphere of radius a is 

[if J f [ifj , 

(no terms containing the product |f , || appear) 
so that the equation of energy Is 

+ 2 77pa^[||j + jff/da^ 


where is the pressure at the level z so that z is measured from a point 33 feet above the sea 
surface, W is the total energy associated with the motion and G(a) Is the work which the gas would 
do on the walls of the bubble If exparided adlabatically from radius a to Infinity. 

Seduction to non-dimensional form . 

For convenience of calculation (2) may be reduced to non-dimensional form by means of the 
length L which is defined Dy 


L * < ^ (3) 

Setting a = a'L, z = a'L t = t' /"I (4) 

a', z' and t* are non-dimensional and (z) and (l) can be written 



It win be seen that if G(a) could be neglected compared with W- (5) would be non-dimensional 
so that tho complete range of solutions of (5) and (d) would only Involve the one variable parareter 
which Is Introduced curing Integration, namely z'^. the value of z* at the level of the explosion. 
For this mason it is necessary to estimate the value of G(a)Al. 


Vg^ue of G(ai It) , 

The ratio G(a)/w can be calculated from the adiabatic relationship for the gaseous products 
of the explosion. Using the adiabatic relationship calculated dy H. Jones and Miller for T.k,T, 

It is found that below a pressure of about 300 atmospheres the gases expand according to the law 

pv^'^® « constant • (l,725 x 10*) (835.3)^’*® • 7.830 x 10* (7) 


where 


- 3 - 


133 







, 1 

h' 

N 

2 



Where V Is the vglyme i grawfte of the products of combustion* 
By definition 


a (a) 


"f 

/V 

05 


pdv 


^ ‘ con»tant) 


kI 2i 


83 X 10’ 

.28 


- 0.25 


wh«re M Is the mass of explosive, 
8 


SInee Mv » j ?r a* (8) may 8e written In the form 


s(») 


2.130 X tO^° M 


( 8 ) 


(9> 


The measurements of the period of the first vloratlon of bubbles formed by T.n.T. oxplosionsv 
show that W ■ o.sg approximately, where 0 Is the total energy released, which for T.M.T, tmy be taken 
as 880 cal. par gam. Thus 


W « M( 0.5 X 880 X U .2 X lo’) » 1.85 X tO% 


Thus 


U183 

W 


L J 


( 10 ) 

(U) 


The corresponding formula for p Is p =■ 1.308 x 10* ^ j dynes/sq.cm, (u) 

-G(a)fW may be expressed In terms of p only; tnus eliminating M/a^ from (tl) and (12) 

■\l /5 


Slal = 1.1S3 

w 


1.308 X to 


0.0177 p^^* where p U expressed In dynes/sq.cm. 


or 2 jIi1 s 0.251 where p Is expressed In atmospheres. (13) 

W 

The exponent 1/5 Is merely the value of wheny • 1.25, 


Conditions under vhich G(a) / V may be neelected . 


It is clear from (l3l that it is only when p Is considerably 1es.s than atnuspherio pressure 
that 8(a) /w could be neglected. When the bubble Is formed at moderate depths the pressure Is less 
than atmospheric pressure over a great part of the period of oscillation. It Is during this period 
that the greater part of the upward momentum associated with the rise of the bubble Is acquired. 

It Is useful, therefore, to solve (5) neglecting S(a)/W when one is concerned with the vertical 
motion. it Is necessary, however, to remember the limitations involved in making this slmpllrication. 
The results will certainly be very Inaccurate near the point of minimum radius, but they serve to 
bring out certain points which are Important In considering whether it is possible to make model 
experiments to explore thu motion of the bubble. 


C aleulation neglecting QIaf/V , 


For purposes of calculation (5) may conveniently be written in the form 



where is tha value of z' at tne depth of the explosion. This equation, together with (5), 
can be integrated step-by-step for any given value of z’^. In the early states whan the bubble 


(s ..... 


'V-'. 


1^ 


I 

1 

I 


i 




Is small ths first term In (i») Is much graatar than th« others so that (W) »ny b« Integrated 
approx Iniitel/ jiving 


t' • I /7w . 1.0025 a’®^^ 

Sjhstltutlng for a* (5) becomes 

_ a*: . * f df . H t« 

df J 11 


which may be Integrated to 


z'o- *• 


Using (W) and (17) up to 1* ■ a 05 and auoseqasfttly solving (w) end («), step-bystep, 
the results tor z’^ * 2.0 are shown In figure l. The radius of the bubble, tine height of Its 
centre and the level of Its highest point are shown. 

it will be seen that the bubble at first rises slowly but that Its centre jumps rapidly 
through a height about equal to twice Its maxlrnum radius during the short period while its radius 
is only half the maximum radius. 

The minimum radius Is given by a* ° 0.211, This seems strange, because If the vertical 
motion Is neglected the approxlmatloh in wnich g{ 3 )/w Is also neglected Implies that the bubble 
collapses completely Into a point; there Is, In fact, no gas pressure to prevent this collapse. 
When the vertical motion Is taken Into account the collapse is (theorotleally) prevented by the 
fact that at a certain minimum radius the whole energy Is concentrated In the flow due to the 
rapid vertical motion of the spherical hollow. 


Similarity on varyim scalsi. 


Since figure l Is non-dimensional It Is possiole to assign to it any des'red linear scale. 
The scale Is, In fact, determlnad only by W and for any given explosive this appears to be 
proportional to M. Thus for T.K.T. the unit length Is, from (» and (10), 


I. go J 


When M Is expressed In gm. of explosive and L Is in cm. 


The depth below a point 3? feet above the sea Is In this case 21. 

a single series of explosions In which f of water ♦ J g feet 1 j, f 

1 (charge dlemeterj* f 


,t 33 feet ai 
In which ^ ! 


Thus Figure l represents 


Is fixed. 


A set of scales af depth and the correspondlng positions of the see surface in relation 
to the explosive are shown at the side of Figure 1. Scales are given for M w 2200 lh„ 300 In,, 

10 lb., and 1 oz. it will be seen that the level of the son surface Is well clear of the bubble 

through the greater part of the motion for N = 2200 1b. at 103 feet and 300 lb, at 50 feet depth 
and that these two explosions might therefore be expected to give similar pressure distributions. 

The 30 lb. charge at ID feet would give a similar bubble during the greater part of Its expansion, 
but near its greatest expansion It would be above the surface and similarity would break down. 

It is evidently Impossible to place a Charge of less than 30 1b. In water sb as to give a regime 
which Is similar to that produced by the two larger charges. On the other hand, if It were 
possible to reduce the atmospheric pressure to less than 33 feet of water, small-scale models could 
be nsde; moreover by using the correct pressure a model experiment could be carried out which would 
represent either of the large explosions. Thus If a 1 oz. charge were exploded at a depth of 
10.0 - 2.U e 7.6 feet below the surface of water and the air pressure above the water surface were 
reduced to that of 2.4 feet of water, the resulting bubble would (in the present approximation) be 


siml lar 


■-1 


3{stP«r ts that of 2200 1b. exploding at 103 feet. Similarly a 1 oz, charge exploding at a depth 
of 10.0 - 4..0 > 6.0 feat below the surface would be similar to a 3oo ib< charge at 50 feet if the 
atmospheric pressure above the 1 oz. charge were reduced to d.o feet of water. 

Co nditior.s juften the bubble is near its jioint c/ ersatsst contractieti . 

The approximation In which 0(a) Is neglected In comparison with w does not give useful 
results near the point of mintmum radius. To Investigate phenciicjiia which depend on conditions 
near this point It ts necessary to use (5) Instead of (id). The calculation must then refer to 
one charge and one depth only, though It Is still convenient to use the non-dimensional form (5). 

As an example a charge of d.66 1b. exploded at depth of 20 feet Is chosen, because In an experiment 
made under these conditions effects were produced whicn might be attributed to the effect of gravity. 
In this cate the value of I Is dus cm. or ld.6 feat, and z'^ • and t' ■ l.dTt 

when t is expressed in secends. 

For csovonlonco in cilculotlon (il) Rtoy !n of 3* and L by tho fcrnulo 


2^ • 0.0177 L®*^® 


and when L « dd5 cm., this gives 


Figure 2 snows the radius and depth of tne centre at time t after the explosion. In the 
experiment above a box containing an alr-oacked plate was (laced at a depth of 6 feet. Id feet 
above the charge. This Is shown In Figure 2, 

Effect of vertical motion on maximum pressure . 

Whan the effect of eompresslbltlty and of vertical motion Is neglected the maximum pressure 
In the bubble occurs when 0(a) ■ W and from (ll) this will be found to correspond with 

4 “ (1.183)“* » 0.81 and 
a' 

p » 1.308 X 10® (0,51)^**® = 5.63 X 10® « 563 atmos. (2l) 

When M > d.66 1b. > 2100 gm, 

a^ » tAtt • dl30 so that a • 16.0 cm. 


and a* “ 


nr • ®-® 3 ® 


It win be seen that the effect of the vertical motion is to prevent the high pressure 
associated with this very small radius from being formed. Thus In Figure 2 the minlmun radius 
win be seen to be a' • 0,120, which is 3.3 times as great as that which would occur in the absence 
of gravity. The pressure corresponding with a = (O.lzo) (dd5) = 53. » cm. is only 5.0 atmos,, so 
that this effect on gravity is to reduce the maximum pressure in this case In the ratio 100:1. 

The amount of energy radiated In tne form of a compressibility wave during the period of 
greatest compress io'i depends on the p,^, the greatest pressure. Conyers Herring and will Is have 
developed formulae giving for the proportion of w which Is radiated the formula 


F *■ Fraction radiated 




Where A depends only ony and c Is the velocity of sound In water. For -y « 1.25, A » l.sr, so 
that {z$) would (|lve as the fraction radiated 


t>67/ 5,63 X 10** 
l.W X 10® 


= 0.31 


136 

If the effect of gravity Is neglected, 


^ 6 - 


or F « • 0.03 (25) 

i.w * 10® 

according to the calculations Illustrated In Figure 2. 

If the same charge had been exploded at a greater depth the bubble would not have expanded 
to so great a radius: it would therefore not have risen so much and F would have been greater, 

it would be Interesting to repeat the calculation for the sane charge for a range of depths, I.e. of 
values of 2 '^. 

Pressurt distribution . 

The pressure in an Incompressible fluid due to the motion of an expanding or contracting 
sphere when the centre naves with velocity u In a straight line Is obtained from Bernoulli I’s 
equation. The velocity potential tp >s 

4> * SJi ♦ i cos S 
r 2 r* 

where 6 Is the angle between the radius vector r and the direction of motion, i written for da/dt, 
referred to axes which move with velocity u so that the pressure la given by 

£ - gz ■ - U - A (u* + V^) (26) 

p 3 t 3x2 

where u and v are the velocity components referred to fixed axes and 3qb/3 t Is the tine rate of 
variation of at a point which Is fixed rolative to the centre of the sphere and therefore moves 
with velocity U. 3<^/3 x is the apace rate of change of In the direction of U. The velocity 
components u and v are radial and tangential, so that 

u--|^=4Sf cos 61 (27) 

3 r r‘ 'T 

■ ‘r H ‘ i ? 

and -2^ • uCos5-vs!n6 • S=5 cos (cos* 5 - i sin* 0) 

3 X r'F ‘ 

The Complete expression for pressure is therefore 


JB - gz • i 2^ (aO f 54u) cos 0 ♦ 4 (cos* 0 - 4 sin* 0) 

p r 2 r' r' * 

- ^ B* + ^ 4U COS0 ♦ A ^ U* (cos*0 ♦ i sln*0)j (29) 

This may be expressed In term* of the non-dimenslonal variables defined In (3) and (4), 


-£- _ z> ■ 

901. 

2a '4'* 

1 

. -I?V. 

w.t 


(a’U' 

■ + sa’u’) 

cos 0 •* 

2 - (cos 0 - I sin* $) 

- 

[iIf 



4'U' 

' cos 0 + 

A sJ 
2 [? 

■j U'* (sos* 0 ♦ sin* 0) 








(30) 

where U' • 

IV 

. _iL. , 

a- Is 22* , 

and 

li- Is AH' 

or - 1! 

^ • 

dV 

/~9<- 

df 


df 

d1 



(30) is now expressed In a form suitable for computation. 


Pressure 


- 7 - 


137 


^ j Pf^ssure at fixed j>oint vertically above an explosion. 

In sow experiments esrried out recently, In which e rectangular alr-tached steel plate 
0.173 Inches thick atil 6 feet x ^ feet sides was used. It was found that a charge of >i.d9 tb, 
fired at a distance of lU feet vertically below the plate [which was then horizontal) produced a 
dlsplaceinent whose fnaxlmum value was w.33 inches. A sleiiiar plate held vertically was ulshed so 
that the maximum dlaplacement was only i.is Inches when the charge was fired at ths earns distance 
(14 feet) but In the same horizontal piano as the plate. This very large difference cannot be 
accounted for by considerations based on the pressure waves which have been measured by piezo- 
electric gauges, for no great differences have been detocted In the pressure waves at points over a 
sphere centred on the explosive. 

It wss to account for these expertmantal results that the preaent Investigation was made 
and ths values of charge weight and depth used were taken to correspond with those of the experiment. 
The expression (30) for calculating pressure contains soma quantities which were not needed in 
calculating the radius and vertical velocity of the bubble. The accelerations 'hT and U* have to 
( ’> be calculated. This was dona graphically, plotting ths values of &* and u' against t'. The 

Initial (non-dimensional) height of the plate above the chergo was 0.97 (Ue. 14 f«et/l). at each 
value of t' values of a', 6*, U‘, O' ano r‘ (• 0.97 - a‘) were tabulated for a range of values near 

the time when the bubble was reaching Its mlnlrtum diameter (I.e. at t* > 0.408), 

Setting these In the expression (30) the following values were ealsulated:- 

TSBLE 1. 


V 

0.380 

0.400 

0.405 

0.407 to 
0.409 
(mesn 
value) 

0.411 

0.U16 

0.426 

0.432 

OeU40 

t (seconds) 

0.268 

0.272 

0.278 

1 

0.280 

8,283 

0.290 

0.293 

0.299 

J(plate) 
(atfflQspherea) 
U«e« 10^ dyna$/ 
sq.Cffl.) 


1.8 

n 

12 

10*5 

■ 

6.2 

1 

■ 




These values are shown graphically in figure 3. 


Comparison uith pressure during the early stages of the expansion of 
the bubble. 


The maximum pressure In the pressure wave Is known from piezo gauge measurements. An 
amplrleal expression representing approximately by results of tneso pressure measurements Is 


where 


P - 

Pg » 4.6 X lo’ 
n ■ 7.5 X 10* 


hi) 


pg being expressed In dynes/sq.cm., M mss in gra. of T.n.T., r distance In cm. from the charge. 

Por a charge of 4.68 lb. n • 6,66 x 10^ seconds and p^ - 138.5 x 10* • 130 almos. at a distance 
r ■ 14 feet • 426 cm. This pressure is very much larger than the maximum pressure calculated for 
the later pressure rise Illustrated In Figure 3, but Its' duration Is so small that It Is difficult 
to show It in a diagram of the scale of Figure 3. The attempt is made, however, on the left hand 
side of the figure. The pressure falls to 10 atmos. In time given by ^-5.8 x lO^t , 10 _ 

l.e,^ In time 0.5 mllllseconde. If this exponential fall In pressure were continued It would fall 
to ^ of an atmosphere In a iittle over a millisecond. In fact this does not happen, the pressure 


falls 


























— Jw I < 1 ^ <V4:w :• Ji',~. ,.^ 


falsa rapidly tin It ISo say ^tft or ^th of Its msMlnwi*' vatuo and than fal’s comparatively vary 
slowly. This gradual daeay of pressure Is shosim In moai. plojo gauge records and was particularly 
ootlceaOla In Hllllsr’s early measurements. It must evidently bo associated with the notion of the 
water round the supandlng btisble and can be treated as associated with radial flow of an Incompressible 
liquid. 


In the early stages when the bubble has not risen appraclobly under the action of gravity, 
the formulae derived by neglecting U, fl and the variation In z In (5) may be used. If a(a)/w Is 
neglected, l.o. If It Is assumed that the whole kinetic energy la given to the bubble at the Instant 
of the explosion and tha work done by the subsequent expansion of the gases Is neglected, the equations 
are then tix>s« used by Raleigh In discussing the collapse of a spherical cavity and by Ramsauer and 
others. In the early stages of such an expansion, l.o. In a time attar the explosion which Is small 
compared with the period of the first exnnslon and contraction of the bubble, the te rm 2/ 3 z’ In (5) 
may be neglected compared with (de*/dt*) . (4) then assumes the form a*^ ^ i ■ 2 it. Inserting 

this value In (JO) the leading term (l.e. the term which Is greater than all tne others except close 
to the bubble) Is 


_E_ _ z‘ 

gpL 


i X .. I Jl[^ 

f' dt’ r* dt' <vTitJ 


r* 


■ . .a 

4 tr r’ai*" 


(J2l 


In terms of t’ this becomes (pee (iS)) 


gpi 


z* • 


1 ( 2/rsi 1 

iw r‘ [ B j 


, 0.0798 f-*'® r*-' 


(J3) 


Restoring the dimensions (JJ) becomes 


P - P, 


0.0798 9^^’ p 


(JH) 


Where p^ Is the pressure at the depth of the explosion. 

Putting g ' 9S1, 0 ^'^® » 62. J7, p« t. (J4) becomes 

p - p^ « 4.98 l“^® r**'® r*^ (J«) 

in the case corresponding with M» 4.66 lb. T.x.T. where L » 44J cm., L^*^® • 2.265 x id® so that 
(JB) becomes 

p - p^ ■ i.iji X t~*^® r“® (36) 

at r • 14 feet * 426 cm. thio is p - p^ * 2*655 X 10® t”®^®. At time t “ 0.006 seconds t”®^® • 
69.6 SC that p - Pj *■ 1.86 X 10® > 1,85 atfflos.; at time t • o.OlO, p - p^ = 1.07 atmos. 

A better approximation Is found by extending the solution of (6) back atep-by-step towards 
the tiine when the bubble is small, using the expression (ll) for taking account of the gas pressure. 
The pressure Is then given by (jO) but it Is found that the vertical motion may be neglected so that 
the simplified expression 


£_gy - is ft L t -af g - i all|! (37) 

p r 2 r* 


may be used. When this calculation is carried out It Is found that the minimum radius a^^ Is attained 
at a time which Is actually oarller than the origin of time used In the step-by-step calculations 
already described. This Is because In starting the calculations the conditions at the beginning of 
the explosion were not required, all that was required was a knowledge of the energy w. The way 
In which this energy was communicated to the water to give It radial motion was limatarlnl ao far 
as the motion during the first contraction was concerned. When, however, It Is desired to calculate 
the pressure round the bubble In Its early stages the exact Instant of explosion has to be determined, 
because otherwise It Is not possible to get any idea of the way In which the shock wave pulse is 
related to the subsequent pressure distribution due to radial expansion. This question la a difficult 
one In any case. If the water were truly Incompressible the pressure would Jump to Its maximum value 
Instantaneously at the nvjment of explnslan. Actually the compressibility delays the first rise In 


pressure 



- 0 - 


139 










pressure at distance r fr&Tc the euptoslon till a tl.ra i^hlch Is spproxiRctely r/c seconds aTter the 
explosion, c being the velocity of sound In mater. TUI sew twtter method Is discovered a fairly 
good approximation can bo made by Imagining that at tima r/c after the explosion tlte pressure pulse 
arrives at radius r and that the praasure Jumps to p^ and then falls according to the atplrleal lam 
(31) till It reaches the pressure which would exist at that time and radius If the water had been 
Incompressible. The subsequent pressure Is then calculated by the methods described above. 

This proposed method la shown In Figure 3, where the pressure pulse and the subsequent slow 
fall due to the radial flow are both shown on the correct scale for a radius r ■ i<t feet from the 
charge. It will be neen that tho two curves cross. It Is assumed that the actual pressure 
distribution is simply determined by whichever Is the greater of the two. This method, though 
necessarily inaccurate near the time when the two curves cut. Is possibly not far from the truth 
over the min part of the rangu. <tie virtual parts of ths curves where the shock wave pressure 

Is small compared with that due to the radial flow and vice versa are shown dotted In Figure 3. 

The results of using the stoth-by-step process and calculating pressure from (30) are shown 
In Figure 3. This calculation must be regarded as provisional and liable to modification when 
better methods have bean developed. It Is included here In order to show the order of magnitude 
gf the pressures to be expected at 14 feet from the 4.S8 lo. charge after the pressure pulse has 
passed. 

Efftct of calculated pressure on plate rigidly held at its edges. 


Pressure wave . 

The effect of tna pressure wave of the form p • e“"* on a steel plate which Is free or 

elastically supported has been discussed In my note *The pressure and Iiripulse of submarine explosion 
waves on plates". It depends on a non-dtmenslonal number £ > /Pc/mn, where m Is tne mass of the 
plate per sq.cm., c the velocity of sound in water, p the dnnsity of water. For plates of thickness 
0.173 Inches m “ ,173 x J.34 x 7,8 « 3.«3. For a charge of 4.65 1b, T,M,T. n • 5.86 x 10“^ and 
c • 1.44 X 10® cm./seeond so that € = 7.2. 


In tne above mentioned report the case where water will not sustain tension Is worked out. 
It Is found that with the above values of £ and n the plate will part from the water after time 


t • — - In e • S.5 X ICT® seconds 

n(e - l) 

and that Ks valocUy will then Oe 

i . !?fi £ . 2(1.365 X 10°) 

mn (3.43) (6.86 x 10"^ 


(38) 


1.3S X lO? cm. /second 

(39) 


The kinetic energy per c.c. of the plate Is then 

?^steel^* ' ^ (7.83H1.38)* X 10* - 7.45 x 10* ergs/c.c, (uo) 


If the water can sustain some tension at the surface of the plate this energy would be 
reduced. If the water could sustain Internal tension though not at the surface of the plate, no 
further energy would be conmunlcated to the plate. If, however, the water could sustain no 
Internal tension, drops would bo projected from the water on to the plate and further energy might 
be conmunicatod to It. 


Dishing of 

The time during which the high pressure (Initially 2 x 133.5 = 277 atmos.l is acting on the 
plate Is short compared with the time taken by tho plate to come to rest In its displaced position 
under tlie action of plastic and elastic stresses. For this reason tne oynatiiics of tne plate can be 
treated separately from that of the water on the assumption that a velocity of 1.38 x 10^ cm./second 
Is instantaneously given to tho plate. In an analysis In which I hope to puhl Ish shortly the dishing 

of a ..... 



140 


- iO - 


of a rectangular plastic plate under uniform pressure, p, acting on one side, Is studied. Though 
the distrlDuilon of the plastic strain In the plane of the plate Is not uniton'j the plate fcohaves, 
so far as displacement normal to the surface Is concerned, like a soap film or ifiSir^rariO with uniform 
tension. This uniform tension Is related to tlie yield stress measured In a tensile teat by a 
factor which differs according to whether u. Mlsos* or Mohr’s theory or plasticity la used and 
varies slightly with the ratio of length to breadth of the plate, but with a plats for which length/ 
breadth ■ 3/2 v, Hisss' theory gives a result which Is t.4 per cent too small and Mohr's a.9 per 
cent too big as compared with a rough theory which simply assumes that the plats behaves like a 
meidbrane stretched to uniform tension fd In all dtrecttons, whore P Is the yield point of the steel 
and d the thickness of the plate. 

using this theory for a plate wh^e dlmanslons are 2b x 4/3b (b Is 3 feet In the particular 
plate to which we are new applylr^ the analysis) I find the following formulaet- 


Maximum displacement 

h • 

0.179 

{ttij 

Mean displacement 

5 >■ 

2 

O.0B70 ^ 

(42) 

so that 

y/h • 

0.4S5 

(43) 


The energy absorbed by the plate during its displacement Is 
lip - L 3.76 P(y)^ d/b^] (area of plata) 
so that work done on unit volume of the material of the plate Is 

Wp » 5.76 y’ p/b* ( 44 ) 

Timf, taken for displacement to reach maximum ^ 

The vibrations of a membrane stretched with tension Pd on a rectangular frame of dimensions 
2b^ and 2bj have been studied by Payleigh who showed that the period of the fundamental Is 



where ps is the density of the membrane. A plastic sheet Is like a mefl*rane during the time when 
it roaches Its naxlmum displacement it only recovers by the amount of elastic recovery which Is In 
the ease here considered small compared with the plastic displacement. Thus a possible novemant 
of a plastic sheet Is identical with the first •^period of the analogous elastic membrane. Thus 
It the plastic sheet Is given initially the normal velocity distribution corresponding with the 
fundamental period It will come to rest In tine 



(«) 


In the case of a plate 6 feet x * feet and density 7.8 this time of dlsplacament r Is 


T • 2.07 X 10"'' seconds if P Is 30 tons/square inch 
or T • 2.S4 X 10“^ seconds If P Is 20 tons/square Inch 


(* 6 ) 


This time r Is marked on Figure 3 so that the relationship between It and the durations of 
the pressures rapy be understood. It will be seen that t Is very long compared with the duration of 
the pressure wave but that tne duration of the long continued pressure associated with the expansion 
and vertical motion of the bubble is longer than r. 


This Is convenient for It necessarily Implies that It Is justifiable to use the inethod of 
the report "The pressure and Impulse of submarine explosion waves on plates" in discussing the 


displacement 




- ll - 


141 


illsplaceniant due to the pressure wave. The pressures Out to the nuccecdlng bubble expansion can, 
so far as their effect on the plate Is concerned, probably be considered as though applied statically, 
though this Is not quite certain because tho pressure due to the first contraction of the bubble 
does rise In tiine less than r. 


Comparison of calculated and oliserved di si>lacements. 


The average energy absorbed by the ploto par unit volume must be equal to the average Kinetic 
energy per unit volume given to It by the pressure wave. From (US), (S3) and (»o) therefore, 

5.76 (0.495)* h*P/b* - 7.45 x 10* crgs/c.c. 


Using P ” 70 tnns/square inch • 3.09 x 10' Oynes/sq.cm. and b =■ 3 foot • 91.4 cm., 
h* = 14.6 sq.cm, or h =■ 3.8 cm. ■= 1.5 m. 


The observed value for the plate In the experiment was 1.15 Inches 
and for a similar plate Out with r.d.x./T.n.t. as explosive It 

was 1.41 inches 


find 

(47) 

(« 8 ) 


In the case of these two expurlments the plate was set vertically with the explosive at 
the same level. The bubble might oe expected to break surface before again contracting In both 

cases so that the subsequent pressure duo to the contraction and second expansion would not be 
expected to appear In any case, but even if the plate and explosive were at such a depth that it 
would occur, it would not produce an effect comparable- with that which occurs between t > 0.275 
seconds and t =• 0.30 seconds when the plate Is placed horizontally 14 feet above the explosive'. 


The effect of a static pressure applied to the alr-backed plate is according to (4l) to 
produce a maximum displacement 


h = 0.179 = 0.179(36 X 2.S4)*p_ 

Pd 20(1.54 X 10“)(0.173 X 2.54) 


1.104 X 10"* p 


cm. 


where P is taken as 20 tons/square Inch and d ■= 0.173 Inches. Thus the plate would be dished 
1.104 cm. per atmos. of applied pressure. 


Since the plate has already been dished to 3.8 cm. by the pressure wave. It will be seen 
from Figure 3 that the pressure due to the kinetic wave which follows immediately after the pressure 
wave and which has a fraximura pressure of about 2 atmos. cannot increase the dishing and U therefore 
ineffective in doing further damage to the plate. 

Plates, which were placed In a horizontal position 14 feet above the charge, wore, according 
to the present analysis', subjected to a long continued pressure which was Ciilculated to begin at 
about t = 0,275 seconds, rise rapidly to a sharp peak, drop to about 11 atmos. and then fall off 
gradually till at about t = 0,30 or O.31 seconds it is again only one or two atmospheres. 

The conditions determining the thickness and intensity of the pulse at t “ 0.2 78 seconds 
(when the bubble reaches its minimum value) are not likely to be In fact as they are described in 
the theory, because the bubble in collapsing will probably be far from spherical. It may well be, 
howevur, that the large and loiig-continuco pressure which occurs between t = 0.278 and t = O.30 
seconds, l.s. during the second expansion, will be produced in the actual explosion bece.use there 
is a strong tendency for the bubble to become spherical while expanding*. It will be seen In 
Figure 3 that a pressure of 9,4 atmos. is maintained for a duration of t ■ 2.54 milliseconds. 

Tnus the maximum displacement of the plate is likely to be at least equal to 

h » (1.104 X 10.0) a 11 cm, a 4.3 Inches 


The observed maximum displacement was 
h a 4.35 Inches 


This close agreement Is almost certainly purely accidental 


Ths obs«rv»d iwxlmum deflect loo with D.P.X./T.N.T. was S.U4 Inches, but the plate was at 
13 feet Instead of W foet and fi.O.X./T.h.T. Is a more powerful explosive than T.n.T. alone. 

Rtferences. 

w U.S. Report No. C4 - sr 20-010, “The theory of the pulsations of tlie 2 f« hi'hOle 

produced by. an underwater explosion". 

i U.S. Report 'A photooraphic study of underwater explosions* 

4 Conyers Herring 


Photographs oy Edgerton. U.S. Report, 






















ON THE CHANGING FORM OF A NEARLY SPHERICAL 
SUBMARINE BUBBLE 


W. G. Penney and A. T. Price 
Imperial College of Science and Technology, London 


British Contribution 


October 1942 




145 


ON THE CHANGING FORM OF A NEARLY SPHERICAL 
SUBMARINE BUBB LE 

W. Q« Penney and A. 7. Price 
October 1S42 

* • i» * « * m * 


Introduction, 

Whin an explosive charge j$ uetonatdd unddr H^tar ihe r^auHlng budPlPf aa it grgiifS in aize* 
raplaty approacnas a apnarical form. The degree of itability of thii tpnerleal form it a matter 
of dome Interest, espeeially In view of the fast that the bubPIc may contract again, as both 
photographic and acoustic observations show. In the oT ivimil charges several pulsations may 
occur, and there is evidence of at least one complete oscillation Doing possible with a targe charge. 
On the other hand, for a very deep explosion, where no done or plums appears, what finally reaches 
the surface it not a gas bubble out an emulsion of gas and water; this suggests that the bubble 
has departed so greatly from tho spherical form that It has entirely broken up and disintegrated. 

The mathematical calculation of the changing form of the bubble would present considerable 
difficulty even if there were no uncertain factors In the problem, h start can, however, be made 
by considering the case of small deviations from the spherical form, and Investigating the tendency 
of these 'to incrooSe or decrease as time proceeds. In the calculations which will new be described, 
a first order porturbatlon theory Is developed for a nearly spherical bubble expanding or pulsating 
In an Infinite Incompressible fluid. The radius vector from the centre of the bubble to Its 
surface Is expressed as a constant plus the sum of spherical harmonic components of different orders 
n, the coefficient of each component varying with the time. Thus the departure frem accurate 
sphericity at any moment Is measured by the magnitudes of these harmonic components. 

^ch harmonic compojient of order n Is found to contain a guasl-perlodle time-factor of the 
form Cj e " f„(t) ♦ Cj c " f„(-t), where f„(t) is a periodic function having the same period T 
as that of the pulsation of the oubble, Is a certain (complex) constant, and Cj and Cj are 
arbitrary constants determined by the Initial conditions. The order of magnltixiB of the function 
f„(t) at any momsnt Is roughly proportional to the reciprocal of the radius of the bubble; 
consequently. In the case of large pulsations, I.e. Intense explosions, the non-spherlcIty is 
greatest when the bubble Is small. Also, in virtue of the exponent ial factor (\^ contains 
In genera! both real and Imaginary parts), the noi; sphericity at any stage, say at the minimum 
size, Inereasoi with eacii pulsation. Indicating that the spherical form Is ultimately unstable. 

The harmonics of high order are found to Increase exponentially at first, and then oscillate 
In negnltude, The higher the order n, the greater Is tho Initial magnification. Hence any needle 
shape Imperfections on the charge will become highly magnified as the charge explodes. This may 
bo the explanation of the prickly appearance sometimes observed In photographs of the early stages 
pr 9U0marinv oxpV^slonss 

The conclusions which can be drawn from these calculations are, however, restricted by the 
general condition that the perturbation must be small and It ts elaarly desirable to extend them so 
as to avoid this restriction. 

Basic ass-^attions. 

Taking tho mean centre of the bubble at time t - 0 as origin, and assuming that the 
departure from sphe.'lclty Is small for all times to be considered, we write the radius vector R 
to the surface of the bubble In the form 

* ' ^ ^ i On I (a < e) (i) 

v;hero Is a surface spharical hartnemte of decree n# and £ Is a small quantity of first order*, 
may bo supposud analysed Into zonal and tesseral hnmcnlcs in the form 



- i ~ 


^ ^r, n ’’r, ^ * 'J'm -.' 


but it should be noted that thy ilrw factor depends only on the dapres n of the harRioftlc» and 
not on ffi* 

Th« valuss of th« bubftU is the lnte«rai of f^l) over a unit sphere, which to the first 
order is apual to 4 ? e?i), in virtue of the otihojonai properties of the hamonics 5^. 

We assume that the bubble is filled with an almost massless perfectly elastic gas; pressure 
waves in this gas are neglected, so that the pressure p at any Instant Is uniform throughout the 
bubble* Assuming the usual adiabatic relation between pressure and volume, we have 

p 8^ • P a„^ tJ) 

where P Is the pressure and the mean radius at time t • d. 

The notion of the surrounding fluid, since It It generated from rest by pressure, will possess 
a velocity potential g6, say* Assuming the fluid Is tnccmpresslble, this potential aatisflei Laplace's 
equation and Is therefore (Mpreaslble in the fom 


^ - r-‘ A(t) + X r"^^ B.(0 S 10, (a) 

ffl n n 

We Shall assume that the surface hamonica In M are Identical with the correaponding harmonics 
in (l), and that the time factors Bj,(t) in (») ars of the same order of smallness as the bj^’s In (i), 
Thaso assumptions are justified by the consistency of the subsequent analysis. 

The leading term on the right of (a) Is separated from the others because It corresponds 
to the case whan the bubble Is accurately spherical, for which the solution Is known. 

spherical hamonics of negative degree only are taken In the expression for^, because the 
velocity, and therefore also vanishes at Infinity, 

The pressure In the fluid Is given by the hydrcdynamleal equation 

S • ij+n-i (grad^^ t f{t) (5) 

fi 2 

Where n le the potential of the extraneous forces, limited In our case to grsvity, so that 
n » - ge. 

The aroltrary function P(t) Is determined In the present case by the conditions at Infinity, 
Taking the pressure at Infinity at the original level al^he bubble to be constant and equal to b, 
we have, since 4> tends to zero, 

F(t) • Qip (S) 

Substituting for0 from (») Into (s), and retaining terms up to the first order, then gives for the 
pressure p In the fluid et distance r from the origin. 




Boundary eonditions . 

The conditions to be satisfied by the velocity and the pressure at Infinity have already 
been considered. We non Investigate the conditions to be satisfied at the surface of the bubble* 

The condition to be satisfied by the velocity is that Its normal component at the aurfaeo 
shell be S(^1 to the normal component of the velocity of the surface as determined by the expression 
(t) for (t. These normal ce^aponents are, to the first order of snail quantities, equal to the 
corresponding nadlal components, and hence, to this order of approximation. 


I 




U1 


R ■ - [ ‘d 4>l^ r ] * 


I t £ S S - * (1 - y h S » ♦ 5; <n + ll B S (S» 

“ n n ' — n n‘ ■ • • n n * • 

an rstaining th« *.ems as far as tn« first ordar only. 

RInca (B) Is truo over a eoniptete sphere, we have on aquatir.g coafftelonts of the 
oorrasponding harntonlcs 


a » a“* A; b|, • - Ja“^ * a”*^* (») 

or. soWlfiu for A and B.> 

' '' n 

A « a®a ; 8„ • (a'^* h„ ♦ a e^)/(n + 1) (to) 

The rennining fioundary condition Is that the pressure Is continuous across the surface of the 
bubble. This requires that the values of p as given by (J) and (7) shall be equal when r • R. 
Substituting from (3) and (t) In (7). and using (lO), we find, to the first order, 

{ P(ag/a)^- g}/p* ga s^ • a'a'f | a* 

^ TT^ { (t - n) a" b„ ♦ 3a b^ * a b*^ } (ll) 

Th t difftrential equations for ait) unU b^ (t) 

Since the spherical harnonlc expansion on the right of (ll) is the expansion of a eonstani 
over e completo sphere, wo have 

aV + I a* • (e a^^ ~ 0 a^) /p a^, la) 

3a bj ♦ a bj - 2ga (13) 

and (i - n) a b|, 3 b b^ ♦ a bjj ■ 0, n 2, (Uj 

Equation (l2] agreea with the known result when the bubble Is accuretely spherical, and It 
may be aolved to give aU) In terms of P, 0, p and y. Substituting this value In (IM), we obtain 

the differential equation to determine l>^(t). Equation (13) Integrates Imnedlotely 

rl rl rl • . . r . 


, f* dt C 3 f* {s(o))^ 

S. ” 2g 2i + , 1 ,-. ot 

a^ a-’ 

Vo Vo Vo 


r ^5 

K 

F! 

5L* 

I 

i 


The first part of this expression Is Herring's formula (a) for the rise of the bubble due to gravity; 
tl« second part represents the effect of an Initial velocity. 

Except for tne caee when 0 0, the solution of (12) makes o(t) a periodic function of t, i.e. 

the bubble pulsates. Consequently tha oquotlon (ik) for b^ Is one In which the coefficients are 
periodic functions, the period (t, say) being equal to that of tho pulsation of the bubble. From 
Floquet's theory of such equations. It Is known that the general solution Is crdinarlly of tha form 

b„ - Cj^* f(t) ♦ ff-t) (15) 

where Cj and are arbitrary constants, and f(t) Is a periodic function of period T, Tne constants 
\ and tho function f(t) are determined by the periodic coefficients of tho equation. 

When, however, the equation possesses a periodic solution, say g(t) (corresponding to 
X. » 0) the general soiutl-.n Is not of the fora (iS); the second solution gj(t) Is not periodic but 
satisfies a functional equation of the type 

92(t + T) » 9j(t) * T 3j(t). 


An example of this occurs In the present calculations for the case n • 1. 


When 


148 


- * - 


r>l Z'J! 1 ^ Imjlnary, b^ «m ultimately Increase to large values. Actue’Iy 

tarty :irtro!“rorot::^i:,i:!^ir3b^ tr" *'’" ^ <* 

Msnee In general tho perturbation Is unstablt *’*' '* 

?^.tfalculation of th e wean radius aft) of the bubble. 

funetIa«"[tr«EellInM «1?5|'c“ l'I;^r^n^in'rii^“"^^^ *“ 


a » a/a^, x - c^t/a^ where 
the equation (12) has a first mtsoral of the form 


{ a~^ - a'^v i^jU fe-J _ i) } 


ano a further Integration gives 


X } [* , 

Ji <^a~^ (y- 1) P'Va'^- i)} 


0 » 0 anuT\l‘'’':n'tM! «sl“[h: ZlT t ' -- 

with t(or i). '* <«'■'«"' •>“* a continually Increases 

terms oft"ls '^u"\lZc" th^^ntt!:! M ^bT “'‘T '- 

(18). bn; of ZTsl Is « r. 1 s7tl 0 her s of the’'l«teg*nd In 
naxlmum raflius to the minimum rablus. '* *>'« '■«''> of tno 

wUlls(3), and H*rtng(4)Vbut'th!irLlnodra«^^ T '* *’>' Buttoroorfh'p) 

dstermlnat Ion of « as a finct Ion of i f ?! 

function of (T-t) wnena is noara rhei/- rd*u*t« <Jeterm I nation »f a/a^ as a 

to the Integral between , a.^Tne;; the s L ?!! ' " “* ePPrXtions 

nwr the singularity a end to the Integral between a and o„ 

ofer «hn'w"l:^::rrcV^^!Lv?;“b^:nt" 

a. this IS probably a fair apprl:!?? :^ .hit " cL*e * ^ 

Most of the Investigations referred to above have aUo dot: ot; I?!: 

»»'«n7.e/3, the expression (is) can be written In the factorised form 


V o a. 


-rr—~y 

tria-l) /{i-g (a *a^* a^)/ {^r)} 


from Which It will be seen that <x^ Is the positive root or 
<1^ + ♦ a « 3P/p. 


S)'^V' In’ :xptL'ln^rrtV:o^t‘r?ow':^''“ 

can be obtained by a simple Uaratlon method; this gives ''® ®’ ^ 


S 1 


Where 



150 


- 6 - 


It alii be saen fron this expression and from (Id) that for small pulsations, the period 

Is 

The equation for 

The etjustlon (IS) for n„ Is reduced to the nomi form by orltlng 


y ■ »„ 


which gives 


(n + |) f + 


Suostituting fora from (U), and for a fron (17) »/}, we obtain 


& -[l!] 


y • F(x)y, 


F(x) = (4n s j) cT* - 3n (1 ■*■ ^) ^ 


The following properties of tnis functlom are easily verified. 


(I) F(x) Is periodic In x, the period being the same as that of a, and therefore ranging from 
Tt for small to approximately for large o„,. 


(II) F( 0 ) - (n r |) (1 - 0/P). 

(III) F(T/2) » (n ♦ j) « 0 o^"^/p). Hence, when the bubble reaches Its maxtmu 

P(x) le negative, tfhen P/g Is large, its value Is approximately -3(n f \) 


maximum site. 


(iv) F(x) Is zero for only one value of a botwoon 1 and o^. When nfQ Is large, this value 
of a Is approximately (dn * l)/dn end thus lies between the narrow limits I 3 and for 
all n. 

(v) P(x) has naxlna when a * i and a • and het a minimum when 


I tt* ♦ 8 n (I ♦ ^ a - («n * t) • 0 . 


For large P/Q, this nlninum occurs approximately whena* (Sn f l)An, end Its value is 
approximately „ n 

(vi) In the case of small pulsations, on substituting from (27) In (?0) and nsgiscting terms 
beyond the first order, we heve 

F(x) » (n + y) « eos 2x. (?i) 

Host of the above properties are shown dlagremmatlcally in Figure 1 , which refers to the 
two extreme cases, P/Q large and f/o • I * €, € small. *s P/Q decreases from large values the 
two minima shown In the figures move together, eventua ly obliterating the msximum at a • and 
tending to the simple cosine function (}i) for small pulsations. 

Solution fo r n ‘ x, f f otion of bubblt in aboenet of exirimtout f orces. 

The first hai'monic term b,(t)Sj. In (1) simply represents a displacement of the bubble, without 
change of form and of amount b^(t) along the exii of the harmonic. Hence the solution when net 
gives the motion of the bubble In the abeonee of sxtrsneeus forces. 

The solution In this ease is easy, and dees not require the preceding analysis. Xe have 
from (13) wUh g • 0, (eampare alto l*a) 


bj, • bj(o)Ai^, 


Which 




151 


7 . 


«thieh show* that th» valoelty of tho eantro of the hubhlt varies Invereely »i the euM of the 
rablui of the bubble, and therefore Increaeee gniatly aa the bubble eontracte. The notion la In 
fact the laae a> that of a oartleie, *!«*» issss !s vafleble shJ prsosrtlons’ to the VBlwae of tni 
bubble. 


If we wflta 


/8^ ' 


and regar*/8. as a function of a, ca obtain from (J2) and (l7) 


th . 


where 


Nance 


da 6 ?^{ o’^ - ♦ (y - i) i>“‘ » (sT* - 1) > 

c . /? <y- t) 

^ * b*' io) Co / I 


(«V 


/3iW 


i + c 


d a 


♦ (•y- i) iv' Q ij) 


(33) 


(3d) 


(38) 


&«) 


and ;9j li therefore an Inveree-pertodlc function of e. 

Mheny • d/3, the Integral nay be evaluated In exactly the same way as the Integral In 
(IS) for X. For values of a near unity, we find an expansion In powers of OJ'* of the form 


/3j(«) 


1 ♦ cfs tan”‘/fc - 1) ♦ 30h"V(o - l) it ♦ - l) ♦ -!)*►♦ ...."j 

^ (37) ■' 


Bhils frcsa nearOj,, 


^j(a) • /3j(a) + 

The graph of ff - i as a function of a U shcpwn In Figure Z, for the case when P/9 ■ 10**, 
and eorraeponding to any initial velocity. When the Initial velocity Is zero the curve shrlnhs 
up to the straight llne/3« 1, corresponding to the obvious fact that the centre of the bubble will 
nHsaln In any displaced position unless It Is given an Initial velocity. 


Nature of the solut i on for large n . 

An interesting question, but one difficult to answer satisfactorily without mechanical aid 
such as the differential analyser. Is that of finding nunerlcal solutions of &b) for large n. 

What In fact one would like to know Is whether Initial small irregularities on the surface become 
bigger or die out as tho notion proceede. There seems no doubt that any Irregularity limited 
Initially to a email solid angle is unstable, and that the Instability Increases rapidly at the 
iflitlsl solid angle decreases. Pemaps the prickly appeantnee of the bubble In Edgerton'a 
photographs Is s manifestation of instability of the type new under discussion: at least there seems 
no more prcbabic alternative. It may bo noted, however, that tho photographs do not show any pits, 
but only needles, whereas the theory in Its present form Indicates both equally. There is clearly 
a close formal similarity between the Instability of very snail Irregularities on the surface of 
the bubble and ripples on the free surface of waler, as the pressure wave strikes. *■ theory of 
the magnification of these ripples has been developed by 6.1. Taylor. 


The function F(x) as defined by (30) has as Its dominant terns, when n Is large, 

F(x) - n(s o'® - 3 a"®) 
so that (ZP) becomes 


H - 

dx 




ny(4 o“® - 3 a"®) 


Thd A***# 




Thi coafflelant of y on R.H.s, i» positiva In th« range 1 <si < H{}, anD is negative for 

a> */3. Hence for large n, y inereanea exponentially In the rang* i.<a< */J, and thereafter 

OBClliates. The interval between suceesalve zeroa. at n la varledi decreaaet like l//n. The 

ratio of the etaxlmun value of y and the value of y at a = l U or the sane order of ntagnitude 

f'o 

X • exp { / n. /(» oT* - 3 «"*) dx } , 

■ 0 

whare x Is the value of x at the zero of the Integrand, keplaclng the Integrand by an approximate 
expression, obtained from 


X (t* 2z/9 r z^/S) / (2z) (LaiMi's substitution) 

It will be found that 

X “ exp { a / n/9 }. 

If n ■ to the magnification X Is *>09; If n ■ 100, X lo ood If n * 1000, x is 1,300,000, 

aptutiew fo r small pulsations of iht bubble. 

The solution of (S) in a simple fone can be obtained for any value of n In the Case of 
snoill pulsations. Although this theory is clearly not applicable to real bubbles because surface 
tension, which would now be relatively Important, Is neglected, we give the theory because it may 
Indicate some features of the more general motion, 

from (20). (29) and (31), the equation to daternina y 1s now 

^ • (n ♦ |) e cos 2x (39) 

which fs a degenerate fonn of Hathleu'e equation, Since e Is small, we seek an approximate soVutlon 
In tho form 

y • A ♦ 8x ♦ e t(x). 

On substituting in the equation and retaining only first order terms we find 


f(x) • - 5 (n ♦ { (a + Bx) eoa X ♦ 8 sin 2x }, 


y •- (a + Bx) { 1 - ^ (n ♦ j) e cos 2x } ♦ ^ (n ♦ ^) € B sin 2x (so) 

Since y • a^^* b^, const, y where a Is given by (27); hence /3^ can bo reduced to the 

form 

0^ • (a' + B'x) < 1 - ^ (n - l) e eos 2x } + J (n ♦ -j) e B' sin 2x (Si) 

This result .-spy bs shssfesd shsn « i from our previous calculation. We have 
/Sj = 3(1) ■ ;§(!) { 1 - I s (1 - cos 2x) }, 

which on integration gives 

/9 • 1 + Const. ■{ (l ^ c) X ♦ I s sin 2x }, 
which agrees with (so) when n * 1, 

At any value of a, a measure of tho noiv-spherlelty associated with the hamonlc S Is afforded 
by the ratio bjj/a; this divided by Its Initial value Is equal Xo 0^h., which is found fro,i («l) 
by dividing by ix, as given by (2T); this gives 


W •• 


- 9- 


153 


^ • (A* ♦ 8'x) { 1 - I (n - a) « CBS ^ (n ♦ |) ff 8’ sin 2x (M) 

Hsnea th» perturbatloi assoelatcB with may Du rejardad as a small oscillation of period T, 
about a moan value which Is continually increasing with t (or x), whan n “ 2, tho amplitude 
of this small oscillation Is constant, out when n > 2 the ampiltude also increases with t. The 
greater the value of n, the greater la the amplitude of this oscillation. This is Illustrated In 
TIgure .t, where the graphs of y?j/a arti for two pulsations of the bubble are shown. 

When^^ is Initially zero, the constant 8* In { 81 } Is zero, and the perturbation Is simple 
harmonic, the amplitude being proportional to (n ~ 

The solution of ths equation (?8) can also ba obtained In the form (is). Adopting Hill's 
imthpd of solving Mathiou'i equation, we letk a solution of (A<) In the form 

y n e ^ 2 e 
r»-« f 

In which one coefficient, say c^, is aroltrary. 

Substituting In and equating coefficients of like powers of e” to zero, we have 
^ ♦ 2rl) * Cr - ^ (ft ♦ ^ *r+l) * ®* 

for all positive and negative integral values of r, 

eliminating the C's gives an Infinlia continuant equated to zero to determined. This 
continuant Is a limiting form of Hill's determinant (Whittaker and Watson, Modern Analysis, ia2o, 
p.si:) and on neglecting terms of order .higher than the first, we find 

X. • ± \ 6t J2 where 6 “ |.(n ♦ e, 
and • c_j ■ - e^, «j ■ ••• ®(«*). 

This leads to a real solution of the font 

^ • ^A* cos ^ ♦ a' sln^j { 1 - I (n - 2) « cos » } (»3) 

For small x this agrees with ths result (at; provided B'/B* • 6h/i, a snail quantity of the first 
order. The apparent difference in ths two results for other values of x Is due to the fact that 
they are both approximations to the first order in e, and It is possible for the ratio of the 
arbitrary constants to have any order of smaUntssu Thus If A*/D* Is assumed to be of the first 
order in e, a further term In would be required to be retained In the coefficient of e^’* In 
the above solution, to ensure that all the terms retalried are of the same order. This would give 
a temi eorrespcndlng to the sin term In (at), 

without going furtner Into these refinements. It will be seen from (<tJ) that, provided the 
condition that/S^/(s - l) remains small is not violated, the perturbation can be regarded as a 
quasl'perlodlc variation of period T |n which both the mean value and the amplitude have a slow 
periodic variation of period T, 

Sol ution for the general case- 

In the general case where j/r has any value < I, ana n Is any positive Integer, It Is 
convenient to derive an equation for/3 In terms of a, snd solve this Instead of dealing with the 
equation (29) directly. This mothod Is somewhat analogous to Llndsmanh's treatment of Mtthelu's 
equation. 

Since a to periodic In t, b„ will be a mtiltl-valued function of a with branch points at 
a - a,j and a « a^. Denote the value of b^ In the half-period ^^T<t<|Tbyb , which Is 
then a single-valued function of a. Thus b„ j Is the value of b^ during the flrst''expsnslon, 

®n, 2 ''*'“8 during the first contraction and so on. 


Considering 






<».l: - jp> - I 


— 


Cgnsiderlftj any half-pofiodi and ehonqlnj tho Indepsnsant varlaDle !n (i») to a, na 


^ *>„a 

da' U a J da aa' 


“n.a * 

and utina (-17), this equation Deeooea 

+ = { f<s) ♦ I } — - (n - 1) ^ { Ho) - 4 } 0 

d o' o 'do or 

wMre the euftlx (nis) Is omitted from ,6 ter the presont, and 

t(o) . -Zi f g 55—-- 

* o“^-o*^*rr fy-1) {a"^ -li 


When > a vj), the equation (46| reduces to 


^ -^tro** (1 + 1 cr) 0-1 . ♦ o (T a* ♦ o - i 

+ {n-il). m 0 


The generel solution of this eqiation Is required for the range l<a< 0 |^». It my be noted 
that when or*' 0 tho equation redueos to a Um6 eqration, and can be transformed to Legendre's equation 
for hp'” (■»)) by the substitution tj » / (l-o). but m, randTjareall complex or tnaglnary. 

The equation res regular singularities at the origin and at Infinity, and also at the four 

zeroa of 

- |cro* ♦ (i + i O' ) a - 1, 

l.e, at 

o • 1 , V J* ' 

At the four latter singularities the indiclal equation is vh* sens, viz, 

2 |i (gs - l) + - 0, 

so that the exponents relative to any one of these singularities Is 0 or 

Hence two fundamental solutions of (AP) can be obtained In the form 

- Vi<*5 * *o" V* */♦ I (SO) 

/3 - ♦ A,^z3« ^ J 

where z <• o, -S;, and o.| la sny one of the above four singularities. 

8y considering tho positions of the singularities in the complex plane, It can be seen that the 
series expressions (SO) relative toa* 1 (l.e, with z • a - l) are convergent for z < 1, i,e, for 
a < 2, and the series with z • a - are convergent for z < - I, l.e. for the whole range (eajept 
fora • t) over which the solution is required. 


The 


- Xi - 


I I f ) 

! i; 


Th« rocurrtnce r«UUon Betnwen th* coefficients In (5o) Is found to be 
(p*l)(»r*l) umr*+ (n-i) 1) k^*{~ (n-l)(an-0 p ♦ ) k^ 

- (r-«)(r>-i) 30cr k*A^j - (i^3)(a^-3) to- k k^^ - 2 a- k^, 

nhere It * 1 or as tlis ease may be, and 


I « 3(J cr k - (3 + O') It + 2) 
m • ^(J 0 <r k** ~ (it d 7) k a «) 
p« a07k^-3-7 


■ 3-{o--i> » 3(3a^»0'a^-«) 

« ''{2 7 '- l )» 2 I * ) 

m »7-3 » 207 0^^ - 3- 7 


ly taking r any positive Integer (iKtudIng zero) In (Si) and taking 1,,^* ft for r > 0, «e obtain 
the coefflelonts of the first series In (so), and by taking r aqjat to half an odd Integer we obtain 
the eoefficlants of the second series. 

It will be noted that the approxlmattoni to /9 near a • t and neara * 01^8111 respeettvety 
0 • Ag {1 ♦ (n-s)fei - 1)} ♦ Aj^^j Aa - 1) (53) 

|8 = A-„<l-^ (x„-a) A-^j/(Oj,-a) (S*) 

In the numerical ease worked out (corresponding toa^ • 30.732) It wss found that the above 
aeries were only suitable for calculation In the regions a • 1 to t,s when z ■ a - t, and a » to 
to 30.73 when z • Oj, - to. Hence to supplement these aeries, the Taylor expansion 


C, ♦ C,z + e,z* ♦ 


of the solution, relative to any ordinary point z « k, was obtained. The recurrence relation for 
the coefficients of this expansion Is 


(rr2)(rM)(Thk*) » (r7t)(2m)(-(k) ♦ {-Umr* ♦ (n-i) q} 

♦ (^•l)(^r-l)(-p) t (n-U (**3 7) C^j -I (r-2)(r-i)(-30 7 k*) 

♦ (r-3)(2r-3)(-< 7 k) ♦ (p-»)(r-Z)(^7) 


n«27k*-(S*2 7)k+S ' 

(# 7 ) 

q - (9 t 3 7) k - 12 
and (, IS and p are given In (52). 

Tht rtlation between 0„ , and 0- . 

^na S Sr i o 

The abcve expressions as a functlo,n of a (with two arbitrary constants) over any 

half-period. If s la odd, 0^ , corresponds to an expanding phase, and 0^ to the euceeedlng 
contracting phase. To obtain' the relation between theee /9's we hove the eonditlone that ^ and 4 
are continuous at a • 0||^, The value of is given by 

/§ » a " ± J— / {a“3 _ a"* - 4 7 loT^ - 1 )) (oe) 

da d a a. ’ 

0 

• ± / (o - a) 4> to), say, 

do 

end the positive or negative value must be taken according as we are considering an expanding or 
contracting phase. 






Hence, If whan s U odd, we write 

“ «o.s ^ ‘ (s») 

^n,»l * %,s+J * «l/2,s+l’^'“m-“) U*)* 

Where g,/^) and j^(z) correspond to the two fundamental sotutions In (so), the continuity or p 
at a « roqulros 

6o,e “ ®o.s*i 


-7^ ’ ®o,s a V*> ^ “1/2.S - “> - JVt^S-rST > 


and therefore 


\a*"- ^ 2,8 


Siinn&r\y 


■* ♦ ? ®l/2.sn jm(") <t> (a) a* <*•* V 

so that the continuity of at a - requires 

° 1 /S,s ” ■ ®l/ 2 ,s+l «l) 

In the sane my the continuity cf and 4 at a • t cstsmlnos the t’elijti,..,. ■.atvi'jiri 
and when s Is even. Writing 

^n,s “ *o,s ♦ \/ 2 ,s * - « - 1 

(«a) 

^n,s+l * *o,$<-l * '‘l/ 2 ,s+l'^^*“ 


rr:iT'"! 

r; 1 ■ ■) 


A o A 

0 |S o»s+l 


*1/2,8 ' ■ *l/2,s+l 


TAe solution tiifien P;Q m zp4, n » 3 

The ease when P/q Is large Is of most Interest In connection with underwater explosions, and 
the shove result has therefore been evaluated for the second harmonic term, with P/q > 10*. The 
second hanmnic Sj Includes (among others) the deviation which deforms a sphere into a prolate or 
oblate spheroid. It will be assumed that (his deformation exists at the Instant of the explosion, 
and that its velocity Is initially zero. 

When P/q ■> lo'^, the value of the maximum radius Is found to be 

«m “ ?0>”' (»S) 

and the period of pulsation of the bubble Is 

T » 0.056 Sjj (T In sec, and a^ In cm.) (66) 

These values are of the same order as those observed for T.H.T, and similar explosives, 

Whon s • « 



157 


- 13 • 


When P/Q Is much greeter IhaniD^, the compresslhlllty of the water will be ImportEnt during 

the (short) time that a Is near to 1> and will give rise to a radiation of acoustic energy and a 

consequent decay of the pulsationsi This effect Is Ignored In the present calculations. 

Nevertheless the result may be expected to Indicate what happens during the greater part of a 

puliation and particularly when a > >. for when n has reigehad the pressur will have fallen to 
a value of order 10 ^0 or less. 


The method adopted In the calculations was to determine first two fundamental solutions of 
equation (Pd) for s oo a function of a, using the expansions (110) and (JP). These solutions, 
denoted respectlvoly'by g, and /(a - l) were chosen to satisfy the conditions. 


<*5. 

g, • 1 , — i *0 at a 

^ da 


( 67 ) 


>^(a- i)j,- 0, ^ - 1 

^ da 


nt a « J. 


( 68 ) 


and thus correspond to the expansions given In (bol when ttj ° 1, and • i, “ -• fbeir 
valuos were detsFRlnr-d froM (50) for tbs range a = i to a • 1.5. These solutions were then 
extended by analytic continuation over the range a • l.S to a •> to by means of Taylor expansions 
of the form (5U) relative to the points k » 1 and 4, For the remainder of the range, a » lO to 
a» ajij, the solutions were continued by means of the expansions (so) with z » a - a^. The various 
expansions used have certain common regions, where their convergence Is sufficiently rapid for 
calculation, so that a number of checks on the results could be marie. The values bf g^ and 
/(a - l) jj found In thin way are given In Table l. 


tabu 1. 


1 

a 

9l 

- 1) ii 

Q, 

8l 

‘'(a - 1) jj 


E. 

t 


1.000 

0 

U.O 

.824 

.673 



i.t 

1.089 

.315 

6.0 

.406 

*456 



1.2 

M64 

.441 

8.0 

.109 

.868 



1.5 

1.30 9 

.647 

9j3 

-.094 

il89 

-.860 

4.47 

i.« 

1.337 

.716 

10.0 

-.103 

#117 

-.671 

4.64 

U7 

1.351 

.782 

“m" ‘® 

-.426 

•^.156 

t,112 

4.37 

1.8 

1.353 

.778 


-.536 

-JI82 

.528 

3.49 

1.9 

i.3»r 

.796 

“m- » 

-.551 

-.350 

.851 

2.01 

2.0 

1.33U 

.808 

ttju - 1 

-.516 

-.373 

.966 

l.OU 

2.1 

1.316 

.815 

- .5 


-.371 

.983 

0.7U 

2.2 

1.29U 

.818 

a_ - .1 

-.469 

-.366 

.997 

0.317 

2.3 

1*269 

*618 

“■m 

-.443 

:-.356 

1,000 

0 

2.4 

1.241 

.eis 

(-30.73) 





2.5 

1. 212 

.810 






3.0 



1.080 

.779 







Apart from the variable a, the functions g, and depend on two parameters, tr (■(//P) and n. As 
far as a Is concerned, the valuas of Pj and Jj given In the table are unaffected up toa » 2 by taking 
any value of cr not oreater than 10~“, and they are not affected by more than about i.% up to a » s. 
Increasing a to lu'^ does not change the values up to a • 3 by more than about l». Consequently 
the early part of Table I will give a fair approximation to the solution of snuatlon (P9) for 
values of a fro- < to 3 (and possibly as far as d), for any value of P/0 likely to arise In connection 
with explosions. On the other hand, altering n has a great effect on the values of g, and j , as 
Is obvious from the approximate expressions (58) Tor {i near a • i, ^ 

The determination of B during a contracting phase, following the known variation during 
the preceding exfranding phase. Is determined, as snown above, by expressing/? in terms of the two 
fundamental solutions and - a) J^. Hence the values of these solutions from a « 9 to 
a • were also calculated and are shown In Table I. 

The relation between the two pairs of fundamental solutions In Table I Is found to bu 

8l = 





138 


- in 




L, 




. -.4434 9^- .086*5v''(aj„-a) -j 

(a - 1) jj = -.3583 9 ^ - . 02641 /(a„ - a) J„ I 


{65) 


The raUtloft betnoen the pairs of arbitrary constants appearlnj In the alternative 
expressions for is obtained Isnssdlatsiy from (at), for If 


^a) a Bo Sm* ®1/ 2 Jm ” *o ^i^ ii- 

and 9 , and are expressed In terms of and by (6f) we have, on comparing coefficients of 
Sm '^m rsauUlhg Identity, 


Bo * - -wn - .3583 -1 

®l/2 " ~ *08626 A^ - .0266 A^^jj J 

Conversely 

*0 • 1.393 Bp - 18.76 

A^/j= -4.516 8^+ 23.22 8^/5 

SInCa the Initial velocity of the deviation Is zero, we have 
^,1 ' ^2,1 “ ® 

Now If 

^2.1 * *o«i" 1) V 

then 

^2.1 - {‘o8'i^S/2(,7(hT) 

** ? *1/2 ^ t ” 2 <t) as a - 1. 


(70) 


(71) 


Hence the above conditions require 

*0 “ ^ *1/2 = *■ 

ttet Is, the value of yS during the first expansion is given by in Table I. 

The coefficients 0^, Bj/^ In the alternative expression for^^ ^ are obtainable from (70), 
and the coefficients in the corresponding expression forjS 2 ^ are thus obtained by simply changing 
the sign of B^/j as in (61). The coefficients for the succeeding expanding and contracting phases 
are then obtained by successive applications of equations (5$) >- (61) and (70), (71). Their values 
for the first two pulsations of the bubble ere given In Table 11. 

TABU II. 


) 


Expansion 

Contraction 

*0 

*1/2 

• ®0 

*1/2 

4.1 


i.o'bo 

0 

9- .44^4 

- .08626 


4,2 

- 2.23T 

+ 4.006 

- .*434 

+ ,08425 

4,3 


- 2 . 3}1 

- 4.006 

+2.427 

+ .2994 


4,4 

+ 8.997 

-17.P10 

+2.427 

^ .2994 


It will be seen that the values of A^ correspond to the successive values of 4 at ei 1, and the 
values of to the successive values of ,6 at a ■> o^. 

The ..... 



ii 






The graph of /9 as a function of a during the first t*o pulsations Is easily constructed from 
Taoie* I and il. This Is shown In figure 4. It will be seen that the variation of ^during the 
second pulsation Is generally of a similar nature to that during the first pulsation, but the sign 
Is changed and the amplitude Is Increased, 

When ^ Is regarded as a function of t. It should be esjpressible In the fann (iS). it -'s 
easily verified that for any function of this form ' ° 

cosh\T = 5 [ /S {t + 2T) 4 ^ (t) 3 //9 (t 4 T), 

M that k Is determined by any 3 values of ^ separated by Intervals of T. Taking the values of 0. 

T and ar as given by In Table II, we find 

cosh X T ss - p.ot, 

whence e . -.3.95, or X . (U38 4 I w)/T,. (72) 

This shows that eventually (It the pulsations could continue) the perturbation ,5 Increa'es 
rour-rold In absolute n«gnltude and changes sign during each complete pulsation. In this sense, 
therdfofs, the perturbation Is unstabla* 

K a, ® fom sphericity Is measured, not by 3, but 

a? STeatly decreases during the expansion. At the end 

Jt oi^sltTsIgT " ° 

int»i 1 '! “ function of t. It decreases vary rapidly at first, due to the rapid 

th!t tL bubMris i“r I "‘"■'"S '•e’a.tively long period 

i!ri» rfft; r f ®"Ses are Illustrated In Figure 5. which shows /3Mx as a function of t 

Seferenees. 

(1) lamb, Phil. Mag., 45, 237. 1»23. 

(2) Butterworth. -Report on the thepretlcal shape of the pressure time curve and on the 
growth of the gas'bubble-, 1923, 

01 Willis, -Underwater explosions. Time Interval hotween successive explosions-, 1941. 

(4) Herring, C4 - SrlO - 010, M.D,R,C., 1941, 







THE BEHAVIOUR OF AN UNDERWATER EXPLOSION BUBBLE 
Approximations based on the theory of 
Professor G. I. Taylor 


A. R. Bryant 

Road Research Laboratory, London 


British Contribution 


December 1942 


THE BEHAVIOUR OF AN UNDERWATER EXFLOSlOW BUBBLE 


163 . 


Approximations based on the theory of 
Professor G. I. Taylor. 

A, R. Bryant 

Peeaaber IS4’3, 


Summary 

Approximate solutions are presented for the equations of motion of the gas bubble produced by 
an underwater explosion, as given by Professor G. t. Taylor, The equations enable the roost important 
features of the bubble motion to he computed approximately with oon^rstlvsly little labour. In 
addition graphs are given which are based upon the approximations, and which enable most of the quantities 
to be read directly as functions of the depth for various charge weights. The effect produced on the 
DuOble motion by changing the charge weight or the depth of the charge may be easily seen from these 
approximations, which should assist in an appreciation of the Velatlons which arise when the scate of 
an experiment Is changed. 

Introduction . 

Professor G. I. Taylor has treated the radial expansion and vertical motion of the bubble of gas 
formed when a charge Is detonated In water*. He has developed equations which require to be integrated 

step by step numerically and has shown that exact staling for different charge weights Is not possible, 
so that the nunerica! integration must be repeated for different values of the charge weight and the 
depth, The following note puts forward approxinate solutions to these equations which enable most of 
the important features of the bubble motion to be computed with comparatively little labour. In order 
that these equations may be readily available they have been listed at the beginning of the note, with 
an explanation of the symbols used, but with their derivation omitted. Although the numerical constants 
for T.N.T. have been employed throughout, the methods of approximation used are applicable to any 
explosive. 

As In the Beport ‘Verticai motion of a spherical buoble and the pressure surrounding It*, free 
or rigid surfaces have been assumed remote enough to cause no disturbance to the motion. Their 
perturbing effect as given by Conyers Herring is discussed in an Appendix. 

The non-dimensional form of the Equations , 

The basic equations of motion of the bubble are used In the non-dimensional /onri given by 
(“rofeasor Taylor. In the list of formulae below, and iq their subsequent derivation, some of the 
equations are best left In this form. To convert to real quantities all non-dimensional lengm 
must be .multiplied by the length scale factor L, and all non-doraensional times multiplied hyv/i, g 
being the acceleration due to gravity. For T.N.T. the value of L la given by ^ 

L (feet) = 10 M* (l) 

where M Is the charge weight In lbs. L Is plotted against charge weight In Figure 1. 

To avoid confusion non-dimensional quantities will be denoted by small letters, while capital 
letters will be used for dimensional quantities (with the exception of the symbols g, and p the density 
of water). Noo-dimenslonal equations will be labelled as such. 


Pant I 


Vertical motion of a spherical bubble and the pressure surrounding it. 


G.1, Taylor. 


164 


- 2 - 




PART I. 

S ummary of Attroxiaaiion Pcmulaa, 

Two non-dimensional variables define th* size, position and motion of tne OuOOIe at any time t, 
via;- the radius of the bubble a ard the depth » of the centre of .the bubble measured frgn a point 
3? feet above sea levol . 

A parameter e appears In the equations, in thoir non-dimensional forms, which Is related to 
the potential energy In the gas; the value of c tor T.K.T. Is given byi- 

t 

e • 0.07S (J) 

whe'e ll is in lbs. 


The ieriod of the first 


U.32 M- 



where 


oscillation - T (seconds) 

(3; 


is the depth of tfio charge In feet below a point 33 feet above 
sea level. 


V. Is thn charge weight in lbs. 

The period Is the time taken for the bubble to expand and contract again to Its minimum radius. 
At a distance d non-dimensional units from a free or rigid surface the period T is altered to 

T* where 

II a (1 T (r.k.S. non-dimensional) (3a) 

T “ 

where the upper sign Is for a free surface, the lower for a rigid surface, and 
a^ Is the maximum radius (see equation (b)). 

'The maximum radius of the bubbl e - 

“m^ “ t ' -ir^z (*• ■ h Sni”* ^ (non-dimensional) (s) 

Zg being the non-dimensional depth of the charge at detonation; a^ Is 
plotted in Figure 2 against Zg, 


The vertical momentum constant - m. 


The non=dimenslonaI constant m Is Involved in several equations ^nd is given by: 


O' 7 a_ 


(non-dimensional) 


(S) 


At a distance d non-d Imens I onal units below a free surface or above a rigid surface the 
vertical momentum constant Is changed to m' where 

3 I 
a z ^ 

m' • m(l - 0.82 -12— —£ — ) (non-dimonsional) (8a) 


This value of m' must be used in all subsequent equations Involving m If the surface effect 
Is appreciable In the case considered. 


The 






161S 


The totai vertical momfintum 0/ the water surrsunt ii ti/? the bubble. 

During th» pnriod whan thi bubbla Is near tts mlnlnuin radius at tha end of the first oscillation 
the vertical momentum of the neater Is approximately constant and equal to 
7 

1M81 ffl Ibs.fest/second. (6) 

where L Is the scale factor In feet and m Is noivdlmenslonal. 


!!Se mtntnum rodtus of the bubble - o. 


a^^(l-ea| ) • 2^ (non-dimenalonal) 


Equation (7) must be solved graphically. Using equations (S) and (4), a, has been plotted In 


figure } against the non-dtmenalonat depth z^. 


ihe ^ygssure luave emitted by the- ce,llai>sine and exbanUing bubble. 


Curing the period when the bubble Is near its minimum radius a presaurn pulse is rsulateu 
outwards with the velocity of sound, for points not too near the bubble the peak pressure (lb./ 
square Inch), the total positive impulse 1 (Ib.-secono/square Inch) and the duration of the positive 
pressure pulse 0 (second) are given by the following formulae:- 


O.UJU 4 71 a, 


(1 - J CBj^ ) (R.H.S. non-dlmcnslonal) (B) 


whore R Is the distance In feet from the point to the centre of the bubble. 

The right-hand side of equation (8) is plotted against Sj. in Figure 4 for a number of charge 
weights. Using Figure 3. may be tabulated for various charge weights snd depths and Is given as 
a function of the depth In Figure 5. 

The Impulse 1 “ (9) 


The Impulse 1 Is plotted In Figure S against charge weight M. 


The duration 0 » 0.218 T 


I < 

I (..> 


JVie maximum vertical velocity of the bubble - U„ feet/ second , 

U m 

• —3 (R.H.S. nom-diraenslonal) (llj 

U^ Is plotted against the depth In Figure 7. 

The, rise of ihe bubble at the end of the first oscillation - li. 

*t the point where the bubole Is at its minimum radius It has risen a distance h (non- 
dimensional) above the point where the charge was detonated 

l.ig 

h • ' g (non-dimenslonal) [ig) 

7? 


The rise h Is plotted against z^ in Figure 8, 



166 


. u - 


U a distance d ndn-d!m«nslona1 units below a free surface or above a rigid surface the 
rise h is altered to h* where 

. a_ I, . . , , , 

),• X h (t - ^ * (noo-Oimensionalf liiai 

Aereement o/ approximations with exact solutions. . 

Recently the full nurrerical Integration of Taylor's equations has been carried Out for a few 
depths and charge weights, covering the range of non-dimensional depths from 1 to u, and charge weights 
fr<yn 2 to «00 Ibs. Comparison of the above approx Inat Ions with these figures, and with some 
unpublished ffguras for a U 02 :. charge at 6 fael depth shows that the agreement over the ertiole range 
is satisfactory. The period of the motion and the irexlmum and minimum redll agree within 1 to 5 per 
cent, the maximum vertical velocity and the rise of the bubble within 5 to 10 per cent, and the peak 
pressure within 7 to <0 per cent. 


PSRTJJ . 


are (l): 


Derivation of the Ajiiroximate Solutions , 

Taylor’s equations of motion of the bubble, when expressed in their non-dimensional foiin. 



(non-dimensional) (13) 


tf is the total energy of the notion, Qa the potential energy of the gas In the bubble 
at radius a. For T.n.T. Taylor expresses the potential energy term (2) 






where e 


0. 07S M' 


1 


( 1 *) 


During the numarlcal integration of equations (l3) by a step by step process It is noticed 
that at different stages of the motion some of the terms In the equations either remain sensibly 
constant or become negligible In comparison with tne remaining terms. The following approximations 
arise from these observst ions. 


The maximum radivs of the bubble - o_ 


During the first holf period of the bubble the vertical motion remains small, so that in 
equation (13) the term z remains substantially equal to ^ (the Initial depth), while the 
1 negligible. The maximum radius of the bubble is obtained by setting ^ • 0 in 

equation (13) aimplified by these assumptions 


ca^”^) (non-diraenslonal) 


(») 


The 

(1) "Vertical motion of a spherical bubble and the pressure surrounding If. Equations S 
and 6. Taylor distinguishes all non-dlnensional quantities by dashes, which are omitted 
for convenience. 

(2) Above report. Equation 19. The exponent J is strictly true only for T.N.T, 


107 


The period of the oscillatior. T 


In the step Dy step Integretion of (13) It is observ,ecl that the time T taken by ths bubble 
to reach its mlnifflisr. radius at the end of the first oscillation is within a few per cent of twice 
the time taken to reach the fflaxlmum rodfus. It Is also found that omission of the term ^ in 
Integrating (13) causes very little difference in the time taken ij reach the maximum radiuSi 
although the actual value of the maxiimm radius is quite considerably altered. The Value of the 
non-dimensional half-period •j* is accordingly obtained from 


(non-dimensional) 


where the integral on the right is taken from a * c to the maximum value of a. 

The value of this Integral has been given by Umb (1) and Conyers Herring (2). Tne result, expressed 
in the non-dimensional variables used in this note is; 

1.135 

tw 5 “ (non-dimensional) ( 16 ) 


This value of the period ( 3 ), converted into real units, is 


The vertical momentum constant 


In the step Oy step integration of equation ( 13 ) it is found that tne non-dimensional 

quantity / a^dl becomes substantially constant when the bubble has contracted to about a half of Its 

maximum radius, and renvslns constant up to and beyond the time when the bubble radius Is a minimum. 

This constant value may be put' equal to m, and is proportional to the vertical momentmn of the water 

surrounding the contracted bubble. it can be shown that the vortical •homentum In Ibs.feot/second 
units is given oy 


Vertical momentum 


Ibfll m Ibs.fcet/second 


* knowledge of the value of ra enables several quantities assoc'ateO with the motion of the 
bubble to be calculated, so that an approximation to ro Is desirable. it is clear that the value of 
m depends meinly on the radius time-curve when the bubble radius Is large. An epproxinote 
evaluation of m may be made if it be assumed that the effect of altering cither the depth z or the 
Charge weight (I.e. the parameter c) Is mainly to alter the noximum radius and the period of the 
motion, without appreciably altering the shape of the radius time curve, at least in that portion 
when the radius is large and the vertical momentuii Is mainly acquired. This is equivalent to 
assuming that the radius time curves could be superposed in this region if the length and time 
scales were adjusted to make the maximim radius and period agree. Mathematical 1y, this assumption 
is that in the equation; 


(c)’ • ' 

for 0 < t < itk 


(non-dimensional) 


the function f (|^) Is Independent of depth and charge weight. 


Hydrodynamics. H. lamb, p.m. 

Theory of the Pulsations of the Cas Bubble Produced by an Underwater Explosion. 

Conyers Herring. 

Tho period may b& exp.essed in a form valid for any explosive. If \ be the energy of 
the motion in calr-rtes per gram^of explosive. M the weight of the charge in lbs. (’ 3 ) becomes 

T (secenda) • C.567 X' 

I (3a) 

This expression enables,', to be calculated from experimental measurements of the period T. 


188 


• 6 - 


Tho final Justification of this assumption Is that tha results to which It leads are In 
reasonable SDrsamsnt with the exact solution of equation (t3). Hence 


™ • /*** a^ dt • t*< a„^ / f(^ ) <1(| ) 

0 Q 


= 


• 3 

or using (iS) *' ®tn 

"> T" 


( nof>-d Irnens Iona) ) 


(18) 


The Constant k* may be obtained from any one step by step solution of the equation of motion 
of the bubble (13), and Its value Is about 0,70, leading to equation (5) 

0.70 a ^ 

'** ’ 5 (non-dimensional) (5) 


The minimum radius - 

In the neighbourhood of the minimum radius the terra •|z in equation (l3) Is nenligible, but 
the term^(^) must be Included. As shown In the last section ^ may be written (when a Is near 
the ffliniffluffl radius) 


dz s 

3T 


2m 

:3 


(non-dlfsansional) 


(Wa) 


and the equation of motion of the bubble becomes: 




pi (1 - ea“*) - I (non-diraenslonal) 
^ a 


( 190 ) 


The minimum radius is given Oy setting || « O, yielding 


(1 - caj 


(non-dimenslonal) 


( 7 ) 


The iressure wave Produced by the coLlafsing bubble, 

During the period when the radius of the bubble Is near its ninimum value, pressures are set 
up In the iurnedlate neighbourhood of the bubble which give rise to a pressure pulse propagated outward 
with tho velocity of sound, Taylor has discussed the pressure distribution close to the bubble, 
neglecting the compressibility of the water, and gives the following equation (l). If P is the 
pressure (In lbs. /square Inch above tne normal hydrostatic pressure at the same depth) at a point 
distant r (non-diroensional units) from the bubble, and if 0 Is the angle between the radius vector 
V and the vertical, then 

^ (a^ a) + (au + Sau) cos 0 + (cos & - sin^ 0) 

- [ (^)" a^ ♦ (|)*aucos0 + ^ (f)‘ u^cos^ 0 ♦ ^ ,in^ 0) ] (go) 

(H.H.5.. non-dimensional) 

vrtiere a • ^ . u « ^ , and u is the vertical velocity in non-dimsnsional units. 

In this 


( 1 ) 


"Vertical notion of a spherical bubble and the pressure surrounding it". Equation (30), 
Taylor’s notation has been modlfiea to conform with the rest of this note. 


li 


In this expression for the pressure the first term is the most inportant a.id will determine 
the pressure at points not too near the Oaoblo. Considering here only this leading terra, the pressure 
at a point H feet from the bubble centre becomes 


iSiiRE . (a* a) 

pi^ dt 


(r.H.S. non-dimensional) 


since the simplified equations of notion (19) are valid near the time of the minimum radius, 
when the pressure disturbance is mainly produced, (20a) may be evaluated to give: 


14HRP_ . 1 

piT U tt a" 


w i !!L (r.h,S, non-dimensional) (21) 

3 a® 


The maximum value of the pressure fra is obtained by insertino in (2l) the value of the minimum 
radius given by (7); hence 

— » 3 — — (l— ica, ^) (R,H,S. noFh^d i mens tonal ) (8) 

0.43U r 4 77 a, ^ 

It may be remarked that in a simibr way all the other terms In (20) may be evaluated as 
functions of the radius a alone, using equations (W) and troir value at the minimum radius coraputed. 

When the radius is a minimum, a and u vanish while u = ” If '* given by (19a). Inserting 
the value a, for the minimum radius, and converting to real units, the maximum vertical velocity U. 
(feet/ second) is 
u 

' -■ U’ - i '- « J5- (r.H.S. noiv-dimensional) (ll) 


Owing to the rise of the bubble points situated some distance vertically at'ove the charge may 
be quite close to the bubble when it is near Its minimum radius, and the peak value of the pressure 
would then require the computation of further terms In ( 20 ), 

The total positive impulse in the pressure wave is got oy integrating (20a) with respect to 
time between the two tines for which ^ 1$ a maximum (l). Conyers Herring has obtained the value 
of this impulse using an equation of motion which Is identical with (l3) with the vertical velocity 
and the gas energy term ea~* neglected. since the value of the Impulse depends only on the maximum 
value of a* which occurs whan the bubble is comparatively large, Herring's neglect of the two 
above mentioned factors will not cause much error. Herring’s value of the total positive Impulse 
(ibs.second/ square Inch units) may be converted into the following useful form. If the simple 
approximation »■ y be used: 


where R is the distance In feet from the bubble centre. 

It can be shown that the duration 0 of this positive pulse Is a Constant traction of the 
period T. Since the times when a^ ^ is a maximum occur when the radius is large, the opproxIaBtlon 
Involved in equation (l7) may be used. Differentiating (17) twice wl-th respect to time, and using in 
equation (20a). 


4 u^^S) 


(r.H.S. non-Olmensional) 


The assumption is made that the expansion taking place at the beginning of the second 
oscillation is similar to the contraction taking place at the end of the first oscillation. 


no 



r 

i" 

r 


E 

E 

L 

i: 


I 

I 

I 





I 

$y 

r. 



The pressure P can only be sero when Is i:ero, I.e. *t Hxed values oT Hence 

the duretlsr. of the posit ivu pulse mist be a constant traction at the total period T, lislnj the 
proportionality factor obtained from tho full stop by step calculation for 1 oz. of T.K.T. at 6 feet 
depth yields equation (to) tor the duration D 

P • 0.21BT (10) 

Tht shape of the pressure uave . 

Equation (2l) giving the pressure as a function of bubble radius and the constant m cannot 
be used directly to plot a pressure-tinie curve. A rough idea of the shape of any given pressure - 
time curve may, however, be obtained for any case by calculating from these equations the 'shape 
factor" This is equal to the ratio of the area under the pressure-time curve to the area of 

w 

a triangular wave form having the same peak pressure and duration. The factor thus gives a measure 
of the "hollowness" of the pressure.time curve. 

Values of P|n, I, D and this "shape factor" for a few charge weights and depths have been 
given In Table 1 to demonstrate the effect of these variables on the "shape factor*. In order to 
visualize the meaning of a particular value of the “stvipo factor" a few iiraa I nary pressure wave forms 
have beon sketr..ed In Figure e and their "shape factors" computed. The curve labelled "0. 24" has 
been drawn to resemble closely tho actual prcasure-t Ims curve calculated tor 1 ozi of T.II.T. at 
o feet depth. 


TABLE 1. 


PEAK PRESSURE, OUPATION. IMPULSE AND SHAPE FACTOB . 

(Pressures and Impulses multiplied by the numerical value of the 
distance In feet from the paint considered to the bubblo centre) 


Charge weight 

Depth belovt 
ses level 

(feet) 

Peak pressure 
P^ (Ibs/sq. in.) 

X 

distance R 
^feet) 

Duration 0 
(seconds) 

Impulse 1 
(ibs.sec./sq.ln) 

X 

distance fl 
(feet) 

■Shape Factor’ 
21 
m 

800 lbs. 

ISO 

il,tl50 

0.U3 

5W 

0.84 

• 

100 

7,090 

0.148 

578 

1.10 

II 

eo 

5.350 

0.169 

5^ 

1.32 

258 lbs. 

ISO 

10,600 

0.078 

256 

0.62 

« 

100 

5.000 

O.iOl 

270 

0.78 


50 

2,950 

O.lSl 

292 

lf31 

16 lbs. 

80 

4,130 

0.046 

«3 

0.46 

■ 

SO 

2,750 

0.0 60 

46 

0.56 

M 

30 

1,800 

0.0T5 

48 

0.70 

1 lbs. 

40 

1,750 

0.0 26 

7.4 

0.32 

" 

20 

1.160 

0.034 

7.B 

0.39 

« 

10 

830 

0.041 

8.0 

0.47 

i 02. 

6 

580 

0.018 

1.3 

, ,■ . 

0.24 


The 




) 



Sf 



i 

I 








- I) - 


171 


The rise of the bubbl e - h . 

An important quantity concerneo with the raotlon of the bubble Is the rise et the end of the 
first oscniatlon, since this determines the position of the bubble when the peak presrure in the wave 
is omitted. None cf the above methods of approsimatlcn enables this quantity to be estimated. 
Inspection of the full calculations published for four depths, and of the unpublished calculations 
tor a i oz. charge at 6 feet show that the rise h due to gravity is given by the following empirical 
formula with an error not exceeding 10 par cent over the range of non-dimanslonal depths from 
^0 = 2 ‘0 *0 - 3 


h » 1.05 t« (non-dimensional) 


Using equation (16) this becomes 


h 


1.19 


5 



(non-dimensional ) 


( 12 ) 


This rise of the bubble has been calculated on the assumption that all free or rigid surfaces 
are remote enough to exert no disturbing effect on the motion. Its value is plotted in Figure 0. 
The correction for the proximity of a surface is dealt with In the Appendix. 


Appendix 


- 10 - 


AEL^taix 

;inity of a Free or Sigid Surface, 


Cony«rs Herring has shown that If the cliarge Is exploded at distance d (noivdlmenelonal units) 
from an Infinite free or rigid surface the bubble acquires a veiocUy v tosards that syrfscOj given 
by 


■ ^ / '4 . a “ (||)2 

0 ‘U 


{non-dimensional) 


where the upper sign is tahen for a free surface, the lower for a rigid surface. Since 
usuilly the two most Important surfaces are horizuntal (a.g. the sea surface and the sca-hed) this 
velocity mey be added to the term - ^ In equation (13) for the motion, I e. the gfavlty term. Us 
effect on the minimum radius, the peak vertical velocity and the peak pressure In the bubble collapse 
pressure wsve may be allowed for by re-defIning the momentum constant m whiun -determines them (see 
e.g. equations (7), (s) and (ll)). Hence if m' is the vertical momentum constant for a charge 
detonated d non-dimensional units below the sea surface or above the sea-bed, then 


m- . / aMl--4 a ] dt 


(non-dimensional ) 


using the method which led to the approximate value of m (equations (16) and (5)) gives 

s 

3 ^ 

m‘ • m (l- 0.52 5l! — 2 _) (non-dimensional) (5a) 

d^ 

where the constant 0,S2 Is determined from one full step by step solution of tho equations 
of motion of the bubble (13), 


As regards the effect of the surface on the rise of the bubble Conyers Herring gives a simple 
approximation which may be expressed In the follewing way:- 

ftlse of the bubble at end of first oscillation. In proximity to the sea’s 
surface or the sea bed 


h- e h(l-i 3s-^) 

5 d^ 


(non-dimenslonal) 


where h is the rise in the absence of the surface effect (see equation (l2)). 

Finally Herring has shown that the presence of a surface alters the period of the notion, 
so that If T* is the period in proximity to a surface, and T is the value given by equation (3), 

T‘ » T(lT-^) (25) 

where a Is the average value of the bubble radius (non-dimensional) over a complete 
oscillation the upper sign refers to a free surface, the lower to a rigid one. using the assumptions 
Involved in computing the vertical nomentum constant m (page *), a is a constant fraction of the 
maximum radius a^, so that (28) may bt written 

r = T (It 0.21 ) (3j) 


charge weight 


LENGTH SCALE FACTOR 








DEPTH BELOW 33ft. .ABOVE SEA LEVEL 'io (NON -DIMENStONAL) 


EIG.2 MAXIMUM PADIUS Of BUBBLE 







Qo (non- 











DEPTH OF EXPLOSION - FEET 


FIG S. PEAK PRESSURE FROM FIRST BUBBLE COLLAPSE 
(when remote from frpc of rigid surface) 







1.. I- I I I . H I 


O Q 


oo O O Q9 O 
O o o o 


<n < 


«M — 


i33J Nl 33NViSia A0 OSHdliinW Nl X3s/ 03S 91 

















NON 








FIG. 9. IMAGINARY PRESSURE - TIME CURVES WITH VARIOUS 

“shape factors’ 


! 


- 


I 



RADIAL MOTION OF WATER SURROUNDING A SPHERE 
OF GAS IN RELATION TO PRESSURE WAVES 


E. H. Kennard 

David W. Taylor Model Basin 


American Contribution 


September 1943 


7.C.-u:dfJMIlUL>B9<UDBucaMi9«l 




NOTATION 


183 




1 


! 

i 





/' 

\ 


R 

12 1 or 

D 

"0 


12* or 
V 
y 

p 

r 

t 

T 

To 

P 

Pa 

Po 

Vo 

Pmut 


C 

c' 

Q 

E 

r 

f 


Radius of the spherical cavity 
Value at the peak of compression 

Value at the point where the gas pressure equals the external 
pressure 

Value at the limit of the first expansion 
Volume of the cavity = 4^12^3 

Ratio of specific heat at constant pressure to that at constant 
volume = 1,4 for air, or 1.3 for TNT gas 

Mass density, ordinarily In dynamical or Ips units 

Radial distance from the center of the bubble to a station in 
the water 

Symbol for time in general 

Period or duration of the motion from minimum to minimum of R 
Value of T when R does not depart widely from Ro 
Symbol for pressure in general 
Any pressure expressed in atmospheres 

Hydrostatic pressure; the pressure on the cavity before the ex- 
plosion, or the pressure in the water at a great distance from 
the charge 

Pressure in the gas, assumed to be uniform 
Value at the peak of compression 

Symbol for particle velocity, zero at great distance from the 
charge 

Velocity of the gas, equal to that in the water at the boundary 
Velocity of sound In homogeneous water 
Speed of sound In water containing bubbles 

The whole energy radiated in one cycle of pulsation, concentrated 
In the phase of peak pressure 

Energy of oscillation, represented by the kinetic energy when 
R = Rg, also by the work done against pg (less the small work 
of the gas) In expanding from Rg to Rg 

Impulse or time-integral' of a pressure 

Fraction of the space occupied by bubbles in water 



184 

) 

» c VT/ ; iV** represents the ratio of the adiabatic volume 
elasticity of v;ater to that of the gas in the bubbles when 
under hydrostatic pressure 

Extinction coefficient (the amplitude of a pressure wave de- 
creases by a factor e~^”^ , where e is the Napierian base, 
in going a distance equal to one wave length as measured 
in homogeneous water) 

Frequency times 2n of sinusoidal waves 

Frequency times 2it for free small oscillations of a bubble 
Coefficient of reflection 


N 




it) 

•^0 

K 



I 





PERSONNEL 

This report was written by .Professor E.H. Kennard; the nec- 
essary numerical integrations and the plotting of certain figures were 
done by S. Pines. 



:j 











185 


DIGEST 

This paper summarizes, and in i;he Appendix derives the main formu- 
las concerned with the radial expansion and compression of spherical gas- 
filled cavities in water. The principal needs for these formulas are twofold, 
in connection with the pulsating motion of the gas globe resulting from an 
underwater explosion, and In connection with the behavior of bubbles of gas 
suspended in the water when subjected to changes in external pressure. 

A sphere of gas in water under hydrostatic pressure, not subject to 
the action of gravity, is capable of oscillating radially with preservation 
of its spherical form. The period of oscillation at small am,.litude is 

when Tj is In seconds and in Inches. As the amplitude Increases the pul- 
sation is slower, and the variation within moderate limits is shown in Figure 
1 , At large amplitudes the formula becomes 

The pulsation of the 
cavity may be described as a cyc- 
lic variation of R/R,,, and ex- 
amples of this are given in Figure 
2. ‘These show the main features 
of the motion, including the in- 
crease of intensity of the pres- 
sure peak and the lengthening of 

the period at greater amplitudes. 

The values of at differ- 
ent values of R^^/R^ are shown 

also in Figure 1 . 

The pressure in the wa- 
ter is equal to that in the gas at 
the boundary between them, to the 
static pressure pg at a great dis- 
tance, and at intermediate positions is affected by the flow as shown in 
Equation [?b], page 6; its peak value is given by the formula 



This digest is a uoiiiensatlon of the texi. of the report, containing a description of all essential 
features and giving the principal results. It Is prepared and included for the benefit of those who 
cannot spare the time to read the whole report. 



Plgui-e 1 - Curves referring to Undamped 
Oscillations of a Bubble or Globe 
of Gas under Water 

Kq is the radius when the gas pressure equals tlie 
hydrostatic pressure, iimu the naximum radius. 
Is the mlninuji radius, T, is the period of 
very small osolllations, T is the period of oscil- 
lation having given value of The curves 

are drawn for r = V3, but y makes little difference. 




Figure 2 - Calculated Time-Displacement Curves for Undamped Oscillation 
of a Bubble or Globe of Gas under Water 

R = radius < = time 

= radius when in equilibrivun T# = period of very snail osoiUatlone 

The time relations of pressure variation are also studied in the 
report, and the conclusion is reached that though the peak of first con5>res» 
Sion is lower than the Initial pressure peak, it is still high enough to give 
ri.se to a wave of compression In the water. Since this first compression 
peak Is broader than the initial shock wave. It may carry with It an Impulse 
exceeding that of the high-pressure part of the primary pressure wave. 

The calculations discussed in this report deal mainly with the hy- 
drodynamic phenomena in an incompressible fluid; however, at each pressure 
peak the compressibility of the water enters to play a part, and nergy is 
radiated in a shock wave. Especial interest attaches to the quantity of en- 
ergy lost in this way because it acts in structural targets in a different 
way from that associated with the slower motion of pulsation. No valid meas- 
urement of the energy in the shock wave Is yet available, and in particular 
its value relative to that of the energy of oscillation E is still a matter 
of opinion. 

The analysis thus applied to the pulsations of large gas globes re- 
sulting from explosions also explains the curious effect known to be caused 
by the presence of small bubbles suspended in water traversed by a shock wave. 
It Is shov^n that these may serve as radiating sources of new shock waves 


















Hdfure 8a - Bafora tha Exploalon 


Figure 8b - After the Exploeion 


Figure 8 - Photographs showing a Shock Wave forming in Bubbly Water 

which in turn cause a layer of bubbles to behave as a dispersive absorbing 
medium. The layer thus reflects an acoustic wave in somewhat the same way 
and fcr somewhat the same reason as a layer of molecules reflects light. 

The only case yet amenable to full analytical treatment is a weak 
sinusoidal wave traversing a field containing many bubbles in each cubic wave 
length. The alteration in the speed of sound caused by the presence of the 
bubbles is shown to be proportional to the fraction of the whole space occu- 
pied by the bubbles, and, in a rather intricate way, on the ratio of the fre- 
quency of the wave to that of the bubbles; see Equation [21], page l8. The 
symbols used in this development are separately listed on page 18. 

It is found that even a small concentration of bubbles may produce 
a surprisingly large reflection coefficient, and where the frequency of the 
wave is near that of the bubbles, not over, say, three times as great, re- 
flection is nearly total. At much higher wave frequencies the reflection 
falls away to near zero. 

It is thought that results similar in a qualitative sense will hold 
in the case of an Incident shock wave, except for the unknown results of cav- 


; 



188 


TABLE OF CONTENTS 

Page 

ABSTRACT 1 

INTRODUCTION 1 

PERIOD OF OSCILLATION 2 

TIME-DISPLACEMENT CURVES 4 

PRESSURE IN THE WATER 5 

RADIATION OP ENERQY 9 

A PRESSURE WAVE AND A GAS BUBBLE = , . 11 

SLOWLY VARYING PRESSURE 12 

IMPULSIVE PRESSURE l4 

THE GENERAL CASE . 15 

A PRESSURE WAVE INCIDENT ON BUBBLY WATER 15 

WEAK SINUSOIDAL WAVES IN FINE-GRAINED BUBBLY WATER 1 7 

REFERENCES . . . . ; ' 23 

APPENDIX 

MATHEMATICAL THEORY OP RADIAL MOTION 
AROUND GAS SPHERES IN WATER 

THE RADIAL MOTION • 24 

PRESSURE IN THE WATER 26 

RADIATION OP ENERGY 28 

BUBBLE UNDER VARIABLE EXTERNAL PRESSURE 30 

EFFECT OF MANY SMALL BUBBLES ON LONG PLANE 

SINUSOIDAL WAVES OP A SMALL AMPLITUDE 31 

COEFFICIENT OP REFLECTION 36 

SCATTERING BY A SINGLE BUBBLE 37 


RADIAL MOTION OF WATER SURROUNDING A SPHERE* OP GAS 
IN RELATION TO PRESSURE WAVES 


189 


ABSTRACT 

In the study of explosive pressure waves, the theory of a sphere of 
gas expanding or contracting under water is needed in two connections - in 
discussing the motion of the gas globe produced by the explosive itself, and 
in considering the effect of bubbles in the water upon the propagation of 
pressure waves. The relevant analytical formulas are collected here and dis- 
cussed. Their deduction is given in an appendix. 

The following topics are treated: 

1 . the period and form of the radial oscillations of a gas globe, and 
the pressure and Impulse thereby generated in the water; 

2. the effect of a pressure wave upon a single gas bubble; 

3 . the Inverse effect of a layer of bubbles in water upon an incident 

wave of pressure, which is partially to reflect or scatter the incident wave, 
and to make the transmitted wave weaker but of longer duration; 

4 . an exact treatment for the analytically simple case of weak waves 
of pressure incident upon watei’ containing bubbles of relatively small size; 

5. scattering by a single bubble. 

INTRODUCTION 

In the study of explosive pressure waves, the theory of the expan- 
sion and contraction of a sphere of gas under water enters at two points: 

First, in considering the motion of the gas globe produced by the explosive, 

which results in secondary impulses of pressure; and second, in considering 
the effect of bubbles of gas in the water upon an Incident pressure wave. 
Therefore it is proposed to collect and extend the relevant analytical formu- 
las pertaining to such motion. Only radial motion will be considered here; 
effects due to gravity or to the presence of obstacles will be reserved for 
discussion elsewhere. Furthermore, the assumption will usually be made that 
compression of the water 3urroundir.g the gas globe can be neglected. 

The relevant mathematical analysis has for the most part already 
been published (1) (2) (3),** but it will all be included for convenience In 
an appendix. 

* In this report a distinction is made between the jo» »(o4e formed by the bulk of the gaseous products 

of an imderwater explosion, and gat bubilct. The word tpkere applies to sither or both. 


Humbers in parentheses indicate references on page 23 of this report. 




190 



PERIOD OP OSCILLATION 

A sphere of gas In water zander hydrostatic pressure p^, not subject 
to the action of gravity. Is capable of oscillating radially with preserva- 
tion of its spherical form. Let the gas be assumed to follow the adiabatic 
law, pV'^ = constant, an assun^tlon that appears to hold well In practical 
cases. Then, for a small amplitude of oscillation, the period Is given by 
Equation [3^] or 



where is the radius of the sphere when In equilibrium under hydrostatic 
pressure (atmospheric pressure included), p Is the density of water, and y 
is the ratio of the specific heat of the gas under constant pressure to Its 
specific heat under constant volume. Por air, y -= 1.4 and the formula can be 
written 

^0 “ill TFT I2a] 

where p^ Is the pressure In atmospheres,* and Is In Inches. For the gas 
globe from an exploded charge, y = 1 .3 more nearly, and 

^0 “ m second [2b] 

In sea water would be 1 .3 per cent greater at the same and p^. 

The Value of J?o for gas globes from charges exploded under water is 
uncertain. Perhaps Rg = R^/2.6 is not far from the truth, where i?™. Is 
the inaxiraum radius. A fair estimate for tetryl Is 

Inches 

where W Is the weight of the charge in pounds and p^ is the hydrostatic pres- 
sure in atmospheres; the value for TfW should not be greatly different. With 
this value of Rg, Equation [2b] becomes 

T, = 0.15-^ [2c] 

With increasing amplitude the period increases; it may be written 

T ^ kTg [3] 

where fe is a dimensionless factor. In Figure 1 the factor k or T/Tg is plot- 
ted against R^^^^/Rg, In the same figure there Is shown, for convenience, the 

The period under on© atiaosphere is thus i?^/129 secona. 





3 


191 


ratio of the minimum radius, , 
to Bq, The values of k were ob- 
tained by numerical integration of 
Equation [35] for several values 
C'; R^/R^ and /R^ were 
found as values of a, and a, from 
Equation [ 36 ] . The value y = 4/3 
was used, in order to simplify the 
calculation; a somewhat different 
value would give nearly the same 
ciujve. 

For large amplitudes, 
perhaps where R^/R^ Is greater 
than 2.25, the formula given in 
Equation [22] on page 48 of THE 
Report 48o (4) may be used 

T “ 1.83 R^ l/^ [4a] 

For a gas globe or bubble in water this may be written 

r - ^ pL second [4b] 

where is the hydrostatic pressure in atmospheres and R^ is in inches. 

For a given mass of gas, hence T is proportional to In 

sea water, 217 Is replaced in Equation [4b] by 214. If use is made of the 
value Just given for Equation [4b] becomes 

r-o.zs-^ [4c] 

Pa* 

These latter formulas may be used to estimate the time of collapse 
of a bubble under suddenly applied steady pressure. If R^ represents the 
initial radius of the bubble and or the suddenly applied pressure, the 
time of collapse is 5'/2, The estimate should be of high accuracy if the ra- 
tio of pressure increase exceeds 2.25* or about 25. 

If the aii5)lltude R„„/Rf is very large, coa^iresBibllity of the wa- 
ter will play an important part. The direct effect of compressibility cjj the 
period will be small, since the high-pressure phase of the motion occupies 
only a very small part of the total period; but a loss of energy occurs by 
acoustic radiation diurlng the time of intense contraction, so that each out- 
ward swing is less in amplitude than the precedlrsg Inward swing. The period 



Figure 1 - Curves referring to Undan$»ed 
Oscillations of a Bubble or Globe 
of Gas under Water 

it. Is ths radius when ths gas prasaure equals the 
hydrostatic prassture, Rm»x the Daxloua radius, 
^mln the minlnun radius, 7*, is the period of 
very sosU oscillations, T is tbs period of oscil- 
lation hnving given value of Aniax/^*’ curves 
are drawn for y a i/3, but y nakes little difference. 







• ■ -H 


between two inlnirauin radii should then be given quite accurately by the formu- 
las if the intervening maxlraiun radius Is used, whereas the interval of time 
between two successive maxima should be the average of the periods as given 
by the formulas for the two successive maximum radii. 

The formulas for the period have been derived from hydrodynamical 
theory but appear to have been confirmed satisfactorily by observation. No 
allowance has been made for the effect of the displacement of a gas globe due 
to gravity, but this effect should be large only under extreme conditions. 

TIME-DISPLACEMENT CURVES 

A number of curves are drawn in Figure 2 which show for several am- 
plitudes, the value of R/Bo during an oscillation as a function of the time. 
The unit for time is the period of small oscillation, Tq ; thus the curves are 
valid for any value of R^. They refer to the case y = 4/3, which is the eas- 
iest to calculate. For air, however, the curves would differ so little that 
it is not worth while to attempt to illustrate the difference. The curves 
were constructed by integrating Equation [32] numerically. As with the peri- 
od, no allowance has been made for gravitational displacement of a gas globe. 



- 1.0 -o.e - 0.6 


Figure 2 - Calculated Time-Displacement Curves for Undanped Oscillation 
of a Bubble or Globe of Gas under Water 

X = radius t ~ time 

= radius whsn in equilibrlvut T, = period of very snail OEClllatloUB 





5 


193 


The maximum radius and the minimum radius are connected 
by the equation 



which for large amplitudes can be shortened approximately to 


I5a] 


^0 I ^0 


[5b] 


see Equation [30], with y = 4/3. 

It Is noteworthy that, as the amplitude Increases, the time- 
displacement curve, which approximates to a sine curve at small amplitudes, 
becomes more and more pointed near the minimum radius. Thus the sphere 
spends very little time at radii below the equilibrium radius JRg when the am- 
plitude is large. This effect arises physically from the diminution in the 
area across which the Inrushlng water Is moving; because of this diminution 
the water has a strong tendency to Increase In velocity, and hence the gas 
meets great difficulty In stopping the motion. 

The curves are calculated on the assianption of Incompressible water. 
For this reason the Incoming and outgoing motions as shown In Figure 2 are 
similar. When the minimum radius becomes extremely small relatively to iZ^, 
however, compresslou of the water begins to play a role, as already stated; 
consequently, the amplitudes of successive oscillations will progressively 
decrease. Each loop of the actual curve, extending from one minimum to tie 
next will be very nearly the same as It would be, at the same maximum radius, 
for Incompressible water. 


PRESSURE IR THE WATER 

Let the pressure in the water at great distances be the hydrostatic 
pressure p,; and let gravity be assumed not to act. At the surface of the 
sphere of gas, the pressure In the water must bo that of the gas or 


P 



[ 6 ] 


by Equation [26], where R Is the instantaneous radius of the sphere. At any 
other point, at a distance r from the center of the sphere, the pressure, if 
the motion is non-con^^ressive, Is found to be 


P = T + 2 - Po) - + p. [7a] 


I 


\ 

/ 


194 


6 


or, for V = 4/3. 


R 


{[(f 


+ i] 

3J \rI 


3J 2 


P»® + Po 


[7b] 


where = fi„j„ , the minimum radius, v Is the particle velocity at the point 
in question, and Vg is the velocity of the surface of the sphere, which is 
also equal to dR/di; see Equations [38] and [39a]* At considerable distances 
the Bernouilli term 1/2 />«* nay be dropped. 

Tlie maximum pressure at any point occurs at the instant at which 
the radius of the sphere is a minimum, without any time delay, in the approx- 
imation in which compression of the water is neglected. At this Instant both 
Vg and V vanish; hence, from Equations [6] and [?a], the maxlinian pressure is 

Pm« " ^ (P, - Po> + Po 





or 


p™. -P.-^ [{£;)” -i]+r. I8a) 

where is the minimum radius of the bubble. For y « 4 / 3 , 

Pm« “ Po ~ + Po 

These formulas should be applicable to the pressure in the water 
that is associated with the oscillations of explosive gas globes. As an ex- 
ample, if =2.6, which appears to represent fairly well the first 

outswing for a Number 8 detonator when p, = 15 pounds per square inch, and 
when is about 5 inches, then i2„,„/jBo= 0 . 1 6 , and at a distance r = l8 
inches from the center of the explosion 

Pn«s " Po “ 15 - l] - 400 pounds per square Inch ) 

However, the pressure varies extremely rapidly near its maxlimm 
value when the amplitude of oscillation is large. Thus a concentrated pulse 
of pressure is emitted d\a>ing the phase of extreme contraction of the globe, 
whereas during most of the time the increment of pressure due to the motion 
is small. In Equations [43] and [37] the following formulas are obtained 
connecting the prsssiire p at a distance r from the center with the radius R 
of the sphere of gas and the time t, in the neighborhood of the time ti at 
which i? = i? 1 = iZ n,i„ , when y = 4/3 ; 

P * (P»«- Po) - I P”* + Po 


[9] 


T 


195 


t - t, 


V2 

i5^ 




[10a] 


For the first contraction of an explosive gas globe, these formulas should 
hold well at least up to = flj/2, and the error in Equation (93 should not 
exceed 5 per cent. The significance of Equation [10a] nay appear more clear- 
ly If it is written 



in terms of a dimensionless factor G, For a decrease of the pressure to half 
of its maximum value, G = 0.4; for a decrease to one quarter, G = 0.8. In 
the example Just described, referring to a Number 8 detonator, where R „„ /Rn 
2.6, Tj « 1/65 second, /itg = O.16, and the two values of t - tj are about 
0.025 and O0O5 millisecond, respectively, the pressure curve is symmetrical 
about its maximum, and the entire time taken from p = p„^A through and 
back to Piatx/4 is about 0.1 millisecond. These results indicate that the 
pressure pulse emitted during the first con?>ression peak should be broader 
than the primary pulse due to the explosion itself, in which the pressure 
should decrease to a quarter of its Initial maximum in less than 0.02 milli- 
second. For 1 ounce of TNT or tetryl, the time from Pm„A Pm.xA in l^he 
pulse due to the first compression peak might be 0.4 millisecond; for 500 
pounds at a depth of 50 feet, 5 milliseconds. 

The total impulse or jp dt in the second pulse, on the other hand, 
may be relatively large. The Impulse from the time of minimum radius up 
to any other time t, when the term p»V2 is negligible. Is found to be, for 
r - 4/3, 


lip r + [111 


where i?i,2 >»3y 4e taken to stand either in both places for R^ - R , the 
minimum radius, or in both places for IZj > maximum radius; see 

Equation [44a]. 

If, in Equation (11), R^^^ " ^m»* ®nd also R « R^, then / = 0. 
This shows that the Impulse during a complete swing is zero, the negative 
part cancels the positive part. The negative Impulse arises from extremely 
small negative pressures, however, and Is for this reason unimportant. The 
positive part may be obtained separately as the maximum value of I as given 
by Equation [11]. The positive impulse emitted during an entire conpression 
and re-expansion when y = 4/5 is thus found to bs 


196 


8 


2 / 4 - = 


2/2 


Po^o 


R, 


■{(!)■ 


see Equation [45]. 

In the example of a Number 8 detonator, where jR, = 2 Inches, 

2.6, Rj/Rf, = 0.l6, Tp * 0.015 second, p<, = 15 pounds per square inch, at r = 
l8 inches from the center of the detonator. Equation [12] gives, for the total 
positive Impulse due to the first compression and re-expansion, 0.059 pound- 
second per square inch. The part of this that arises from the central peak, 
in which the pressure exceeds a quarter of the maximum pressure, as found by 
substituting R = 2Ri In Equation [11] and multiplying I by 2, is about 0.024 
pound-second per square inch. This accounts for rather less than half of the 
total. Even so, it probably exceeds the Impulse due to the high-pressure part 
of the primary pressure-wave, which should not exceed 0.02 pound-second per 
square inch . 

In Figure 3 the pressure p is shown as a function of the time t, 
for a contraction from a itiaxlmum radius = 2.5 Rf and a subsequent re- 
expansion. The ordinates represent values of p/p^ ; the maximum pressure 
is given by Equation [8b]. The abscissa represents t/T, where T is the 
period of oscillation of the gas globe. Part of the curve is repeated on an 
expanded scale. The curve is independent of the quantity of the gas, which 
deberaines the value of R,. 



f/T for Curvo A 


Figure 3 - Curve showing the Calculated Pressiu’e p developed in the Water 
during the Oscillation of a Bubble or Gas Globe under Water, 
in Terms of the Jfexlmua Pressure p„„ 

Acoustic radiation of eneror is lgnored« Tisa t is plcttsd in terms of the period 
of oscillation T» Part of the curve is repeated on an enlarged tine scale* 






In this discussion no accovait has been taken of the effects of 
acoustic radiation of energy. This radiation will cause each expansion to be 
somewhat less energetic than the preceding contractions so that the emitted 
pressure and Impulse will be somewhat less. No attempt will be made here to 
develop a more accurate theory of this phenomenon, but the total radiation of 
energy associated with compressibility In the water can be estimated roughly. 

RADIATION OP ENERGY 

At considerable distances from the center of the gas sphere, where 
pw* is negligible, the pressure as given by Equation [7a] or [7b] falls off 
with increasing r according to the same law that holds for spherical waves. 
This observation leads to the surmise that moderate cotapresslon of the water 
will not greatly alter the magnitude of the pressure p at any distant point 
but will introduce the following feattu?es as characteristics of spherical 
waves in contrast to non-compressive motion: 

1 . a time lag corresponding to the finite speed of propagation of 
sound waves, and 

2. a congjonent p/pe in the particle velocity, added to the velocity 
as derived from non-compressive theory. 

This surmise is confirmed for small anplltudes of oscillation by acoustic 
theory. 

In order to form a rough estimate of the energy radiated, therefore, 
the pressure as derived from non-compressive theory may be combined with the 
acoustic formula for the energy that is carried off to infinity, in spite of 
the fact that a strict use of non-compressive theory leads to no loss of en- 
ergy to infinity at all, in acoustic theory, the emission of radiation re- 
sults from the component p/pc in the particle velocity and amounts to p^/pc 
per vinit area per second. Hence, to find the total amount of radiation, it 
is only necessary to integrate p^/pe twice, first over a large spherical sur- 
face drawn about the gas sphere, and then with respect to the time. Further- 
more, the pressure falls so rapidly from its maximum value that the emission 
of energy occurs almost entirely while the pressure is in the neighborhood of 
its maximum, or while the sphere is near its ralniraum radius; hence a good ap- 
proximate value can be obtained by using an approximate value for the pressiu’e 
that holds near Its maximum. 

The amount of energy radiated per cycle by a sphere of gas for which 
y ■> 4/3 is thus found to be. In the notation already employed, 

o= [14] 

® ^0 '"mm' 



198 


10 


■where c denotes the velocity of sound in water; see Equation [*^8]. To make 
this formula approxlRstely correct, the amplitude of oscillation must be 
large enough to make the peak of emitted pressure a sharp one, but not so 
large that great compression of the water occurs; the range of Its validity 
may be something like 


< o. 1.6 < 


< 2.76 


A more interesting quantity is the dimensionless ratio of the 
energy emitted in a cycle to the total energy of vibration, E. The excess 
energy that is present as a result of the oscillatory motion is the same as 
the kinetic energy of the water at the instant at which R «= since, if 
this energy were suddenly removed at that instant, the sphere would remain in 
equilibrium. As the gas expands to maximum radius, this kinetic energy is 
expended In doing work against the difference between the hydrostatic pres- 
sure and the pressure of the gas, and the latter work is readily calculated.* 
In this way the energy of vibration is found to be 

£ - + 1©’- 1] = v[© + 1{0 - 1] n5i 

provided y= 4/5; compare Equation (49a ]. Thus for the first cycle 

7T* ^ fRoX^ 

a _ W cT„\rJ g 

iJj 3 \Ro> 3 

or, after inserting c = 48lO x 12 inches per second and using Equation [2b], 


' 4- ^ _ i 


[l6a] 


where is the hydrostatic pressure measured in atmospheres. 

Measurements of the radiated energy qre not available, but a com- 
parison may be made between the calculated loss by radiation and the total 
observed loss of oscillatory energy, which is easily found from the progres- 
sive decrease in the maximum radius for successive oscillations. Prom Equa- 
tion [ 15 ], the change in energy is 


* The water is driven, so to speak, by two springs, the gas inside and the hydrostatic pressure outside. 
As it oscillates, one spring loses energy while the other gains energy; the excess of the gain by one 
spring over the loss by the other, as the radius changes frOio its squilibriua value S* to a value R, 
represents the potential energy of vibration. 


i 




n 


199 


4 » 


or, if the relatively sraall term Sq/S^ is dropped, 

AE^A(lnp,Rfl 


[ 17 ] 


[17a] 


This last formula represents the change in the energy as approximately equal 
to the change in the work required to produce the cavity of maximum size, 
which can be calculated without making any assumption concerning the equilib- 
rium size of the gas globe. 

For the gas globe formed by Number 8 detonators exploded just far- 
enough under the surface of the water to avoid blowing through, an average 
value for the first expansion, as inferred from the periods of oscillation, 
seems to be about R ^ /Rn = 2.6. This corresponds, by Eauatlon [5a], to 
For this case, by Equation [l6a], ^ 0.74. 

The observed decrease in energy during the first contraction, calculated in 
the manner Just described, is about 40 per cent. 

The discrepancy between 0.74 and 0.40 la in the right direction and 
may well be due to compression of the water. At minimum radius, Equation [6] 
makes p, equal to 6.Z* = 1500 atmospheres, which would compress the water by 
about 7 P®r cent. An attempt to estimate the amount of compresslonal energy 
that would exist in the water leads, however, to a divergent integral, which 
merely Indicates that the non-compressive approximation to p is inadequate 
for the purpose. It is clear, however, that, if the gas at minimum radius 
absorbs only part of the energy of motion, its minimum radius will be greater 
than it has been calculated to be on the assun 5 >tlon that the gas takes up the 
whole energy, and the pressure peak will accordingly be lower and will result 
in a considerably smaller radiant emission of energy. Thus the true value of 
Sl/E a&y easily be 0.40 instead of 0.74. 

The estimate of the radiated energy has been based, as have all of 
the preceding formulas, upon the assumption of perfectly symmetrical radial 
motion. Further losses of the energy of the radial motion may result in ac- 
tual cases from turbulence caused by departures from radial symmetry, or from 
conversion of the oscillatory energy into energy of translation due to grav- 
ity or to the proximity of obstacles. 


A PRESSURE WAVE AND A GAS BUBBLE 

It is of special interest to investigate the propagation of explo- 
sive pressure waves through water containing bubbles of air, since a screen 
of bubbles has been proposed as a protective device. Let it be assumed that 




200 


12 


the wave begins with a very steep front behind which the pressure falls off, 
and that the bubbles ahead of the wave are in equilibrium under hydrostatic 
pressure. The behavior of a bubble under such a wave will vary according to 
circumstances. Special cases of this phenomenon will now be considered. 


SLOWLY VARYING PRESSIJRE 

Suppose the pressure falls off slowly as compared with the time of 
contraction of the bubble under the maximum pressure in the wave. This 
should be the case when the shock wave from a large charge enters water con- 
taining small bubbles. In this case 
the pressure on the bubble is prac- 
tically steady during the process of 
compression. 

Estimates of the rapidity 
of heat exchange between the gas in 
the bubble and the water Indicate 
that the gas should follow the adia- 
batic law, pV’’® constant, since 
V « R*, the new equlllbrluja radius 
under pressure p will be 

R = [l8] 

where S, is the radius under hydrostatic pressure p^. For air, y ® 1 .4 and 



Figure 4 - Curve illustrating Behavior 
of a Bubble under a Slowly 
Varying Pressure Wave 


Thus the equilibrium radius changes but slowly in comparison with the pres- 
sure. If p/po = 200, corresponding to a rise of pressure from atmospheric to 
3000 pounds per square inch, jB/Kq = 1/3.5; if p/Po ® corresponding to 
6000 pounds per square inch, R/R^ ® 1/4.2. 

The time required for a bubble of initial radius to contract 
when the pressure is suddenly raised to a high value and then held steady may 
be estimated as half of T as given by Equation [4a] or [4b] with B„„ re- 
placed by Bq. For bubbles in water, 


1 0.0023gn 

2 vp: 


[19] 


where Bo is in inches and p^ is the applied pressure in atmospheres. Thus, 
even if Bo is as large as 0.1 inch and the pressure no greater than 150 
pounds per square inch, so that p^ = 10, the bubble collapses in less than 


13 


201 


1/12 millisecond, which is a short time relative to the duration of the pres- 
surs wave from a large charge. If Rq - 0.1 inch and p ^ 1000 pounds or = 
67 , T/2 Is less than l/30 millisecond, which is a short time even for the 
wave from a pound of explosive. Smaller bubbles will collapse more quickly 
in proportion to their smaller radius. 

In collapsing, the bubble will overshoot its new position of equi- 
librium under the increased pressure, and will then re-expand. If the bubble 
lost no energy, and if the pressure remained constant, then the bubble would 
actually expand to Its initial size, after which it would collapse again; It 
I*" would, in fact, execute undamped oscillations about its equilibrium radius 

Under the Increased pressure. The bubble would be analogous to a mass on the 
end of a spring; if, when the mass is at rest, a constant force suddenly be- 
gins to act on it, the mass oscillates about a new position of equilibrium 
and, in doing so, returns periodically to its Initial position. 

The period of the oscillations will be much shorter, however, than 
those which the same bubble would execute under normal pressure. Under a 
pressure of 1000 pounds per square inch, for example, the equilibrium radius 

Is reduced from its value under one atmosphere in the ratio 

The period of oscillation, which is proportional by Equation [1] both to p^~i 

and to the equilibrium radius, would then be 2.7 “ 22 times less 

than under one atmosphere. Under 3000 poimds per square inch, the period 
would be 50 times less than under one atmosphere, for the same relative am- 
plitude of oscillation. 

Con?)ression of the water cannot be Ignored in these cases. Calcu- 
lation of the pressure in the bubble when at its minimum radius gives fan- 
tastically high values. This means that, because of couipression of the water, 
the minimum radius will actually be several times larger, and the maximum 
pressure many times smaller, than the values derived from non-coBjpresslve 
^ 5 theory. Furthermore, it is certain that much of the kinetic energy acquired 

by the bubble as It contracts will be radiated away during the phase of ex- 
treme compression. The oscillations of the bubble about its new equilibrium 
radius will thus be highly damped. 

An upper limit can easily be set to the amount of energy that can 
be radiated away by such a bubble in collapsing. The total work done by the 
applied pressure as the bubble collapses is equal to the product of the pres- 
sure itjto the change of volume of the bubble. With any explosive wave of 
consequence, however, the final volume is relatively small. For example, 
even if p is only 300 pounds per square inch as against an initial pressure 
( } jjj = 13 , from the relation ratio of the corresponding volumes 




202 


14 




la, for air, V/V^* (Pa/p)^^'^ = 1/8-5* Hence, the work can be 

calculated from the initial volume only and is nearly equal to 

where B(, is the initial radius. 

It may now be asserted that the radiated energy cannot exceed W 
It may well be almost equal to W, however. For the part of W that la spent 
in compressing the gas is approximately, or exactly if y * 4/5, equal to 




47TP,JJ/ - i7TP, Ro^{^ - l) = inpR,^ [(W - 


P' P 

by Equation [27a]. This is less than if P is greater than 20 p„. Loss of 
energy due to friction should also be small, unless departures from symmetry 
cause appreciable turbulence. 

After the bubble has settled into its new position of equilibrium, 
it may contract somewhat further as it loses heat of compression, and as the 
gas dissolves in the water. If the pressure slowly decreases, the bubble 
will re-expand without executing sarked oscillations. 


IMPULSIVE PRESSURE 

At the opposite extreme from the case of steady pressure stands the 
impulsive case. Let the pressure be applied suddenly and let it disappear 
again before the bubble has had time to change appreciably in size. Then the 

bubble will begin contracting at a 
certain inward radial velocity vq. 

If compressibility of the water can 
be neglected, the analysis gives 

J_ 

pRo 

where p Is the density of water and 
I Jpdt, the applied impulse; see 
Equation [51]. If / is in pound- 
seconds per square inch and S© in 
inches, 



v„== - 


[20] 


Figure 5 - Curve illustrating Behavior 
of a Bubble under a Pressure Wave 
of Very Short Duration 


10,700 -jj- Inches per second 
«u 


[20a] 


It is clear from this formula that enormous velocities are easily 
produced, while the Inertia effect on the bubble motion is relatively small. 
From a Number 8 detonator at l8 inches, for examnle, the shock-wave Impulse 


\ 

, 1 ' 



15 


203 


may be about 0.013 pound-second, so that, even if Rq is as large as 0.1 inch, 
Vj according to Equation [20a] equals l400 Inches per second. Moving at this 
velocity, the bubble would shrink to nothing in l/l4 millisecond. Since the 
duration of the shock wave scarcely exceeds I/50 rsilllsecond, the value ob- 
tained for Vo for of collapse should be roughly correct. 

For heavier charges, however, this analysis ceases to apply. Thus 
for the same bubble at a distance of l8 inches from 1 ounce of TNT or tetryl, 
I = 0.2 pound-second and Vg »= 21,000 Inches per second; at this velocity the 
bubble would collapse in 1/210 millisecond, whereas the shock wave lasts per- 
haps 1/10 millisecond, in such cases a bettex' estimate of the time of col- 
lapse is obtained from Equation [191 • If in this equation Rq = 0.1 inch, 

« 4000/i 4.7, representing a peak pressure of 4000 pounds per square Inch, 
the value T/2 = I/72 millisecond is obtained for the time of collapse. Even 
this latter value la probably considerably in error, but it serves to confirm 
the conclusion that the bubble will collapse long before the shock wave has 
disappeared. 

After collapsing, the bubble will re-expand. If the time of col- 
lapse exceeds the duration of the shock wave, so that the expansion occurs 
under the original low pressure, the bubble may overshoot its original size. 
The time required to reach the original dimensions may be of the same order 
as the time of collapse; for the shortening of the time that results from the 
loss of energy by radiation will be offset somewhat by a lengthening due to 
the fact that the expansion occurs against a lower pressure. 

THE GENERAL CASE 

Between the two simple cases of relatively steady pressure and of 
ln«)ul8lve action there lies an intermediate range in which analytical treat- 
ment is laborious. Qualitatively, light can be thrown upon these situations 
with the aid of estimates based on the formulas pertaining to the simple ex- 
tremes, but quantitative results can be obtained only by numerical integra- 
tion. 

The foregoing discussion of the effect of a pressure wave on a 
single bubble may now be followed by consideration of cases involving more 
than one bubble. 

A PRESSURE WAVE INCIDENT OK BUBBLY WATER 

A problem of great interest is that of a plane wave of pressure 
entering at normal incidence a layer of water containing bubbles of air or 
other gas. 

The principal qualitative features of the effect of the bubbles 
upon the pressure wave are easily inferred. 


204 


16 


The bubbles make the water efi''ectively much more compressible, 
hence the velocity of propagation will be greatly reduced. An isolated pulse 
of pressure may, for this reason, be retarded in its passage through the bub- 
bly water. 

Furthermore, If there la a definite boundary between the homogene- 
ous and the bubbly water, partial reflection of the wave nay be expected at 
the boundary. The reflection may, however, be reduced in amount by the oc- 
currence of cavitation at the boundary of the layer of bubbly water. The 
wave reflected from the first surface of this layer will necessarily be one 
of tension, since the bubbles reduce the acoustic impedance of the water. If 
the v;ater cannot stand the requisite tension, cavitation will occur in the 
homogeneous water, and in this case the reflected tension wave will be partly 
or wholly absent. A layer of cavltated vjater should then advance against the 
bubbly water, and subsequently move back again; the ln?)act of this layer 
against other water may give rise to a secondary reflected wave of positive 
pressure. 

It would be expected that high-pressure waves would be less effec- 
tively reflected than low-pressure waves. For, if the pressvupe is great 
enough to cause the bubbles to collapse almost completely, further increase 
of pressure will not cause materially greater amplitude of motion of the bub- 
bles, so that the reflecting action cannot increase in proportion to the in- 
cident pressure. 

Additional conplications, perhaps resembling resonance effects, may 
result from the inertia of the water surrounding the bubbles. Furthermore, 
loss of energy due to scattering of the wave by the oscillating bubbles, or 
to other causes, will result in a weakening of the wave. 

Because of these effects, the bubbly water will behave as a disper- 
sive, absorbing medium. The dispersive action, signifying that the various 
harmonic components of the wave travel at different speeds, will cause the 
wave to increase in length as it passes through the bubbly layer. If the du- 
ration of the wave is short enough relative to the time of vibration of the 
bubbles, the lengthening may be so great that it is best described as a re- 
emission of pressure by the cooqpressed bubbles as they expand again. 

The wave of pressure that emerges on the far side of the layer of 
bubbles will thus be likely to be weaker but of longer duration than the 
original Incident wave. There are, furthermore, other effects that lengthen 
the transmitted wave. Repeated reflections from the boundaries of the layer 
may occur. A single entering pulse may thus emerge as a series of repeated 
pulses of rapidly diminishing amplitude, which will blend together more or 
less completely into a transmitted wave of increased length. 


) 





17 


205 


Then, .finally, there are the wavelets scattered in all directions 
by the bubbles. In part, effects of these scattered wavelets have already 
been taken into account, for they actually constitute the physical mechanism 
by which the Incident wave is weakened and partly reflected. But the scat- 
tered wavelets will also appear independently as an additional wave of pres- 
sure scattered in all directions. In a similar way the waves of light 
scattered by the molecules of the atmosphere, which, on the one hand, cause 
a refraction and a weakening of the sun's rays, also appear Independently as 
the blue light that comes from the sky. Scattered wavelets coming from more 
and more distant parts of the bubbly layer may prolong the transmitted wave 
as observed in regions beyond the layer. 

The momentum carried by the waves, on the other hand, should be re- 
duced only if a reflected wave of tension occurs. Such a reflected wave may 
carry back a large part of the incident momentum. If, however, the reflec- 
tion is prevented by the oceui'rence of cavitation, all of the incident mo- 
mentum must appear somehow in the transmitted wave. 

The transmitted momentum might, as a matter of fact, exceed the in- 
cident momentum. In such a case the conservation of momentum might be pre- 
served in either of two ways. Partial reflection from the farther side of 
the bubbly layer, occurring .in the medium of lesser acoustic linpedance, may 
cause momentum reversed in direction to be carried back toward the source of 
the waves. Or, momentum of the same sort may be carried back by the rear 
halves of wavelets scattered off in all directions. 

Several features corresponding to those Just described have been 
observed in experiments at the David W. Taylor Model Basin, which are to be 
described In other reports. 

The quantitative treatment of these phenomena, unfortunately, en-. 
counters great difficulty, as does any problem in highly non-linear wave 
motion. The analysis can be effected readily, in fact, only for the extreme- 
ly simple case of very weak waves of sinusoidal form, passing through water 
that contains many bubbles in each cubic wave length. After this case has 
been solved, a weak wave of arbitrary form can be treated, if desired, by 
means of Fourier analysis. Although devoid of direct bearing on the topic 
of explosion waves, the analytical results for weak waves may be sxiggestlve 
enough to be quoted here. Their deduction is given in the Appendix. 

WEAK SINUSOIDAL WAVES IN FINE-GRAINED BUBBLY WATER 

Let the following assumptions be made: 

1 . The pressure is so weak that linear acoustic theory can be applied. 
This ln?3lles that the bubbles change size only slightly as the waves pass. 



206 


18 


2. The spacing of the bubbles is large relatively to their own diam- 
eter, but yet small relative to the wave length of the waves in the bubbly 
watei’. Let 

c be the speed of sound in homogeneous vjater, 
c’ be the speed of sound in the bubbly water, 

be the extinction coefficient, with the significance that 
the amplitude of the pressure decreases by a factor e 
as the wave traverses in the bubbly water a distance equal 
to Its wave length in homogeneous water, 

/ be the fraction of space occupied by the gas in the bubbles, 

Rfl be the equilibrium radius of the bubbles, assumed the same 
for all, 

w = 2tt/T, where T is the period of the waves. 

Wo a 2n/To, where To is the natural period of small radial 
oscillation of a bubble, 

jyf g Here JV* represents the ratio of pc*, the volume 

elasticity of water, to ypo, the volume elasticity of the 
gas contained in the bubbles. 

Then, according to Equations 1 66 ] and [6?], the analytical treat- 
ment yields the equations: 


-.2 


- = 1 + fN^ 


1 - 

Wo 


/ 3 w® 


121 ] 


2P 


c' 


/3 /N 


(J 




' Wo*l ' N^- Wo® 


[ 22 ] 


At very low frequency, ^ = 0 approxl«iately and 


c = 


YTTJW 


[25] 


Since, as in Equation [l], ^ the quantity N is in reality 

independent of the size of the bubbles. For air in wa.ter at atmospheric 


prsssur 


LU VllW - - 

Where o = 58,000 Inches per second and by Equation [2a], Wj = 


\n/To = 27T X l29/«op ^ = 58,000 l^/258»r= 124. 


19 


2n>7 

■MW « 


The value of N is so large that at very low frequencies, in the ab- 
sence of all resonance effects, a ainall ajsount of air cauec-a a large decrease 
in the wave velocity. Thus if / - 0.1 per cent, c = c/Vl + 124 x 0.1?4 = 
0.25 c; if / = 1 per cent, c' = O.08 c. 

The coefficient of reflection, or fraction of the Incident energy 
that is reflected, is given by Equation [70] or 




At very low frequencies this becomes, approximately, 


fr = 


[l-f] 

2 

V\ + /AT* - 1 



V\ + fN^ + 1 


[24] 


•[25] 


from Equation [23], a formula that Is easily obtained from a much simpler 
calculation. The latter formula gives, for / = 0.1 per cent of air, 

« - (f)‘. 0.36 

and for 1 per cent of air, 

11 4 ® 

Curves are shown in Figures 6 and 7 for 0.1 per cent of air in 
water, or for iV= 124 and / » 0.001. In Figure 6 the ratio of velocities 



Figure 6 - Refractive Index relative to 
Homogeneous Water, c/c', and Extinction 
Coefficient p for Sinusoidal Waves in 
Water containing 0.1 per cent 
of Air in Fine Bubbles 

e' = nV8 speed 

e = eave speed In honogeneous water 

w = 2>r tioes wave frequency 

Up = 2ir tines natural frequency of radial 
oscillation of the bubbles 

- eictlnetlon coefficient 

Pressure decreases by factor as wave 

progresses a distance X = 2nc/u. 



208 


20 


c/c', or refractive index of bubbly 
water relative to homogeneous v/ater, 
and the quantity ^ are plotted 
against w/wq, which represents the 
ratio of the wave frequency to the 
natural frequency of the bubbles. In 
Figure 7 18 plotted the reflection 
coefficient K of the bubbly water. 

The curves are valid for any bubble 
size that is not too large; the size 
of the bubbles determines Wq. 

A strong resonance effect 
is brought into evidence by these 
curves. Especially striking is the 
persistence of this effect as u> in- 
creases above Wq. The decrease in 
wave speed c ' that is caused by the 
bubbles at low frequencies is re- 
placed, as w begins appreciably to 
exceed by an increase in wave speed; for /= 0.001 and w = 2.5a>o» c' = 

22 c, and even at = 5«o, c' » 1.7 c. Furthermore, the scattering, which la 
proportional to shows a strong persistence at values of w/«o up to 2 or >. 
As a consequence, there is a strong band of nearly total reflection from 
w = Wo to w = 3wo« Above w = 5^0. ^^e other hand, reflection becomes In- 

appreciable; at such frequencies, the inertia of the bubbles prevents them 
from following the vibrations of the incident wave to any considerable degree. 

If w is increased to very high values, however, a point is ulti- 



Pigure 7 - Reflection Coefficient K 

K Is the fraction of the incident energy that 
is reflected, for the bnbhly water 
to wMch Figure 6 refers. 


mately reached at which the assumptions underlying the analysis no longer 
apply, because the incident wave length is no longer large as compared with 
the spacing of the bubbles. 

Observations have been reported on the scattering of sound by bub- 


bly water, but they do not seem to lend themselves to a test of these equa- 
tions. The analytical results may be employed, however, to throw some light 
upon the effect to be expected when a shock wave enters bubbly water. A 
photograph of this phenomenon is shown in Figure 8. 

If the effective length of the shock wave is relatively great, or 
at least not less than a third as great as the wave length corresponding to 
the average natural frequency of the bubbles {w/a»o < 3), then it may be con- 
cluded with safety that the reflection will exceed the value given by 









209 



ngura 8a - Before the Btploalon Flgtire 8b - After the Explosion 

Figure 8 - Photographs showing a Shock Wave forming in Bubbly Water 


Equation [25]. In this statement, the wave length as it exists In homogene- 
ous water is meant. If the effective length of the shock wave is actually 
comparable with the bubble wave length, the reflection should be materially 
increased by resonance effects, and at the same time the wave in the bubbly 
water should be heavily damped, in consequence of the scattering of the in- 
cident wave. If the length of the shock wave is progressively decreased, 
however, until it becomes several times smaller than the bubble wave length, 
the reflection should fall off rapidly, and as the shock wave is further 
shortened both reflection and scattering should tend toward small values. 

The statements regarding high reflection are conditioned by the 
assumption that cavitation does not occur. As stated in the previous section, 
cavitation occurring in the homogeneous water lying next to the bubbly layer 
may decrease the reflection or markedly alter its character. 

The part of the incident energy that is not reflected but enters 
the bubbly water is gradually scattered by the bubbles as the waves progress; 
this process accounts for the progressive weakening of the waves. In real- 
ity there will be also a certain dissipation of energy due to friction and 
heat conduction, but estimates indicate that this dissipation ought to be 
comparatively small. 



210 


22 



s 

s 


The scattered waves are hard to treat analytically because they are 
themselves subject to continual re-scattering. It my be of interest, how- 
ever, to consider for comparison the scattering by a single bubble, it is 


found that the bubble should scatter as much energy as is transported in the 



incident waves across a certain area A, which may be called the scattering 
cross section of the bubble. The value of A for waves much longer than the 
diameter of the bubble, as given in Equation 175], is 



At w = 0, = 0; from w » 0.6wj well up toward such high frequencies that the 

validity of the formula becomes doubtful, A exceeds the actual cross- 

sectional area of the bubble itself. For air in water, N = 124. For such 
bubbles, A approximates 4a-iJ(,®, the superficial area of the bubble, when u 
lies within the range 

4<jp < w < 20Wo 

An extremely sharp resonance effect occurs. At ts» ■ w,, A «= 
20 , 500 >rSt*; the bubble scatters more than 20,000 times as much energy as 
would fall on it directly. If « differs from Wj by 2 per cent, however, the 
scattering is only a ninth of its maximum value. The half-value width, or 
width of the resonance peak between points on the curve at which A has half 
of its maximum value, is 0.014 Ug* 

The energy scattered by a group of bubbles should be Just the sum 
of the energies scattered by the individual bubbles, provided the bubbles are 
distributed at random, and provided differences in the intensity of the in- 
cident waves may be neglected. If the bubbles are not distributed at random, 
however, interference effects may occur. The reflected beam from a bubbly 
region of water having a sharp boundary arises from constructive interference 
of the waves scattered by the individual bubbles, and it is for this reason 
that the resonance peak in reflection is so much broader than the peak in the 
scattering curve for a single bubble. If there Is no sharp boundary but the 
density of bubbles varies gradually, the reflection will be weaker; if the 
density is nearly uniform within any distance equal to one wave length of the 
incident waves, the process becomes essentially one of scattering with little 
resemblance to regular reflection. 




23 


211 


BEmENGES 

(1) "Underwater EKplosligma, Tima Interval between Successive 
Explosions j" by Willis, February 1941» Volume II of this compendixun, 

(2) "Theory of the Pulsations of the Gas Bubble Produced by an 
Underwater Explosion," by Confers Herring* Volume II of this ccxnpendium. 

(3) "Stability of Air Bubbles in the Sea," by P* Epstein, NDRG 
Report G4-sr 30-027, September 1941* 

(4) "Report on Underwater Ebcplosions , " by E.H, Kannard, Voliaae I 
of this conpendium. 



212 


APPENDIX 

MATHEMATICAL THEORY OP RADIAL MOTION 
AROUND GAS SPHERES IN WATER 


Consider a sphere of gas behaving adlabatically, so that, If and 
V are its pressure and volume respectively, or R its radius, and if Vo and Rf, 
denote values under the hydrostatic pressure p^, then 

PoVo 


p, po(-| Jr/Jo**] 

whence 

[26] 

Here y is the ratio of the specific heat of the gas at constant pressure to 
its specific heat at constant volume. The energy of such a gas is 


or, if y ■= 4/3, 



Att 

3(y ~1) 


Po 




[ 2 ?] 


W - 47TPo 



[2?a] 


THE RADIAL MOTION 

By inserting the value of W from Equation [2?] and also r„ = i? in 
Equation [14] on page 46 of TMB Report 48o (4), the fundamental equation of 
radial motion for the sphere of gas in incompressible water vinder steady 
hydrostatic pressure p^ is obtained in the form 


fdR'? 


2 1 Po (Rof'' 


' dt 1 

R^ 

3 y - 1 p ^ R! 

3 


where p denotes the density of water and Cj a constant whose value depends 
upon initial conditions. 

The maximum and minimum radii of the oscillating sphere, and 
respectively, occur when dR/dt = 0 and hence are those values of R which make 
the right-hand member of Equation [28] vanish. Hence and are connected 
with each other and with Cj as follows: 


2 «. „ 

Cl = 

a p ‘ 


[- 7^ (I:)’'] = I ? 7^ (- 


l^o \ 


Sy 1 

j 


[29J 


25 


213 


whence, after dividing by 2j}j i?oV3p. 



1 


1 

' i? ' 

1 


y - 1 


The relation between the radius and the time t may be found by 
solving Equation [28] for dt and integrating: 

/-rs _ i _ ? £.l-i 

J J I i V - I p ^ R ’ 3pJ 


or, in terms of 

^ = ^ofx^dx[cx- X* - ^ [32] 

For Inflniteaiiaal amplitudes of oscillation about 12 = iZj or * = 1 , 
the integration is easily effected by writing 

» = 1 + w. C = 1 + — + b 
y - 1 

where 6 Is a small positive constant, expanding powers of (1 + lo) by the bi- 
nomial theorem, and dropping all tei'ms whose effect on t becomes negligibly 
small as b 0. Then Eqxzatlon [32] becomes 

* = lvo/(6+6«;- fyw*) * dw = 




For a half osclllatlou, the limits for w are the roots of the quantity in 
parentheses in the Integral; these roots give to the sine of the angle the 
values +1 and -1 , respectively, as is easily verified. Thus for a complete 
OBClllatio". the sin"’ contributes a factor 2tr, and the period is 


or, for y = 4/3, 


== ^0 2 ^ - 2nRo |/^ 


To - rrSo y 


For larger amplitudes numerical integration is necessary. This is 
slmplesi^ wh6n y * 30 that E^uavion [32] boCcni$3 


214 


26 


and 



[351 


e == X,* + 



[36] 


in terms of the minimum or maximum values of *, »j or ® 2 . For purposes of 
numerical calculation, the substitution, * = »j(1 + «*) is useful near »i, 
and X = *j(l - V*) near 

When the an^jlltude of oscillation is large, a useful analytical 
approximation for t can be obtained which is valid near the tine at which 
the radius R takes on its minimum value R^, When R is near ft,, x* is rela- 
tively small and can be dropped without much error; if x,® is similarly 
dropped in Equation [36], so that C = 3/»i, the Integral in Equation [35] » 
taken between the limits x, and x, becomes 




1.JL + 1 

16 Xj 5 Xj® ' 


It is convenient, also, to eliminate p and by means of as given by 
Equation [34a], Then Equation [35] gives, since *, » ft,/ftj. 


t - ti = 


JT Iftoi 'ft, ^ 1 16 16ft, ^ 6 ft,® / 


[37] 


This formula should hold well so long as x* is small as coiiq>ared to 3, per- 
haps up to ft = Rq/2. 


PRESSURE IN THE WATER 

The pressure In Incompressible water around a sphere of gas exe- 
cuting radial oscillations is given by Equation [8] on page 46 of TJ© Report 
480 (4), provided r, is replaced by ft and u, by dR/dt. This gives, with % 
replaced by v for the particle velocity of the water, 


P = f + (df) “ Po ~ i + Po [38] 


for the pressure p at a distance r from the center of the bubble. Here p is 
the density of water and pg is the hydrostatic pressure. Or, if substitution 
is made for p^ from Equation [26], for dR/dt from Equation [28], and for C, 
from Equation [29], 


P = 


Po 

3 


ft 

r 





[39] 

1 „ 

2 Pt)2 + Pg 


} 




27 


215 


and for y = 4/3, 


P = Po 




+ Po 


[59a] 


The maximum pressure, occurring at the instant at which R = and the water 
comes to rest. Is, If y = 4/3, 


P™. = Po ^ [(|^) - l] + Po ['♦0] 

If RjR^ Is small, only the terms in need be kept. Then, 

approximately, for y ^ i|/3. 


P 


m«x 




[41] 


or 


P = Po 




(^:) Q) 




Po 


[42] 





+ Po 


[43] 


An examination of Equation [39a] shows that for any R< RjZ, which implies 
that R^ < RjZ also, the error in Equations [42] and [43] is not over 7 per 
cent. 

The effective impulse, fip - Po)dt, may be found by direct Inte- 
gration, but It Is most easily found by using Equation [ 9 ] on page 46 of TMB 
Report 48o (4) 


where R and v replace r, and w in the original. If <j, the instant of min- 
imian radius, is taken as the lower limit, at which dR/di = 0, a{R^ dR/dt) be- 
comes simply the value of R^dR/dt at the upper limit t. Hence, from Equations 
[28] and [31b] 




3p. 

2Po 


C^R - 




or, using Equation [29] and eliminating p by means of Equation [34a], 


216 


28 



In which iniiicates that J?j may be inserted in both places, or R 2 may be 
employed Instead of For y = 4/3, 


1 






ip Jv^dt [44a] 


As R varies from R^ to R 2 , I as given by Equation [44] or [44a] 
rises to a maximum and returns to zero. Its maximum value represents the 
contribution of positive pressures and may be denoted by this value oc- 
curs when p - = dl/dt = 0. If the Bemouilli term pvV2 may be neglected. 

Equation [39a 3 gives 



and elimination of R between this equation and Equation [44a] with = R^ 
gives, for y = 4/3, 






[453 


RADIATION OF ENERGY 

As stated on page 9 the pressure field at a distance from the gas 
sphere is essentially an acoustic field and involves the usual radiation of 
energy to infinity. The intensity of a sound wave, or the energy conveyed 
across unit area per second, is (p - Pj)Vpcj the amount conveyed per second 
across a large sphere of radius r concentric with the bubble 1s, therefore, 

pc 

sines p is uniform over such a sphere, and the total energy emitted will be 

J pc 


In this integration the lag in time caused by the finite rate of propagation 
of sound waves can be Ignored. 


29 


217 


Because the energy depends on the square of the quantity p - Pq, 
radiation will occur almost exclusively near the peak of the wave, provided 
the ainplitude of oscillation is large. At large distances, furthermore, the 
term pv^/2 may be dropped. Hence, for y = 4/>, Equation [42] may be employed 
for p, or, in terras of x ~ R/R^, 


Po “ P(l 


Ro 


Prom Equation [35] 


r 

2Pu (Cl _ x< - 3)^ 


Hence, approximately, for y » 4/3, 


Q = 



Rq r ^ 

x^iCx - X* - 3 )^ 


[ 46 ] 


in which C is given by. Equation [56]. 

If the amplitude is large, x* may be dropped and C may be replaced 
4y 3/*i, as in obtaining Equation [3?]. In integrating up to a large x, fur- 
thermore, the limit can be replaced by <*> without much error, because of the 
rapid decrease In the Integrand. Hence, if y = 4/3 and the amplitude of os- 
cillation is large, the energy emitted during a compression and subsequent 
re-expansion has the approximate value 





zf ^ 

*1^ xf 


The integral equals 7 t/(2xj®^*). Hence 


^ 2V2ni _X 
= — :: — p 2 



or, if we also eliminate p by means of Equation [34a], 


[47] 


(|t)* ,48) 

For comparison, the total energy of oscillation E is equal to the 
kinetic energy in the water at the Instant at which the radius R = R^, since, 
if the water were suddenly arrested at this instant, the sphere would remain 
in equilibrium. As R decreases to its minimum value, i?j, this energy, to- 
gether with the work done by hydrostatic pressure as the radius decreases, 
becomes expended in work done in compressing the gas. Hence, by Equation [27] 




218 


30 






whence 


f 1 


3{V-l) 


' y 1 

Ly- 1 


-r 


y - 1 J 


in which, according to Equation [30], Ro might be substituted for R.^', or, if 
y •= 4/3, 

E = 47rp„ [|fi + i(|i) _ i] = 4,rp, fi/ [|^ 4- | (^) - i] f 49a] 


BUBBLE UNDER VARIABLE EXTERNAL PRESSURE 

Up to this point the hydrostatic pressure p^ has been assumed to be 
constant. Let it now be assumed to vary with the time. Such variation nay 
be caused by the action of a piston upon the water; if compression of the wa- 
ter is negligible, this pressure will be transmitted instantly to all parts 
of the water. The theory developed for this case should also hold approxi- 
mately for the action of a shock wave upon a bubble, provided the bubble is 
much smaller than the effective length of the shock wave in the water. 

Examination of the deduction of Equations [1] to [10] on pages 45 
and 46 of TMB Report 480 (4) shows that all of these equations remain valid 
if Pg varies with the time t. If in Equation [10] on page 46, R is written 
for Tg, the radius of a spherical gas bubble, the eqmtlon becomes 


•mf 


Only the simple case of an Impulsive variation of pj will be 
treated here. Let take on large values during a very short time ij. In- 
tegrating the last equation during this interval, 

ft 

Now Pg is nearly constant during the short time t,, hence fpgdt is very small 
and may be dropped in comparison with jpgdt. In the second integral, R is 
nearly constant, whereas dR/dt may undergo considerable change; hence, ap- 
proximately. 




I ^ 


i 1) 

m 


I €) 


31 


219 


- si 3 t) - « If 

where A denotes the change of a quantity during the time tj. Thus 


dR 


- -7S 


[50] 


dt pR . 

If dR/dt = 0 Initially, the velocity prddueed by the impulsive variation of 
Pj Is, therefore. 


dR 

dt 




[51] 


EFFECT OF MANY SMALL BUBBLES ON LONG PLANE 
SINUSOIDAL WAVES OF A SMALL AMPLITUDE 

In order to deal accurately with the effect of bubbles upon waves 
of pressure. It is necessary to make full allowance for the compres^slbllity 
of the water. The analysis then becomes very difficult unless it Is re- 
stricted to very small variations of pressure, so that acoustic theory can 
be employed. This restriction will now be made. Even so, only the case of 
sinusoidal waves can be handled readily; waves of other forms may then be 
treated if necessary, with the help of Fourier analysis. 

It will be assumed that the spacing of the bubbles, although large 
relatively to their diameter, is small relatively to the wave length of the 
wave, either in the bubbly water or in homogeneous water. This assumption 
will be taken to imply. In particular, that the average pressure in the wa- 
ter at any Instant Is sensibly the same as the pressure at points midway 
between the bubbles, and also that the local pressure field around each bub- 
ble Is sensibly the same as it would be If this field were exactly spherical- 
ly symmetrical and had a value at infinity equal to the actual mean pressure 
between the bubbles. 

Let there be « bubbles per unit volume, all having radius when 
in equilibrium under the hydrostatic pressure p^. Let x denote distance in 
the direction of propagation of the waves. 

The equations of propagation are easily obtained »in the usual wa/, 
by considering an element of volume having the form of a cylinder of length 
dx and of unit cross-sectional area. Let v denote the average particle ve- 
locity of the water in the direction of x. Then the voliune of the mixture of 
bubbles and ^water that is in the element at any instant increases during the 
next interval tfi by 

^dxdt 

Ox 


— j*. 


220 


32 


This change in volume is supplied partly by a change in the volume of the wa- 
ter itself, partly by a change in the volume of the bubbles. As the elas- 
ticity of water is equal to pc®, where p Is Its density and f the speed of 
sound in it, the increase in volume of the water is 




where p denotes the average pressure in the water surrounding the bubbles. 
The increase in volume of the ndx bubbles in the element is 


Hence 


Ti dz ^ dt = AnnR^ ^ dx dt 


^ dx dt = 
dx 


dx dt + 4wnR^ dx dt 
pc^ ot at 


9jp 

dt 


- pc 


2 


dv 

az 


+ Airnpc^R^ 


dt 


[52] 


During the same time the momentum in the layer has been changed by 
p(|^d<) dz = - U dxdt 


since d*,(- dp/dx) represents the net force on the element, whence 


dv 1 dp 

dl^--^ ^ [ 53 ] 


If Equation [52] is differentiated with respect to t and Equation 
[53] with respect to *, and if d^v/dtdx is then eliminated between the two 
equations, the equation of propagation for p is obtained 


^ ,2 

dt^ dx^- 


+ Annpe^R^ 


d^R 

dt^ 


[54] 


Here a term in [dR/dt)^ has been dropped as being of the second order. In 
the same way p was treated as constant in deducing Equation [53]; and the de- 
crease in mass due to the presence of bubbles was also Ignored as being very 
small. 

Equation [54] is unusual in form among wave equations in that it 
contains two dependent variables, p and R, A second equation is, therefore, 
necessary, and it may be obtained by analyzing the motion of the bubbles. 

This motion can be handled conveniently with the help of the prin- 
ciple of superposition. If the bubbles did not change in radius, the inci- 
dent wave, according to the assumptions made, would cause the pressure p near 


33 


221 


a bubble to vary in time without the occurrence of nmrked inequalities of 
pressure. The effect of a radial motion of the bubble will then be to super- 
pose upon this incident pressure field an eniltted train of spherical waves. 

If is the pressure due to these waves, the local pressure at any point 
near the bubble will be p + p,. 

The average pressure in the water can be written 

P = p,e'''*cosw(« - ^,) + p,, [55] 

where pj, u and a are constants and c' is the speed of propagation of the 
waves through the bubbly water. The factor is introduced to allow for 
damping due to the scattering action of the bubbles. The corresponding par- 
ticle velocity, obtained by calculating dp/dx, substituting in Equation [53], 
and integrating with respect to (, is 



The mean pressure near a bubble at ® = 0 will then be 

p =» p, coswt + pj [57] 

Under the influence of this pressure, the bubble will execute 
forced harmonic vibrations. Because of this vibratory motion, it will emit 
a train of spherical waves which, according to our assumptions are to be re- 
garded as superposed upon the average pressure represented by Equation [55]. 
The pressure in the emitted waves at a distance r from the center of the bub- 
ble can be written 


P. = ^ P 2 cosw[f - f = t + b 

in which Rq is the radius of the bubble when undisturbed, c is the speed of 
sound in water, and p^, w and b are constants. The corresponding particle 
velocity, taken positive outward, is 


V, 


R 


per 


P 2 


COS 


W 1 1 ' — 


c I u>r 


sinu 


(‘'~ 



as can be verified from Equation [3] on page 38 of TMB Report 48o (4). At 
the surface of the bubble, where r can be set equal to In constructing a 
first-order theory, 

p^ == Pg cos w f ' 


Iz 

pc 


^COS (Ait' T 


c . 

— jr- Sin (VC 1 
uRo / 


V, 


dR 

dt 




and the displacement and acceleration of the surface of the bxibble are, re- 
spectively, 

Ji — jRf, = f dt = -22-/ ^ coswt' + sinwt') 

® J di pc<j V uR^ / 


coawt' -wainwt') 
dt* pc 'Ko ' 

It is easily verified from these equations that the value of at the sur- 
face of the bubble can be written in terms of J? as follows 

P d‘R P ,,2o 2 iR 

^^Odti + c" IT 

P- ^58] 


Here u has reference to the incident waves, so that if X is their 
wave length in homogeneous water, 

c K c X 

But according to our assumptions, Ro/k must be small. Hence the term 
can be dropped and Equation [ 58 ] can be written 


D j. £. 2 n 2 di? 

P. + -^0 d7 


The total pressure is now the sum, p + p,; and at the bubble this 
must equal the pressure of the gas. The volume strain of the bubbles is 
dV/V= d(/?*)/«*= 3di?//?= 3(i? - Hence, if E' is the elasticity of 

the gas, the pressure of the gas is 


P.-3E 


, R Rn 


Hence 


, _ _ -ZE'[R-R^) , 
R-, 


and by Equations [ 57 ] and [591. 

r, d R p 2 2 dR 3E / . 

pRo 'jjT 7 -^0 77 ^R - Rq) ^ - Pi cos wt 

The corresponding eqi^tloii for non-ooiipresslvs theory is obtained by letting 
c become infinite. If pj is also replaced by zero, the usual equation for 
free oscillation is obtained, namely, 

d^R . _3£l/o._ R , 0 

j,2 + „p 2 «n) - 0 




35 


223 




where u>j2-n Is the frequency of free undamped oscillation as deduced from 
non-compressive theory. Hence the preceding equation can be written 


A particular solution of the last equation is easily verified to be 


R - Rn = - 


, (Wo^ - 0i'‘ 


r + R, 


~ sintjtj [ 62 ] 

T^J 


This solution represents steady forced oscillations of the bubble. 

The occurrence of sin &>t in Equation [62], or of dR/dt in Equation 
[6l], represents the effect of radiation damping or of scattering of the in- 
cident wave by the bubble. The complementary solution obtained by solving 
Equation [6l] with Pj = 0, represents a superposed damped free oscillation 
that soon dies out. 

Values of the derivatives that occur in Equation [54] may now be 
calculated and inserted from Equations [62] and [55]. with * set equal to 0. 
Furthermore, R may be replaced by R^, for a first-order theory. Since the 
equation must hold at all times, the cosine terms must balance Independently 
of the sine terms on the two sides of the equation. Thus are obtained two 
equations for the determination of c' and a 


2 C" 

c'^ 


2 2 2 
- € a ^ 0 


. 47rnC^Rn U)^i.U>n^ — 


+ 


2„4_i 


innRu u 

- 6 , 2)2 + I 


A more interesting form is obtained, however, by Introducing, first, the 
fraction of the space that is occupied by gas, or 

/=|n7rEg^ [ 63 ] 

second, the ratio of the elasticity of water to the elasticity of the gas, 
which will be denoted by N\ where 






224 


36 


y Equation [60], and, lastly, the "extinction coefficient" 

_ etc _ ak 
^ ~ w ~ 2 jt 


[ 65 ] 


there X is the wave length of the incident waves in homogeneous water. Then 

.2 


-4 - = 1 + fN^ 

c 


1 - 


g> 

Wa 


/ w'f , 3 £J“- 


= X 


[ 66 ] 




<1/ 


2^ = /3 /N -5-^ S r 

“ i:;?) ^ 


= y 


[67] 


The values of c' and of ^ can also be written separately in terms of the 
quantities denoted by X and Y as 




[ 68 ) 

169 ) 


COEFFICIENT OP REFLECTION 

Suppose that, under the condi- 
tions specified in the foregoing, a train 
of plane sinusoidal waves in homogeneous 
water falls at normal incidence upon the 
plane face of a layer of, uniformly bubbly 
water. Then there will be a reflected 
train of waves in the homogeneous water 
and a transmitted train in the bubbly wa- 
ter; see Figure 9. The pressures and par 
tide velocities in these three trains 
may be written as follows, in the notation just employed: 

Incident : 


Transmitted : 


0 0 0 

0 ^ ^ 

0 o o O 

^ C p ^ 0 

0 O ^5 O 
o o o o 

O O o 

o 
0 


O O Q 

^ o ^ 
0 0 
O o 


t> 


Figure 9 - Sketch illustrating 
the Reflection of a Wave 
from Bubbly Water 


= Pi cosw - 1) 

4t 

II 


= Pa cos w + ^ + 

rj V 

I 

II 


= Pg e~“* cos w - 




= ^ c-“* [cosw ( 

> L 

i-ir*r') 

1 + sin 6j (1 

' ti) \ 

t - -fr + r')l 

c /J 


225 





1 


r 

t- 

[■ 

V 

i- 

f 

! 

i 

I 

i- 

} ' 
k. 

I 


I 











^7 


y I 


The last two of these equations are adapted from Equations [55] and [56]. in 
writing the pressures, hydrostatic pressure is omitted. 

In these equations p, may be regarded as given, whereas p^, pj and 
the phase shifts r and t' are determined by the boundary conditions. At the 
Interface, at which as = 0 , the pressure and the particle velocity must be 
continuous. By writing down the two equations that cXplfOSS tilG se conditions, 
and putting in them, first, t = 0, then wt = tt/ 2, four equations are obtained 
From these equations p^, p^, r', r can be found. It will suffice to write 
down the following formula, obtained by eliminating all unknowns but Pg from 
the four equations: 



j 


^ “ - [{' " r) + 


The coefficient of reflection K, or fraction of the incident energy 
that is reflected, is equal to since reflected and 3 .ncldent waves 

travel in the same medium and their intensity is, therefore, proportional to 
p*. Hence 



in view of Equation [65]. 



[70] 


SCATTERING BY A SINGLE BUBBLE 

The amplitude of pressure at a distance r in the waves scattered by 
an isolated bubble is p^Rjr terms of the amplitude Pg at the surface of 
the bubble, whose radius is Where r is large, the waves are sensibly 
plane, and the average energy transmitted by them across unit area per second, 
if they are sinusoidal, is 


1 

2pc 



[71] 


in which the factor l/2 represents the effect of averaging over the square of 
a sinusoidal fiuiction of the time; see TMB Report 480 ( 4 ), page 39 , Equation 
[ 5 ]. The total energy scattered to Infinity per second by the bubble is thus 



47rr * 


pc 


[72] 


How upon substituting derivatives from Equation [62] in Equation 
[59] and combining the resulting terms into a single sinusoidal term, it is 
found that 


226 


38 


1 + R 


2 W' 


7^ 




(wo“ - + R 


2 

7^1 


sin(o>4 4- y) 


where y Is a phase angle of no present Importance.* The coefficient of 
sin (wt+ y) represents p^, the amplitude of p,; In this coefficient the term 
can again be dropped in comparison with unity. With the value of pj 
thus obtained. Equation [72] becomes 


Q = 


27iRq^ 

pe 







[ 73 ] 


It is more useful, however, to express Q in terms of the intensity 
of the incident waves, or the energy transported by them across unit area per 
second, ivhich is, in analogy with Equation [71]. 



In terms of N as defined in Equation [64], 


[74] 


where 


Q = A1 


A = 




nRn 


[ 75 ] 


) 





e 


Thus the bubble scatters as much energy as falls on an area A placed perpen- 
dicularly to the direction of propagation of the Incident waves. 




THE MOTION AND SHAPE OF THE HOLLOW PRODUCED BY AN 
EXPLOSION IN A LIQUID 


G. I. Taylor and R. M. Davies 
Cambridge University 


British Contribution 


February 1943 




227 


L 

I 


i-. 

f 






y ■ 

I ) 


a. 


{ 



THE MOTION AND SHAPE OF THE HOLLOW PRODUCED BY AN 
EXPLOSION IN A LIQUID 


G. I. T«ylor and R, M. Davieiii 
February 194a 


Summary , 

It l> shown In a previous paper by one of us that the gas^flUed hollows produced by explosions 
in water cannot be reproduced on a ental) scale unless the pressure above the water Is correspondingly 
reduced, in the experiments here described, which are on a very email scale, bubbles were produced by 
sparking under a liquid contained In an evacuated vessel, in this way it seaas that we can reproduce, on 
a very snail scale, the chances In form and motion of the gas bubble produced by a large explosion, it 
la worth noticing that, In small-scale explosions at atmospheric pressure, the effect of the free surface 
Is SO great that the Cuoble hardly rises at all, even when It gets to 3 feet diameter, yet these bubbles 
Of 6 cm. maximum diameter showed the rise under gravity which Is expectud in large-scale explosions. 

One of the objects of these experiments was to see how far the essumptlon that the explosion gas 
bubble Is spherical 1$ justified. This assinptlon has been made to facilitate calculation of the rate 
of rise of the bubble, and our experiments show that It Is correct over tho greater psrt of the first 
pulsstlon of the bubble. The photographs slso verify the theorotlcal conclusion that the surface of the 
bubble Should be smooth and stable during the first expansion and the early part of the first contraction, 
but should become unstahle when the bubble ma contracted to near Its minimum volume. 


Introduction , 


in Report (•Vortical motion of a spherical bubble and the pressure surrounding It'^, hereafter 
called Report a, the effect of gravity on a bubble Is calculated, assuming that the bubble is maintained 
spherical by internal constraints which do no work on the surrounding fluid apart from that whicn Is done 
by the gas contained In It during the expansion and contraction of Its volume. This assumption Imposes 
a severe limitation on the accuracy with which the calculated results may be expected to apply to real 
explosion gas bubbles, on the other hand it does enable calculations to be made which may be expected to '' 

be reasonably significant In cases where analysis in which tho whole motion is taken as a small perturbation 
ef a pulsating bubble referred to a fixed centre cannot be used. 


in the calculation of Report a the effect of tne free surface or of a hcriaontal rigid surface on 
the motion of the bubble was not considered though, as Conyers Herring’ showed, it may In some cases be 
comparable or even greater than that of gravity. photographa, taken by Edgertem i at the Taylor Model 
Sasin, of the bubble produced by exploding s detonator cap at a small depth below a free surface do in 
fact show that during its first period, at any rale, the effect of the free surface is greater than that 
of gravity, so that the bubble sinks Instead of rising. This Is. qualitatively at any rate. In accordance 
with the theoretical prediction of Herring. 


in Report a it Is shown that large upward displacements of the bubble, due to gravity, may be 
expected when its maximum diameter is comparable with the head of water necessary to produce the hydrostatic 
pressure at tho level of Its centre. This condition occurs when an explosion produces In water a bubble * 


whose 


U,5. Report NO. Ck-sr 20-010, ’The theory of the pulsations ,‘f the gas bubble produced 
by an underwater explosion", October i9#i. 


t u.s. Report, 


'A photographic study of uRda.-a.atcr explcsl-jns*, October tom 


228 


" 2 - 


Whose diameter Is comparable with 33 feet, but not when a detonator cap produces a bubble only 10 Inches 
diameter. a small-scale explosion under water with a free surface at atmospheric pressure cannot therefore 
represent dynamically the events accompanying 2 full-scale explosion, but it the explosion could bo made 
under water with a free surface at reducad pressure similarity In this respoci would be obtained. 

Experiments, in which bubbles were produced by sparking under water and other liquids in an evacuated vessel, 
were therefore undertaken with a view to seeing, on a small sc.’1e, what may be expected to happen to the 
bubble in large-scale explosions. Photographs of a series of simil ir bubbles at succsssivc stages of their 
development were first taken on stationary plates, but later a revolving drum camera enabled the history of 
a single bubble to be traced. A preliminary report on these experiinents is given below. 

a. The similariiy re iationsh j-p s. 

with ideal fluids, free from viscosity, the conditions that two bubbles whose linear scales are In 
the ratio i:k shall be similar, arej- 

(1) hj = Hhj 

where hj^ Is the depth of the large-scale charge below the surface and hj is that of the small one. 

(2) = NPj/pj 

where p, and Pj are the pressures of the air above the. surfaces in the two cases, and 

and Pj the densities of the fluids. 

(3) W./Pj = 

where and w^ are the energies which the explosives give up when the products of combustion 

expand adiaoat ically to infinity. 

(u) G(aj)/pj = N“G(a2)/P2 

where G(a) represents the work which would be done by the gas in expanding adlabatleally from 

radius a to inf i nity. 

If the liquids considered in the two experiments are viscous, their viscosities yiCj, must be 
related by 

(5) Pj/Pi * N ^/2 

This condition ensures that the Reynold's numbers of the two bubbles shall be the same at any stage- 

in the cases whore the pressure in the bubble falls below thr- saturation vapour pressure (s.v.p.) at 
the temperature of the liquid, it is necessary, in order that both liquids may boil at the same stage of the 
expansion, that 

(6) (s.v.p. of liquid i)/p^ = N. (s.V.p. of liquid 2 )/p^ 

5. Limitation to applicability of scale relationshits . 

It is unlikely that it would ever be possible to satisfy all the six above relationships simultane- 
ously. some of them, however, are of little importance compared with others. 

After fixing arbitrarily a value for n, (1) can be satisfied by setting the charge at ihu correct 
depth and (2) by exhausting the air in the chamber where the nodol experiment is to be Carried out; (3) can 
can be satisfied by varying the energy of the explosive charge or the spark which produces the bubble. 


condition 


- 3 - 


229 


I? 


; i 


I I 




o 


r 


Condition (u) Is none difficult to satisfy. It could be satisfied if (a) the model and full-scale 
bubbles were both produced by explosion of jases contained in thin spherical envelopes such as rubber 
balloons, provided the initial pressures of the unexploded gases in the two cases were the same as (or any 
given multiple of) those of the surrounding fluids, and (b) the densities of the two fluids are the same, 
Ue., ~ Pj« Such experimental conditions viould ensure that conditions (3) and (s) were both satisfied, 
but it would be difficult to devise a model experiment so that {s) was satisfied when the full-scale bubble 
is due to the explosion of a sol It. explosive and the small bubble t-.. a spark. Fortunately, however, there 
Is not much difference between the bubble produced by releasing all the eitergy at a very snail radius and 
that produced by releasing most of it at very small radius and the rest during the expansion. Dr, Conrle's 
cu.-vss showing the radius a as a function of time are very insensitive to the constant *C' which determines 
the value of G(a)/w, 

The value of 6(a)/M for the bubble produced by a mass M of a given explosive depends only on the 
absolute pressure In the bubble. |f similarity of bubbles on two scales is obtained by choosing the 
atmospheric pressure, veight of explosive and depth so that conditions (l), (a) and (3) are satisfied, 

G(a)/M will be less at a given stage of the expansion in the small-scale experiment than In the large- 
scale explosion, owing to the fact that the pressure is less. it seems that. In comparing large-scale 
underwater explosions with small-scale sparks in a liquid under reduced pressure, one is comparing bubbles 
In which the constant c in Corarie's calculation varies from 0.0« up to O.l with the Case where C is so small 
as to be negligible. The difference between the two cases Is not great. 

The effect of vi snosity , 

To satisfy condition (3) in a snail-scale experiment designed to represent a bubble In water would 
require a liquid whose viscosity is very much less than that of water, no such liquid is available. 
Fortunately, however, it Is not necessary to satisfy this condition, because the loss of energy due to the 
viscous forces opposing the expansion or contraction of the bubble is very small compared with the whole 
energy of the bubble, even when a liquid of much greater viscosity than water is used. 


The rate of dissipation of energy owing to viscosity is 

]ds 
;§rjr,a 


F « ipSS 


i.1) 


the integral being taken over the surface of the sphere, and q being the radial velocity of the fluid, 
since q a ke^/r^, equation (7) gives 
F • 8 7T pai} 

and the total unorgy, v^, dissipated during time t is 

‘ .2 r® . 

a BTtpL j aa‘ dt a airp-j aa da. 

Equation (j) of Report a is now modified to the form 

~iT/Oa^gz + sr pa^ + itr/ja^ z* + evpj a& da « W - G(a) • (8) 

Reducing to the non-dimensional form of equation (5) of Report a equation (s) becomes 


ff!lV . .4 - 

\df ^ 2 77 a‘* I w j 6 V^df / 


2 

— Z' 
? 


where 


(pVg) 5 0 


df 


( 9 ) 


(1.0) 


\ } 


If 


230 


4 


If * is small the relationship between a‘ and t’ will be little altered by viscosity, so that the 
value of l.ie integral in equation (lO) c .n be estimated usin^ the v„lues calculated when //, « i). |n the 
case calculated in Report A, whore ^ - 2, the value of r‘ /da'V^j, taken over the whole of the 

•^o \dF/ 

first period of the bubble is 0.6, so that the value of i> at the end of the first periivl is 


U.ft 7T tt. 



(U) 


In soniB of our experiments, in which transformer oil of viscosity 0.3 poises and density ?.5’5 
gm,/cc, was used, was of the order of 2; the value of 6 calculatad from equation (ii! was o.bos, so 
that less than one per cent of the energy of the bubble is dissipated by viscous forces during the first 
period, if the bubble remains spherical. 


5. The choice of operating liquid . 

with explosions in open water the pressure at points far distant from the bubble is never less 
than 1 atmos: the minimum pressure in the bubble, though considerably less than t atmos., does not fall 

as low as the saturation vapour pressure (s.v.p.) of water, in experiments with water under a surface 
pressure which is low compared with normal atmospheric pressure but well above the s.v.P. the minimum 
pressure near the surface of the bubbl * may be less th,;.n the s.v.P. The water then boils near the bubble. 

Figure i shows two superposed photographs, on the same plate, of a bubble made by a spark under 
water at surface pressure rather greater than the S.V.P., (l) at an early stage of the expansion, when it 
is smooth and spherical, and (2) at a later stage, when the small vapiiur bubbles due to bailing have had 
time to develop. it will be seen that the surface has becumc pitted by the boiling. 

In vrder to av.-.id boiling when experimenting at low pressures it is necessary to use a liquid with 
a low s.v.P. For this reason transformer oil was used. Though the viscosity of this liquid Is about 
30 times that of water, it has been shown that the expansion of the gas bubble Is not likely to be materially 
different from what It would oe In water, at any rate while it is approxiirr.toly spherical, 

6, D eecription of the apparatus . 

The airtight vessel in which the bubbles are produced is shown in elevation in Figure 2! It 
consists of a vertical glass cylinder, a, 12 inches diameter, ^ inch wall, 15 Inches high, cemented between 
two horizontal steel ond-plales, 6 and C, 18 Inches diameter, t inch thick. This vessel is filled with 

water or oil to a depth of about 11 inches, and It is connected to a vacuun pump and a pressure gauge 
through the tube 0. 

A Dubble is produced in the liquid by the discharge of a condenser, charged to a potential difference 
of 11,000 volts, across a gap of about lO/lOOOth inches between the rounded ends of the horizontal brass 
rods E and F, ^th inch diameter and 3 inches long. The electrode E Is connected electrically to the upper 
steel plate 8, which is earthed; the electrode F is insulated and is connected to the condenser and the 
timing pendulum By a lead passing through the insulated bush, G. The electrodes are fixed to the brass 
frameworks shown in the diagram so that they can be moved independently, without admitting air to the vessel, 
by means of a brass push tube and two brass push rods pissing through stuffing boxes fixed to the upper 
steel plate. 

The framework supporting the etectrode f can be moved verticalTy or rotated by means of the push 
rod passing through the right stuffing box shown in the diagram, A hinge allows the electrode to fly 
back during an explosion, whilst the horizontal member of the framework is supported and is prevented from 
moving at right angles to the plane of the diagram by a wire staple fixed io a block of insulating 
material attached to the push rod. 

The framework supporting the electrode E is connected through the hinged joint to the push 
tube passing through the lower stuffing box, shown cn tho left in the diagram; a push red passes through 
this tube and through the upper stuffing 0i,x and carries at its 1 wer end a horizontal brass bar to which 


) 



) 


1 




- 5 - 231 

is flxeO a knife-eage k ana a wire staple passing over the brass bar attachea tc when the upper 

stutting box is tight and the lower one is slightly sl.ackt-ncd the trarnowork can be moved vertically as a 
whole or rotated, when the upper stui'ting box Is tight and the l.;wur one Is slightly slackened the 
framework can be moved vertically as a whole «r rotated, when the upper stuffing bi.x is slackened and 
the lower one is tight the central push rod car. bo mi.vod up or d.-wn relative to the push tube, thus 
raising or lowering the knife-edge k and ir,ere; 5 slng or decreasing the gap between the ends of t and F. 

when photographing the bubble. It is essential t,. correct the optical distortion caused by the 
cylindrical form of the bubble tank. F.r this purpose, t«. parallelizing tanks, L, consisting of plate 
glass plates cemented to metal frames which in turn are cefiicnted to the outside of the tank, are placed 
Opposite one another so that the glass plates are vertical and parallel; water .r oil Is p.ured into 
the parallelizing tanks so that the level I'f the liquid is the same on the twi. sides of the glass cylinder. 

The inset in Figure y shews the non-adjustabln spark gap which was used In the early experiments 
with water; this gap »as attached to a rod passing through a stuffing box in the lower steel plate, C, 
and it Is seen in the photograph of Figure i, it was discarded because it offers too large an obstruction 
near the spark and because the gap was n;t adjustable. 

The general layout ,-f the apparatus is shown in Figure 3, in this diagram, A represents the bubble 
tank in plan (with the end plates, etc., omitted), E end F the electredes forming the under-llquid spark gap 
and L the glass-fronted tanks for correcting the distortion due to the cylindrical form of the bubble tank. 

H represents the illuminating spark gap with electrodes set about jo/ioooth Irenes apart; In experiments 
with water, the electrodes were made of magnesium, which gives an intense spark with most of the energy 
in the blue region of the spectrum, and blue-sensitive (’ordinary') photographic plates or films were used. 

In experiments with our oil, which transmits only the green end yellow regions nf the spectrum, 2lnc 
electrodes wore used in conjunction with fast orthochromatic pl.atcs or films. 

The spark gap mIs placed at the focus of 4 condenser lens, N, 6 inches diameter and 6 Inches focal 
length. The light from the illuminating spark thus passes through the bubble tank as a parallel beam; 
after emerging from the Vessel, It passes through a second lens P of high optical quality, which is set so 
as to bring the light to a focus on the centre of the camera objective, o* The camera can be used either 
as a stationary plate camera, ur, as shewn In the diagrjvti, as a revolving drum camera. The camera lens is 
adjusted so that the under-liquid spark gap is sharply focussed •■■n the ph.:tographlc pt.-ite ur film. 

Figure 3 also shows the timing pendulum and the electrical circuits used to produce the explosion 
and Illuminating sparks. The pendulum, whose period is one second, consists of a steel rod fixed to a 
horizontal rod mounted between accurately machined centres; at Its lower end, the pendulum carries an 
extension shaped as shown In the diagram. This extension swings above a number of brass screws with rounded 
upper ends, the gap between the screws -and the pendulum being adjusted to u/ioooth inches; these screws are 
mounted on an are of insulating material and they are spaced so that the pendulum' takes b milliseconds to 
pass from any one screw to the next when the pendulum Is released from the horizontal. The first screw, 
labelled 0 In the diagram, determines the discharge of the condenser Cg used to produce the under-liquid 
spark; the remaining screws, labelled i, 2, 3,... In the diagram, determine the discharge of the condenser 
C| (of which only four are shown and only one labelled) used to produce the sequence of illuminating sparks 
for photographing the bubble. 

AS the diagram shows, these condensers are connected in series with resistances across a large 
reservoir condenser c^. charged through a valve rectifier to a potential difference of ttOOO volts; the 
junctions of the condensers and the series rest stances are each connected to an electrode screw. 

When the pendulum swings over the screw 0. the gap between the screw and the pendulum Is momentarily 
in series with the under-liquid spark gap; the two gaps break down and the condenser Cg discharges across 
them. Similarly, the first condenser Cj discharges across the illuminating gap and the gap between the 
pendulum and the screw 1 when the pendulum moves over this screw, and so on. The 5 megohm resistances in 
series with Cg In effect Isolate this condenser and prevent more than one spark under the liquid during 
the course of an experiment. The resistances R| are such that the time-constants R|Cj and RjC^, are large 
enough to prevent the reservoir condenser c,. and the l.ater C|'s from discharging completely during the early 
Illuminating sparks and, at the same time, small enough tu allow a certain amount of charge to flow into 
the condensers Cj from C^, 




232 


- 6 - 


f. Experime n ts under vacuum . 

Though experiments under a vacuum do not represent any possiole full-scale explosion In open water, 
they are of interest for two reasons 

(1) with a given value of the non-dimenslonal paran>otor jfQ*. the effect of the free surface on the 

motion of the bubble is as small as It can be, so that comparison with the formulae of Report A 

is as justifiable as it can be; 

(2) The Instability of the free surface, which occurs when It is accelerated downwards under the 

action of external pressure, is absent. 

Before the revolving drum camera had been constructed, 0 number of single photographs were taken 
of the bubbles produced by sparks In oil using a 0.4 pi. F. condenser charged to 4000 volts. These photographs 
were timed to occur at times t ^ xo, IS, 20 up to iRO mill I seconds after the Initiating spark. 

Figure u shows the bubble at t • 0.025 seconds. It will be seen that it Is a very smooth and perfect sphere. 
It remains spherical during its expansion, but during its contraction Its vertical dimension decreases more 
rapidly than its transverse diameter so that It becomes flattened. This flattening Is more pronounced on 
the under side than the upper side. Figure s shows the bubble at t ■ o.os seconds. The flattening Is here 
very apparent, 

AS the bubble decreases in diameter the flattening becomes more tmd more accentuated, till the 
underside becomes concave and the bubble assumes the shape of a mushroom. Figure 6 shows the bubble at 
t « 0.08 seconds. This is somewhere near the minimum size at the end of the first period of pulsation. 

Figure 7 shows the radius, a, and the rise, z^ - z, of the centre uf the bubble at times up to 80 

milliseconds after Its formation. To compare these with calculations based on the assumption that the bubble 

remains spherical, it Is necessary to assume a valua fer the energy, w. which the spark gives to the oil. 

It will be seen that. In Figure 7, If a smooth curve is drawn through the points representing the radius of 

the bubble its maximum value, will be about 3.7 cm., and since the total depth z^ of the liquid from 
the level iWiero the pressure is zero is 6.05 cm., ■ 3.7/6.05 • 0.61. 

in comrie's calculated curves in Report a for C » 0, l.e , with no gas inside the bubble, It will be 
found that » 0.63 <’i»r z'^ = 1.0, 0.25 for z*^ » 2.0, 0.14 fer z’^ ■ 3.0 and 0.090 for « 4.0. 

It appears therefore that the bubbles represented in Figure 7 correspond very closely with the condition 
z’g • 1.0. when z'^ « 1.0, the scale-length, L, Is the same as the depth z^ of the explosion below tha 
level of zero pressure, so that l • 6,05 cm. This corresponds with energy W » » X.X6 x 10* ergs, when 

p « 0.875 gm./em.3 

The non-dimensional time scale t* in which the calculations are expressed Is related to the true time 
scale t by the relation t « t' /|i/g) « 0.0786 t' seconds, when L - 6.05 cm. Multiplying the ordinates of 
cemrie's curve for z'^ • 1,0 by 6,05, and the abscissae by 0.0786, the calculated relationships between a, 

(Zg - z) and t are found. These have been plotted In Figure 7. It will bo seen that, until the top of tha 
bubble is very near the free surface, its observed radius and rise are close to what was calculated 
theoretically, 

S. Beperiments under surface pressure of 6.fi cw. of oil . 

The depth of the apark was 6.05 cm. so that experiments in which the surface pressure was that due 
to 5.S cm. of olt are equivalent to explosions at a depth of (33 x 6.06) /6. 5 ■ 30,8 feet In open water. 


In one 


z’g • Zjj/L where z^ * 'plps, p is the pressure at the level of the explosion and l Is the 
scale length defined by the equation (3) of Report A, namely L • (w/gp)^ 


' Ji. 


In ow'j set of experiments under these conditions the drum camera was used and the duDbles were 
produced by discharging a o,4/u- r. condenser charged to 4050 volts. Two films, due to two separate sparks, 
were taken and together they covered the period t * o to t = uo milliseconds. The first film, covering 
the period t = o to t = 60 milliseconds, is shown in Figures sa and gfc: the numbers to the loft of each 
photograph are (i) the number (v. Figure g) of the particular contact of the timing pendulum which is 
operative In the photograph; and (z) the value of l, calculated I'rom the seperation of the photographs on 
the film. The dark band, lying on the right-hand side of the photographs, is due to one of the leads to 
the under-oil sparl^gap; iha faulty reproduction of photograph no. 2 Is due to the fatt that it lies on the 
join of the two ends of the film on the camera drum. The second film covered the period t » 60 milliseconds 
to t • 120 milliseconds and photographs ho. 17 U- no. 22 are reproduced In Figure Sc. The values cf t arc 
net exactly spaced at Intervals of S mill Isec'.nds eating to the fact that the timing pendulum was n.>t always 
in exactly the same position relatively to the lower point of the gap when the spark went off. Some of the 
sparks were as much as 1 millisecond late and one of them was nearly 1.5 milliseconds late. 

Owing to optical difficulties the field of view was not large enough to take in the bottom of the 
bubble In its early stages. The radius was, however, taken as half the horiiontal diameter, and the rise 

of the centreabove tho spark gap was taken as (the height of the top of the bubOlc above the spark gap) 

minus (half the horizontal diameter). Figure b shows the radii (Indicated by points surrounded by triangles) 
and the heights of rise (indicated by crosses) obtained in this way. The points are numbered 1, 2, .... 
on the same system as the photographs of Figure 8. and the points given by the second spark are distinguished 
from those given by the first by the letter 'a*. The points 12 and 12a do not coincide, showing .that the 

energy of the spark was different in the two cases. it will be noticed that the two sets of points lie very 

well on smooth curves.’ 


The maximum radius of the bubble was 3.16 cm., and since z 


6.5 + 6,05 = 12.55 cm., 


It has alroady been mentioned that this happens to be the value which corresponds with z 


2.0# SO that 


1. = » 6.27 cm. The corresponding value of w Is w » 981 x 0.875 x 6.27** = 1.32 x 10* ergs. 
The factors by which Comrie's calculated values for z'^ > 2.0 must be multiplied are given by the equations 
a • 6.27 a’ and t » 0.080 t'. 


For z'p • 2.0, Comrie gives the first minimum contraction at f a o,67 which corresponds with 0.0535 
seconds. This is considerably larger than the observed value shown In Figure 9, namely 0.045 seconds. 

The difference may largely be explained by taking aceognt of the effect of the free surface, according to 
Herring, the period is reduced by the proximity of the free surface In the ratio / r \ 

(1 - ^ 1 l. 

Where a is the mean value of a during the pulsation and h Is the depth of the explosion below the free 
surface. Taking a as 2.7 cm. (v. Figure 9) he 6.05 cm., the predicted period of the bubble is 0.0535 
(1 ” 2.7/24.2) = 0.0475 seconds. Which Is close to the observed period of 0.045 seconds. 

Comi>arison of rate of rise with theoretical estimate. 


The calculation of the rate of rise, neglecting the effect of the free surface, cannot usefully be 
compared with the observations because the critical point at which tho minimum radius and consequent rapid 
rise occurs is, as has been seen, 0.009 seconds earlier than that predicted by the simple theory. To take 
account of the effect of the free surface, using the equations given in the Appendix, involves a -great deal 
of laborious calculation. Two simpler methods can be used, one is to take Comrie’s calculated values of 
2* - z' for observed values of a, r-ather than for the values of t'; another is to calculate U » -dz/dt 
directly from the observed values of a and t, using equation (1) of Report a. and then finding z by Integrat- 
ion, The values obtained by these methods are indistinguishable in the range shown In Figure 9. It will be 
seen that until the bubble begins to contract the observed rise is not very far from the calculated rise, 
but th.at the observed rate of rise becomes less than that c.alculated shortly after this. When the radius 

begins to contract rapidly at t • 0,035 seconds the bubble begins to rise rapidly, but the rate of rise is much 
less than that calculated on the theory which assumes that the bubble remains spherical. The probable 
reason for this seems to be that the bubble ceases to be even approximately spherical as it gets near its 
mlnumum radius. It flattens on the underside and even becomes hollow there before reaching the minimum, as 
Figures 5, 6 and photographs No, 7 to 10 of Figure sb show. 


Comparing Figures 7 and 9 It will be seen that the scatter of the points In Figure 7 is a 
true measure of tho variability of the energy of the ouboies proouceo by successive sparks. 


234 


- 8 - 


Tfte hcillow which appsars In photograph no. 9 of Figure 8b subsequently disappears as the bubble 
expands a second time but reappears on its second contraction. Photograph No. 19 of Figure 8c appears 
to have a flat undersurface but the fact that It is roally only a thin hollow shell will be seen in no. 20i 
where the bubble is at its second minimum (see Figure 9) ano the shell has contracted to vanishing thick- 
ness over a large part of Its area. 

The very rapid rise near the minimum radius, which Is so characteristic a feature of the simple 
theory, depends on the fact that the virtual momentum of a spherical hollow a^g • iu (mass of 

. i 2 

liquid displaceu^i that a decf^asu Ift corf^spunds With a rap id increase in u. If the bubble becomes 
flattened, the virtual Inertia rapidly increases if It fl.atttnt without changing radius, The virtual 
Inertia of an infinitely thin dish is actually rather greater than that of a sphere of the same diameter. 
Thus if the •radius* of the bubble is taken as 4 (the horizontal diameter) Instead of 4 (horizontal 
diameter * vertical diameter), the measured curve will not reach such a low minimum and the calculated 
vertical velocity will be reduced. Figure 9 shows both methods of estimating a near the minimum; the 
points calculated by the former method are surrounded with a triangle, and by the latter with a circle. 

Even making this assumption as to a, the theoretical calculation predicts a larger rise of the 
bubble than Is observed in this experiment. some, but not all, of this can be accounted for as being 
due to the effect of the free surface. It seems probable that a wake is formed behind the bubble shortly 
after it beginj to contract and that this greatly decreases tha vertical velocity,- so that at the time the 
contraction of radius becomes very rapid the bubble has far less virtual inertia In a form in which It can 
be concentrated by the contraction than it has according to theory. it may be that the effect is 
accentuated by the high viscosity of the oil used in these experiments. Further light is thrown on this 
by experiments made with a smaller bubble. 

10, Ex-pi rim ents mtk a smaller bubble . 

Another set of measurements was made with tha pressure at the surface equivalent to 6.5 cm. of oil 
and the spark at 6.05 cm. below the surface, but with a spark of smaller energy, namely that given by the 
discharge of a 0,2 m condenser charged to a potential difference of 4000 volts. The stationary plate 
camera was used, giving a set of single photographs of the bubbles produced by the different sparks, The 
results are shown in Figure lO. it will be seen that » 2.1 cm., so that ‘ 2.l/(6,5 ♦ 5.05) ■ 

0,167. This lies between the value 0.14 for z'^ * 3.0 and 0,26 for z'^ » 2,0. By interpolation the value 
of z' corresponding with “ 0,t67 is z' • 2.8. For this value of z' Interpolation from comrie’s 

curves gives ® so that L » 2.1/0.44 * 4.16 cm. For this value of l the time scale >s given 

by t/t* • /(4. 76/981) ■ 0.070. The duration of the first period of oscillation for • 2.8 Is given by 
Interpolation as t* » o.5l, so that I » o.51 x 0.070 » 0.035 sernnds. This Is larger than what is observed, 
namely about 0.030 seconds. The effect of the surface, however. Is to decrease the period In the ratio 

1 arid taking a ° 1.9 cm. this ratio is 0.92, so that the Calculated period would be 0.0326 seconds, 

. Uh/ 

No better agreement with observation could have been expected in view of the fact that each point In this 
series refers to a different bubble. 

Referring to Figure 10 It will be seen that, after the bubble has ceased to rise at the rate 

calculated on the assumption that It remains spherical while pulsating, the rate of rise fluctuates with the 

pulsations but attains roughly constant mean velocity of about 00 cm./sec. If the resistance of the 
bubble is expressed in the form R « k 7r a^ (^^) where k is a drag co-efficient, R Is equal to the buoy- 
ancy, so that tv a^/Og » k 77 (4 yOU*) or k » sag/su^, anO since the mean radius during the period while 

the bubble is rising approximately -uniformly Is 1 cm., k • (s x 1 x 98’) . (3 x 80^) > 0.41. This Is of 
the order of magnitude that might be expected. The drag co-efficient of a solid sphere at the same value 
of Reynold's number is 0.56. 

It Is worth noticing that the drag co-efficient of a spherical bubble, for the small values of 
Reynold's numbers in the region where stokes' Law holds, is tv»o-thlrds tne drag co-effIcSent of a solid 
sphere of the same radius, if this relationship holds at the larger value of Reynold's number appropriate 
to our rising bubble, the drag co-efficient would be 0.374. 


- Q - 


235 


1 


I 


13 . The stabil it y of the surface of the bubble near the posit i on of minimu m 
cont ractio n. 






wfisn the Bubula contains gas the radius contracts at increasing rate until the volume decreases 
sufficiently to raise the internal pressure aOove the undisturbed hydrostatic pressure ot the level of the 
centra. The radial velocity then decreases and reverses sign at the minimum radius. The curvature of 
the (a. t) Curve Is concave towards the ixis of t during the greatBr part of the first oscillation so that 
the acceleration of the surface is directed from the liquid inwards towards the gas. During this part of 
the pulsation small disturbances of the surface should be stable. When the pressure of the gas rises the 
radial velocity decreases and the {a, t) curvo oecomes convex to the t axis, the surface oeing accelerated 
outwards from the gas towards the liquid. Under this condition the surface may be expected to be unstable 
In the same way that a horliontal liquid surface Is unstable when the liquid Is sbeve and the air below, so 
that gravity acts from the liquid toward the air. 

These conclusions are borne out by the photographs of a bubble taken near its minimum radius. 

For example, in the photographs shown in Figure eb, it will be seen that up to photograph no. 8 the surface 
of the bubble is very smooth. Indicating stability of small disturbances, and Figure 9 shows that when 
photograph no. 8 was taken, the surface had net begun to accelerate outwards. ihe part of the (a, t) curve 
of Figure 9 which is Convex dbwnwardjoccurs just before photograph Nc. 9 was taken, so that instability might 
be expected to show Itself In the photograph. it will be seen that in photograph no . 9 the surface has in 
fact become ragged owing to the Instability. 

It will be noticed in photographs nos. 9, lO, ii and 12 of Figure 8b that the instability which has 
appeared in No, 9 seems to disappear very rapidly on tne lower part of the bubble but continues and increases 
at the top. 

12 . The ai>i>earance oi the free surface , 

in the photographs 0 / Figures sa. b and c, the free surface appears rather out of focus above the 
bubble: in photographs Nos. 1 to 12 , the free surface has moved very little, but in nos. 17 to 22, it nas 

risen to a considerable height. 

^ 3 * Appendix, The effect of a f ree or rigid su rfac e on th e notion of a stheric al 
buhile^ 


In order to compare the results of these experiments with calculation of the observed rate of rise 
and pulsations of the bubble, It became necessary to extend the calculations of deport * so as to include 
terms representing the effect of the free surface. Th- condition which enables the rise of the bubble 

under gravity in one pulsation to bo comparable with its maximum diameter necessarily involves a free 
surface effect which is comparable with the gravity effect. 


Herring gave a formula for the effect of a distant free or rigid surface on the position of the 
centre of a bubble. if U is the velocity towards the surface his formula is 



(A.l) 


where 

a = the radius of the bubble at time t, 
h « the depth of the explosion, 

* rtoU) Co 

U » * l_ and Rj is defined in Herring's paper, 

h* dt 

and the upper sign refers to a rigid surface while the lower refers to a free surface. since only terms 
of order i/h'^ are retained It seems that equation (A.i) may be written 


d 

dt 


{a? U) “ * 



(a. 2 ) 



I 


The 





236 


- iO - 


The fomtuia (A.l) »as given by Herring as applicable only to cases sthsn u Is dmII cooiparad w|tb da/$t 
but It applies equally well when U Is comparable with da/dt provided It is assumed that the bubble is 
constrained to remain spherical and terms containing powc^rs of l/h higher than l/h^ are neglected. With 
these assunptions the velocity potential of the flow Is 

V " ^1. ♦ ^ t 4 ua^ cos^? ’ ♦ ii!i f (*.3) 

r jr* ^ r' ar** !th* ^ 2 f^ !\ 

whore r> and &' are the co-ordinates of any point referred to tne Image centre aa origin and the line 
joining the centre of the sphere with Its Image as ■ 0, The upper sign refers to e rigid plane and 
the lower to a free surfacei 


Hear the sphere may be written 

(p ■ e ua^ cos $ j ( a^a + a^&r coa 6 ^ ^ ^ a^a cos & I 

r jr* I 2 h 4h^ 8h* 8h*r* J 

The kinetic energy T of the whole motion is on»>ha1f the kinetic energy of the Mow surrounding the spkefa 
and Its Image so that U > - // <p(2^ S ds, the Integral being taken over the sphere only, 

P \3 r / 


a®a cos 6 1 

8h*r* J 


Substituting fp and 'h<U'i r from equation (s.a) 


JL » 2Tt (l ± — \+ 

^ I H 


The energy equation 1$ therefore 


■ a^ Tf a^iu / a* 

— * — (17 


(-) 


4 a a.j/8\ w ..fpa^a*, 

— wp 8' g* + 2 vrp S'' a‘ f 1 4— \ ♦ — p U 4——— —x- all » W - G(a) (8.6} 

J \ 3 2 h* 

where w and 6(a) have the same meaning as In section 2 above. 

The condition that no resultant ektemal force acts on the sphere Implies that the eo-4fficient 
of cos 0 In the expression for the pressure at the surface of the sphere shall vanish. 


d t J fixed point 


- ♦ ga - gr cos 0 


where z is the depth of the centre below the level where the pressure Is zero. Also If 9^/d t represents 
the rate of variation of at a point which moves vertically at the seme velocity as the centre of the 
sphere 


I d t j fixed point 


P c 

■“t ■ ‘’|T7 


cos 0 - i sin 6* 

r B<?j 


Substituting from equation (A.g) In equations (a.8 arwl 7) the tern containing cos 0 In p/p 
Is found to be 


f 1 . 3 / 3 

J dU + all ” Qa i •— *t* I 

1 2 2 Mh* I 2 


a^ a 4 ^ a* cos ^ 


Multiplying this by and equating to zero It Is found that 


i-(a3u, .. 2 90^ 

dt 4h^ dt ^ at I 


This 







238 



PIG. 1. 

5ro photographs of the saiae bubble ia water under Icar 
«urifaei> pnsigure. 

(1) at OLB early stage, irtien it la smooth aad spherioali 

(2) at & later stage, mhea the bubble has beoos» 

pitted by boillKig near the surfaoe. 



KIO, 4. 


PIG. S. 


PIG. 9. 


Photographs of bubbles l a ^?3SS?a-illS!^51LjlS3£ 

the bubble beeomes distorted aa its age laereases. 


Pig. 4. Spherical bubble, age 2S mllliseo. 

Pig. 5. gobble with pronounced flattening on underside, 
age 60 mllliseo. 

Fig. 6. liuahroom-ahaped bubble, age 80 mlllisee. 











M.ILLUMltOATlWC 


PlH.O»ICW 

SHMWaW. 



























STUDIES ON THE GAS BUBBLE 
RESULTING FROM UNDERWATER EXPLOSIONS 

ON THE BEST LOCATION OF A MINE 
NEAR THE SEA BED 


M. Shiffman and B. Friedman 
Institute for Mathematics and Mechanics 
New York University 


American Contribution 


May 1944 


“1 


245 


TABLE OP CONTENTS 


Introduction 


Part I» Discussion of the Resultss 

1* Preliminary remarks 
2* The principle of stahilizatlon 

S« The location of the mine 

4* An e: 2 can 9 le 


Part II«Ktethematical Study of ih© Sscondery Pressm:*e 
Pulse. 


1« Introduction 11 

2. The energy equation 12 

3* Non-dimensional variahles 14 

4. The scaling factors and the internal energy of 15 

the gas for T*N«T« 

5. The equations of motion for small values of the 17 

radius a 

6. The minimum radius 16 

7« The presswe pulse 19 

8. The optimum peak pressure 22 

9« The Ir^ulse 23 

Part IlI.The Mechanism of Stahillzatlon by gravity and 
the Sea Bed- 

1* The exact equations of motion 26 

2* The approximate evaluation of the period and the 29 

momentvon 

3. The stabilized position 32 

4* oaie migration of the bubble 34 

5. The correction due to the free surface 35 

Appendix !• 

The Nvenerloal Integration of the Differential Equations. 38 


t p j 




246 


- 11 - 


Appendix TI* 

P^e 


The Velocity Potential for a Pulsating Sphere Moving 
Perpendicularly to a Wall* 


1. 

Statement of the problem 

41 

8* 

Some theorems on images 

42 

Ott 

The coxistructlcn of 

44 

4* 

The construction of 

49 

5* 

The kinetic energy of the water 

51 

6* 

Approximate theory for a sphere pulsating between 
a rigid wall and a free surface 

54 


Appendix III* 

Numerical Evaluation of Some Definite InteRpals. 57 

TABLE OP GRAPHS 


Figure 1* Scaling factor L 61 

Figure 8* Distance from bottom to produce isaslimam pressure 63 

Figure 3* Pressure factor for mine of 1500 lbs* of T«N*T* 65 
In water 150 ft* deep 

Figure 4* Pressure factor at sea level for mine of 1500 67 

lbs* of T* N* T* In water 150 ft* deep 

Figure 6* Dependence _of Internal energy (u) on 69 

momentum (s) 

Figure 6* Dependence^ of preasixre factor (q) on 71 

momentum (if) 

Figure 7» The displacement of the bubble 73 

Figure 8* Radius of bubble; case 7 75 

Figure 9* Position of bubble j case 1 77 

Figure 10. Position of bubble j case 3 79 

Figure 11* Position of bubblei oase 4 81 

Summary of Formulas 83 

Bibliography . 85 



- 1 - 


247 


( i 


I 

i' 

r 


S 

fr., 


ii 




( 


Studies on the Qas Bubble Resulting Prom 
Underwater Explosions 

OH THE BEST LOCATION OP A MINE NEAR THE SEA BED 

At the request of the David Taylor Model Basin of 
the Biareau of Ships and of Division Re 2 of the Bureau of 
Ordnaaoe, a series of investigations has been undertaken 
by the New York University Group of the Applied Mathematics 
Panel of the N«D«R»C« with the object of analyzing the 
phenomena associated with the gas bubble produced by an 
underwater explosion. 

The present study, carried out by Dr. Max Shiffman 
and Dr. Bernard Friedman, with the cooperation of the 
Mathematical Tables Project In the extensive numerical work, 
la concerned with the following problems If a mine of given 
weight of explosive is to be placed near the sea bed, what 
position should It have to cause maximum damage to a target 
at the surface? 

It seems appropriate to make a few Introductory re- 
marks about the broader research program of which this report 
Is a part- Often the destructive effect of an underwater 
explosion Is not wholly duo to the high presstire shock of the 
explosion. After the initial shock wave has passed with 
enormous speed, a oon^)aratlvely slow pulsation of the gas 
bubble (consisting of the burned gases) takes place. In the 
second and sometimes even third pulse of this motion of the 
bubble, a pressure pulse of considerable strength la emitted. 
While the peak of the shock pressure from the explosion is 
far above tliat of the later pulses (in typical cases, six 
to ten times the pressure of the second pulse), .the dxiration 
of these later pulses Is much longer (about twenty times as 


248 


"S*» 

long) I so tJsat the momentvcra l^arted and the destruction 
caused may he comparable to that of the first shock# The 

ssQond. pulse has been cenfiT>med more and 
more not onlv by modal experiments, but also by analysis 
of actual damage to allied ships by mines [1], [2]# 

Just as significant as the long duration of the seo- 
end pressure pulse , is the migration of the gas bubble 
during the pulsation# The migration oan increase the effect 
of the second pressure pulse because, under suitable conditions, 
the center of this pulse might be much ©loser to the target / 

than the explosion# Factors Influencing the motion of the 
bubble are: rigid walla such as the hull of a ship or the 
sea bed, which attract the bubhlei the free surface of the 
water, which has a repulsive effeeti and the buoyant force 
of gravity, which causes the bubble to rise towards the sur- 
face# 

Understanding and controlling the Oomplloated Inter- 
play of these effects it of sufficient practical in^ortance 
to warrant oomprohenal’;'*; resosirch# Unfortunately, a great 
yaplcby of expex’lments wxth full sis© charges Is hardly 
feasible, and. experiments with small charges cannot easily 
be reinterpreted for large oh8U*ges# Such difficulties call 
for theoretloal investigation as a guide to experiment# 

Historically, the earliest theoretical problem, uatis- 
faotorlly analyzed in older studies, is that of a single 
spherical gas bubble immersed in an infinite body of water; 
in particular, the pressure as a function of time and distance, 
the period of pulsation of the bubble oan be explicitly 
determined# 

For the past few years an extensive research program 
on underwater explosions has been pxirsued by 0# S# Taylor 


* Bracket references refer to the bibl5.ography# 



-3 


249 





and his aasoclatos in England# In this country, theoretical 
work was started by C. Herring (Division 6, N»D*R.C») and 
is being carried on at the David Taylor Model Basin (in par- 
ticular by Captain W. P, Hoop, E. H. Kennard, 0-. E. Hudson) 
and by the N.Y.U.Croup of the A.M.P* at the request of the 
Model Basin and of the Bureau of Ordnanoe; oonaiderable experi- 
mental research is being done at the Underwater Sound Labora- 
tory of Division 0, N.D.R.C. 

Taylor [3] and others have determined the upward motion 
of the bubble under the buoyant force of gravity, a motion 
taking place in jerks. Herring [4] and others have studied 
the manner in which rigid walls attract the bubble* assuming 
that the bubble is rather far from the wall. Shiffman [5] 
has developed an improved method which permits numerical 
analysis of the motion when the bubble is close to the wall, 
and even when it touches the wall. The Influence of non- 
rigid walls and the repulsive effect of the free surface of 
the water on the bubble have also been studied. Much of the 
material is condensed and supplemented in the con 5 >rehensive 
work by Konnard[6]» 

A very ln 5 >ortant aspect of the problem is concerned 
with plastic-elastic deformations of the target in inter- 
action with the motion of the water. The process of daiaage 
can be understood only by a careful analysis of this inter- 
action. Extended research by J. 6. Kirkwood [7] has 6stab= 
ll.shed that the damage to a structure depends on the ratio of 
the diiratlon time of the incoming pressure pulse to the "time 
constant" of the structure, l.e., the span of time during 
which the structure is "receptive" to the impinging pressure 
effects. If this ratio is small, the impulse Is the more 
lit 5 >ortant factor for damage, while if it is large, the peak 
pressure is more important. 

Another problem that should be mentioned is that of 


( 


250 


-4 


tlie dhang© In the shape of the gas bubble* The migrating gas 
bubble does not retain its spherical shape, but flattens out 
in a plan© perpendicular to the direction of the motion* 

The result is a deceleration of the bubble and, as a oons©“ 
quenoe, an intensif ioation of the pressure pulse* Prelim- 
inary studies of this effect have been carried out independently 
in England and at N*Y*TJ* by Dr* Shlffman [8]* 

Finally, the still unsolved problems presented by the 
striking phenomena occurring at the water surface must be 
mentioned* A proper interpretation of the "domes'* and 
"plumes" rising into the air above the explosion should yield 
much information oonoernlng the process taking place under 
water* 


The effect of gravity in moving the bubble closer to 
the target is not the only factor which leads to Inci’eased 
damage by the second pulse* Another element of major im- 
portance, emphasized by Shlffman and Px’ledman, is that the 
peak pressure of the second pulse depends greatly on the 
state of motion of the bubble at the moment of the first 
contraction* It la shown that the peak pressure of the sec- 
ond pulse possesses a decided maximum if the gas bubble is 
stationary at the moment of contraction* For depths of 
water and weights of explosives actually used, it is indeed 
possible to keep the bubble stationary by counterbalancing 
the gravitational force with the attractive force of the 
sea bed and the repulsive force of the water surface* 

principle of stabilization forms the main point 
of the following report* It is assumed that the mine is to 
be placed fairly close to the sea bed, but not necessarily 
directly on it, and it la then found (roughly stated) that 
for mines of about 500 to 2000 lbs* of explosive in water 
about 70 to 150 ft* deep, the optimal position is about one 


-5- 


251 


maxliflum Isubble radius above the see. bedj this means that 
the mines should be moored approximately 30 ft# above the 
sea beds More precisely, for water of given depth and for a 
mine of given weight of explosive, there is a definite optl- 
niuin location for the detonation, that looation being given 
by the principle of stabilizations 

Of course, such a statement makes sense only if, at 
the outset, the position of the mine Is restricted to a 
place deep xindsr water, iae«, fairly distant from the target# 

Similar problems concerning the optimal position of 
an explosion also occur for charges nearer the surface, and 
for Charges dropped from fast flying planes and thus having 
an initial horizontal and vertical momentum at the Instant 
of the detonation# The discussion of such questions is 
planned for later reports# 


Richard Courant 

Contractor «8 Teohnioal Representative 



2S2 


'6 


Part 1> PlsGUsaloa of the Results 


la Preliminary Remarks a 

TOie present report la concerned with the following 
problems what Is the best location of a mine near the bot- 
tom of the sea so that dtirlng the secondary puls© the peak 
liressure is a maxlmom? The result obtained is: for weights 
of 500 to 2000 Ibsa of T.NaTa, in water of depth 100 to 
150 ft# g the mine should be placed about 20 fte above the 
sea beda More exact locations are contained In table 1, 
p# 9« 

It Is significant to compare different mine positions 
as to their effects on the secondary pulse* This report demon- 
strates that, during the secondary pulse, the highest possible 
peak pressure at the surface Is obtained when the mine is 
placed In a definite "beat** position. The theory Indicates 
that If the mine la placed even a few feet away from Its 
best location, the peak presatu*e might be as ‘little as one- 
half of that eorreaponding to the "best" location.* Further- 
more, when the mine la placed directly on the sea bed, experi- 
mental evidence is available showing that the secondary pulse 
la weak and erratlo*C9] 

Axicther point of interest Is that the "best" position 
also increases the effectiveness of the shook wave oon^ared 
with a location on the sea bod Itself. In the latter case, 
some experiments Indicate that the presence of the sea bed 
has no effect on the shock wave, and others, that It results 
In an Increase of about 12 percent in the peak pressure and 


« See the example discussed In section 4 below. 


Impulaer On the other hand, when a mine is at the "best" 
location (about 20 it« above the sea bed for a depth, say, 
of 100 fta), then the shock wave peeik pressure at the sur- 
face Is Increased about 25 percent over that of the sea -bed 
position since the pressure at the surface Is Inversely 
proportional to the distance from the explosion* 

2* The prlnolple of stabilization * 

The best location of the mine Is determined by the 
following Principle of Stabilization: For a given mass of 
explosive, the maximum peak pressure in the secondary pulse 
is obtained if the gaa bubble produced by the explosion la 
kept motionless at the time of its mlniniiiw aiste . This 
stabilization can be attained for mines by suitably balancing 

the upward buoyant force on the bubble against the downward 
attractive force due to the sea bed* In other cases, the 
repulsive force of the surface, as well as additional factors, 
could be utilized* 

In part II, a Tnathematioal demonstration of the principle 
of stabilization is given* The following plausible argument 
also points to the same result. In general, the total energy 
of the bubble and water is divided into two parts, namely, 
the kinetic energy of the water surrounding the bubble, and 
the internal energy of the gas inside the bubble* If the 
bubble Is moving at the time of Its minimum size, then some 
of the total energy la diverted into kinetic energy of the 
water and thus leas la available for the Internal energy 

[9] reports no change In the shock wave for a mud bottom* 
Experiments in Woods Hole for a sand bottom show an Increase 
of 12 percent, according to an oral discussion* Theoretically, 
If the sea bed were perfectly rigid, the peek pressure would 
be multiplied by the factor ^ , which means an Increase of 
26 percent* Actually, however, the explosion tears a hole In 
the sea bed and also transmits a shook Into it* 


254 


' 8 ' 


of tiv© gas* ©lis would reduce the pressure insid© the gas 
and also the pressure In the water. To obtain the optizoua 
pressure, therefore, this bubble naiat be motionless at the 
time of its minlHEisi alae«* 

For an appraisal of the value of stabilization, the 
damage due to the secondary pulse must be studied. All 
theories of damage show that for a pressure pulse which 
lasts ?Lon.2 relative to the time oonstant of the atruoture 
to be damaged, the damage is approximately proportional to 
the peak pressure and not to the Impulse or energy. Since 
the duration of the secondary pulse la approximately twenty 
times that of the shook wave, it would seem that maximizing 
the peak pressure of the secondary pulse increases the damage. 


3« location of the mine. 


Let W be the weight of the explosive in pounds, 

(the numerical values refer to T*K»T* for which the charac- 
teristic constants were available to the authors), D the 
dlstiiioe in feet from the bottom of the sea to a point 33 
feet abov(} sea level (allowing for the pressure of the atmos- 
phere), and Bg the distance in feet from the center of the 
mine to the aea bed® The problem is to determine the best 
value for if W and D are given. Ihis is more easily 

expressed in terms of the non-dimensional quantities which 
are used in part II. For the unit of length select 


( 1 . 1 ) 


L = 


13.2 



f 


which r«'presents (approximately) the maximum radius of the 


* Ehis argument is not completely rigorous since contri- 
butions to the pressure in the water occur from the motion 
of the water as well as from the gas pressure inside the 
bubble. Ihe fact that the gas pressure is the more lu^iortant 
contribution arises from the nathsraatlcal analysis of the 
Interplay of the two effects. 






gas bubnl© at the first expansion. Set 


( 1 . 2 ) 


^ =T. * Dr, 


0 i, 


so that d and h^ represent the distance D and B 
measured in ’’bubble radii”. 


Sea leueL 


1^6 best position of the mine is determined approxi- 
mately from the equation 


(1.3) 


d ® 6»2'b + »4 • 

o o 


(See derivation in part III). (Jraphs of (1.1) and (1.3) 
are drawn in figures 1, 8 at the end. 

On the basis of (1.1) and (1.3), the following table 
can be constructed showing the best location: 


Weight of T.N.T. 
Tin lbs. ) 


1500 


1000 


1000 


Table 1 

Depth of Sea Bed 
Prom Sea Level 
(in ft. ) 


Best Location of Mine 
from sea Bed 
{ in ft. ) 


20.2 


17.0 


10.0 










m 


10 - 


33i 6 following limitations and qualifications should 
h© noted* First, the affect of the free surface has been 
Ignored, so that these results are valid if the alns is 
sufflcleutiy far from the sea lev^el, say, more than 3*5 L ft* 
Second, it is aastuned that the bubble does not meet the 
sea bed in the course of Its motion* This Is approximately 
equivalent to > •8L* Third, the sea bed la asstuaed 
to act like a rigid wall* 

The validity of this last assumption should he tested 
by experiment* But there are plausible reasons for supposing 
that as far as the balancing phenomenon is concerned, the 
sea bed does act very much like a rigid wall* Daring the 
largest part of the time of pulsation of the bubble, the 
bubble is large and the pressure In the water Is ?ow| there« 
fore, a sand bottom could be considered as rigid* The 
balancing effect is determined In the main by this portion 
of the period of pulsation* 

4* An example * 

Consider a mine with 1500 lbs* of T*N*T, in water 
of depth 150 ft* Figure 3 la a graph showing the relative 
magnitudes of the peak pressure In the secondary pulse 
when the mine is placed at varying distances from the 
bottom of the sea# The best location of the mine Is 21*0 ft* 
from the bottom} the resulting peak pressure at any point 
in the water, as ooa 5 >uted In part II, Is 2015/R atmospheres, 
whbre R is the distance in feet from the point to the center 
of the contracted bubble* If the mine is Initially placed 
at any other distance from the bottom, this value for the 
peak pressure i?s to be multiplied by the factor whose graph 
la drawn In figure 5* ^ 

This graph demonstrates a remarkable sharpness* In the 
peak pressure curve as a function of the distance of the 

* The correcitTon ^e'e to the i^ree stirfaco is treated in 
part HI, seotion 

* In realH^y the graph will not be quite as sharp because 
of deviations from the assumptions listed on page 1S» 


11 » 


257 


R 

S 


I:; 


l; 

i'-; 


'A'- 


^ I 


sins frora the bottoia* Fos* exaiaple, if the mine Is plaeea 
either 1? ft. or 29 ft. from the bottora, the peak presaure 
is only one-half of the maxliBum.-. 

K one is interested in the damage to objects on the 
surface of the water, it is necessary to take into account 
the migration of the gas bubble. Again considering the above 
example, the peak pressure at the surface directly overhead 
is 15.6 atmcsphsrss (excluding the reflection from the bottom) 
when the mine Is placed 21,0 ft. from the bottom. For any 
other location of the mine, this value Is to be multiplied 
by the factor whose graph Is drawn in figure 4. The graph 
still exhibits a considerable sharpness, but if one places 
the mine somewhat higher than 81.0 ft. from the sea bed, the 
effect on the surface would be Improved slightly. This Is 
due to the upward migration of the bubble. Note, however, 
that If the mine is placed too high, say 87 ft. from the 
bottom, the favorable effect of the upward migration la 
sharply oounterb^lanced by the resulting weakness of the 
secondary pulse. 


Part II. Mathematical Study of the Secondary 
Pressure Pulse 




1. Introduction . 

High pressure pulses are produced only when the size 
of the gas bubble is near its minimum. During this time, 
the buoyant force due to gravity is small. Likewise, the 
proximity of rigid walls or free surfaces will not materially 
affect the motion, since their Influence depends on the ratio 
of the bubble radius to the distance from the bubble, and 
Is small if this ratio is small. Therefore, during the stage 
of minimum sizs, the bubble can bs oonsldsrsd as liamer-sed 
in an infinite body of water and subject to no outside forces. 
The pressure pulse produced by the bubble under these 





2S8 


12 


CljL^OXl^S oaHCSS WxH lnVo3 uxgn uOu« 

On the other hand, all these ‘outside influences 
do n-Pfenfc the bubhle in the stage when It Is not small. 
Consequently, in contracting to its mtinimumi size, the 
bubble and surrounding water have a linear momentum which 
remains practically constant while the bubble passes through 
the rainlraura size stage when the outside influences can be 
ueglectede The pressure pvilse produced by the bubble depends 
on the linear moraentum acquired by the bubble. The dependence 
of the peak pressure on this linear momentiim Is the main objec- 
tlve of this investigation and la represented in figure 6. 

For the derivation of this result, the following aaaump- 
tlona are made: 

1. The water is an ideal incompressible fluid. 

2. The bubble remains spherical in shape. 

3. The gas inside the bubble Is in thermal 
equilibrium at each instant and follows the 
adiabatic law. 

These assumptions are reasonable for the xnajor portion of 
the period of pulsation, and are violated only In the very 
short time interval when the radius of the bubble is small. 

The violations are in the nat'ure of corrections to the theory, 
and will not materially affect the principle of stabilization. 
We shall not enter into a discussion of the assumptions, but 
merely refer to [4]. 

2. The Energy Equation . 

Let A be the radius of the bubble at any time, B 
the vertical distance of Its center from some horizontal 
level, and the hydrostatic pressure of the water at the 

center of the bubble. The motion of the bubble is described 
by specifying A, B as functions of the time T* 






-13. 


259 



The velocity potential $ describing the flow around 
a moving, pulsating sphere in an infinite body of water is 
easily obtained from olaaslcal hydrodynamics. It is 

( 2 . 1 ) $ .. 

where R,& are coordinates as Indicated in the diagram above, 
and the ’’prime” denotes a time derivative. The kinetic energy 
i of the water is then 

(2-2) f = ar^A®tA.® + Ib.^I 

where p is the density of the water. For these classical 
results, see, for example, [3] or Appendix II. 

The potential energy of the system of gas bubble 
and water is 


(2.3) 


U = + 0(A) 




where the first ten\i is the gravitational potential energy 


260 


-14= 


of tlie bubble due to the absence of water and the second 
term G(A) Is the Internal energy of the gas* If the 
gas obeys the adiabatic law, the Internal energy is 

3(A) = jfrr^T 

where ^ is the adiabatic exponent, K is a constant depend- 
ing on the explosive, and M is the mass of the explosive 
usedo 

The energy equation is 
(2.4) ‘J’+U=E 

where E is the constant total energy in the system (after 
the passage of the initial shock wave due to the explosion). 

3. Non-dimenalonal variables . 

A considerable simplification in the writing of the 
equations is obtained by introducing non-dimensional quantities 
with appropriate scaling factors. Likewise, In part III, it 
la even more Important to Introduce non-dimensional quantities. 
For the purpose of coit^arlng the formulas developed in this 
part with those in part III, we shall use the scaling factors 
convenient for part III. It should be mentioned that these are 
not the most convenient factors to use if the results of part 
II were the sole objective. 

Set 


(2.5) A = La, B = Lb, T = Ct, 

A 

'W 

where L,0 are scaling factors with the diraanslons of length, 
time, respectively, and a, b, t are non-dimensional variables. 
(Henceforth, capital letters will indicate dimensional quantities 
and small letters non— dimensional quantities.) The scaling factors 


■15 


261 


whi^i we shall use are 




where Is the hydrostatic pressure at the center of 
the explosion, and p Is the density of water** These 
scaling factors are advantageous because they have a simple 
physical lntei»pretatlon intimately connected with the motion 
of the bubble* It will be seen In part III that L Is the 
maximum radius of the bubble If the internal energy of the 
gas could be neglected* Actually the maximum radius is 
approximately *9SL* Also, it will be shown In part III that 
the time scaling factor 0 represents 2/3 of the period of 
pulsation of the bubble if no outside influences such as rigid 
walls or free surfaces are present* 

The energy equation In terms of the non-dimensional 
variables becomes 

♦ 1 ^ 8 ]* .5 + 2 ^ = 1 , 

where the "dot- ” denotes differentiation with respect to the 
non-dimensional time t* 

4* The scaling factors and the Internal energy of the gas for 
T. N. T* 

It Is desirable to express the scaling factors L,C 
In terms of the mass of explosive and the depth of the explosion. 


w These scaling factors differ from those used by G* I* Taylor 
In [3]* 


(2*7) a® 


262 


-16- 


distance of the center of the s:;plosion. froa 
a point 33 ft* above sea level, so that s pgD » It is 
still necessary to express the energy constant E in terms 
of the mass of exploalve. For the ease of T* N* T«, sxperi'^ 
mental results Indicate that E is approximately one—half of 
the total chemical energy released by the explosion. See [4]. 
Slight changes in E will not materially affect the results 
S appO£ia?s In i/ii@ l'o3T'i!i Ha Uslng 'tlis givsn 

by Q* I, Taylor in 1 3 ] , P« 4 , and converting to the English 
system of measurements, one finds that 


(S.8) 



Where W Is the weight of T. H. T. in pounds, nnd is 
the distance in feet of the center of the explosion from a 
point 33 ft. above sea level. 

Likewise, making use of the experimental values given 
by a. I. Taylor in [3], p. 4 . the -quantity for T» N. T. 
Is, in n 


where M 
non-dime 


(2.9) 


where k 


Cihanglng to Esgllab uzilts and {2«8}, one finds 

that 

(8.10) k * •O^rrt , 


vhore has the sane m^inlng as in (2.8) • Eg[uatlons 
(8.9) ttftd (8.10) are tho desired expressions of the Internal 
energy for T.N.T. 


6. The eouatlona of awtlon for small valnes of the radlna 

^n the 

Thft ^eoondarr high pressure pulse is emitted r , ^ 

ellftl A 

huhble near its miniTsam size, and henceforth ** ^ 

additive 

oons.ider the case irtien the radius is small. 

- 3 .... vUen nsgllgiole 

in th. «>»er «matlon (2.7) J* tlon b.oon.. 

ooc 9 >ared vltlx the cuotoer and the r 


\ i 




The Lagrangean 


(8.12) 


. ..quations of motion are 
. 0 

dt 34 ^ 


4 

(2.13' ' dE 83 

where X* ^ heo'Tae 

(S.14) ^ 


.2. 


3 k 


4 (2 a^) - 3a*("^^ + 5 f » 


-18- 

:284'K'^ 

a' 2«15) shows that whon. the bubble Is near 
I!ho o /•! rae linear momentun ^ a®fc remains constant- 
Its mlnlmur ec .^ - '-nt value by s • Then 
Denote *•’ 

,{ 2 . 16 ), b = , 

' a® 

axid the energy equation (2«ll) yields 


(2.17) 


« 1 

‘ -7 


k 

7 ^ 


,«2 

08 


The aqua tloitf (2.16), (2«17) completely del?*. 'nine. In *rn:)3 
of s, the motion of the bubble during Its moj’t Inber-iiatlog 
phase when the radius a If. amall. An Integrator-* .f (2.17) 
and then of (2.16), ylel^i? a, b as functions of t. 

Equation (2. 14) <^lves an e:;cpression for the quantity 
(a®l)*, which Is iKpirtant for the determination of the pressure: 

(2.18) (a^a)* * 

Substituting (2.16), (2.17) In (2.18), we obtain 


(2.19) 


(.®D* 




6. The minimum radius. 

The value of a quantity q at the time when the bubble 
Is exactly at its minimum size will be denoted by q. The 
iw4Ti<w.»'m radius a of the bubble is obtained by iiettlng 
i « 0 in (2.17). The resulting equation la slarjllfled by Intro- 
ducing, in place of a, the (non-dimensional) inbernal energy 




u of the gas, defined by 


s /(' 


y, T' 


it k 


il • i 

,V I Si 


■I r « 
if 4 


(S.20) 


^ » or a 


The qviauiilty u represents the portion of the total energy 
of the syateai which exists as internal energy of the gas at 
the time of minimum size* From (2.17), we get the following 
equation which determines u: 

(2.21) = 'i (■%) t (Oraphed in Pig. 5). 

u - \k / 

'J^e values of various other quantities at the time of 
mlniTmm size Of the bubble are of lntei*3at« Prom (2.16) we 
have 


( 2 « 22 ) 


and (2.19) yields 


b = 


3 s „ 3 a r4 

rr “ “iT 


(2.23) 


o. . % 8/3 

(a**a) = — 4rw u (4 - 3u) . 


Thus, given s, the Internal energy u at mlnlMunj 
size is obta ined from equation (2.21). All the other quantities 
a, (a^a)* can then be found from (2.20), (2.22), (2.23). 

V. The pressure pulse . 

Sections 6 and 6 give a ooiaplete description of the 
motion of the bubble when It is near its minimum size. We 
shall now Investigate the pressure pulse delivered to the 
surrounding water. 


!'■ I 


266 


-= 20 - 


The potential function ^ for the flow of the 
surrounding water has been constructed In equation (2*1). 
!Ths pressure in the water can be obtainod from Bernoulli »g 
equation as follows* 


(0a84) £ + I (grad «!> )^ - (||- - B» If-) = 0. 


where P Is the excess over hydrostatic pressure, Z = R cos 0, 
and the term B» appears because of the moving coordinate 
system* A substitution of (2*1) In (2*24) yields 


(2.25) 


lAfADi 

P 


+ terms in higher powers of 


1 




For points’ in water not too near the bubble, the first term 
on the rl^t hand aide of (2*25) is dominant and the other 
terms may be dropped* Introducing non-dimonaional variables, 
we obtain 


(2*26) 


p s isfail a ^ , i(a^a)*, 

R 3 R 


The preaswo in the water, therefoi>e, depends essentially 
on the quantity (a^a)* • 

!Hie formula (2*18) for(a &)* oan be given an Inter- 

ft > 

estlng Interpretation. The last term - represents 

the (non-dimens Iona 1) pressure of the gas inside the bubble, 
and (2*26) then shows that the pressure at any point In the 
water is oosr^osed of two parts 8 the internal pressure of the 
gas, and the dynamic pressure due to the motion of the water* 
This Is more clearly seen If equations (2*26), (2*18) are re- 
written in terras of the original dlraenslonal variables* The 




,// in-., 


« 21 «» 


267 


result is 


(2.27) ^ = ff 1 f-n^^ + ?(A)] . 

where p(A) la the pressure of gas Inside the ’ •.ilrble 
when Its radius Is A. 

Ihe principle of atahlllz'tlon *8serts that th« most 
Important term In the expression ;..ci;7) gai prc .?.upe 

tern p(A). To obtain the maximum possible o:<.« 

should maximize p(A) even at the expense of complete «i iir/P'?," 
tlon of the dynamic terms -g pA*^ + ^ pBi^ , This Is not 
obvious and requires a mathematical proof. 

Returning now to the expression (2.26) for P, we 
wish to determine the time when the pressure pulse reaches 
a peak value. It la to be expected that the peak pressure 
oeoura when the bubble la of minimum size. Although this 
la not obvious from the expression (2.18) for (a^a)*. It 
Is a simple consequence of the eq[uatlon (2.19). By differ- 
entiating (2.19) with respect to a and by using (2.17), we 
have 




;2 


-2 

- 

“ST 


- 21 

35 


k 

T57? 


This shows that is negative and therefore attains 

its maximum at the smallest possible value of a. Hence 
the peak pressige occurs at the time of the mini™™ gize of 
the bubble . 

The value of the peak pressure P can now be obtained 
by substituting (2.23) in (2.26)| the resulting expression 

la 


1 

IT 




5^- 




(2.28) 


u ®/®(4 - 3u) 



22 


Thla Indlce. tea the depend .nee of the • eak presaune f on 
the Intern/il energy u •! s the time o. minimum alzey and 
through (2«iSl)s on the^l.^near momentum a. 

The Quantity u ' (4 - 35) wllj he called the 
" pressure fa'.otor " and ’- ill he denoted h; the symbol q: 


(£•29) 


0/3 

5 (4 - 35 ) • 


It depends on^ a by virtue of (2*21)» 
8* The optltguA pea ^ : pressure * 


ffe are r?,o>? In a position to find thal linear momentum 
t wh'.ch the maximum peak pressure* The peak pressure 

P, by (2»28), proportional to the pressure factor q In 

(2*29), v"Mch in turn depends on s by virtue of (2*21)* A 
graph of . 4 H i funct-lon of is drawn In i'lgure 6, and 

demons t’’t* tee the iver^ '.cant fact that q ta largest 

when ? Is practically 0 (or u i rrctiu,; ,l2y 1'-^ Aotually, 
by (2« .r9), q Is a\ maximum when u a or hv i ^*81; , when 
f m •■- 51 k^» But ^ince k Is small, this dlff 
from X a 0 that l|; can be neglected* See figure 

Thus, the fallowing general Principle of SUhlllzatlon 
has oeen demons tratijd; For a given mass of explr>siv .~ . the 
opt /,; :aan peak pressur e In the secondary pulse la cl; t . Ined by 
ke > r.lng the bubble m<^\tlonless at the time of its n)>' r ; mum size* 
The value of tha optimum peak pressure ^opt< 

Ciitalned from (2«28) b*^ setting u = 1, with the resul ’ 

p\ _1 PoL 


f 

TIsing (2«7) and (2.10), and expressing In atnosphex- 

by means of P^ = Bq/ 33 atmcgpheres, where Is the 

distance In feet of the cente> of the explosion from a pc. it 



-23- 


269 


33 ft* above sea levels we have 


aiimospjueres 


where W is the weight of T* N® T® In pounds and R is 
the distance from the center of the bubble In feet* 

^his may be compared with the peak pressure of the 
primary shook wava^ whose value, experimentally obtained, is 


1yV3 

^shook = -IT- atmospheres® 


The optimum peak pressure of the secondary pulse Is approxi- 
mately 15 percent of the shock wave peak pressure, but the 
duration is much longer, as will be seen in the next section* 

9* The impulse ® 

The ln5)ulse I per unit area, carried by the secondary 
pulse, S.S the time Integral of the pressure, 

I B y PdT ss 0 y* Pdt, 

* 

whore the limits of Integration are the times when P « 0? 

By (2.26), 

o# • 

where the limits of Integration are the times when (a a) *=0® 
using the fact that the motion Is approximately symmetric about 


«■ See [3) page 13, where this experimental result is quoted 
ir metric units* 

% The significance of this value of the Impulse will be 
discussed later* 


270 


-24- 


the tim© of the mlnlimim size of the bubble, the abov’e 
integral becomes 


12.51) 


4PoC 


!T 


[a^a]. 


2* 2* ^ 
\7here la a] Is to be evaluated at the time when (a a) = 0, 

or a^a is a inaxlinum. 

Equations (2.8), (2.9) and (2.16) yield 


-fi 


a a = a - ka 


V4 . 3a . 




and 


^ (a^a^) = 1 - 


k 


2 


^ 3s 


- 4a‘ 



Setting the last expression equal to zero In order to find 
the jnsixlimm of (a a) , we obtain an equation for a* If 
both k: and, s were equal to zero, the solution of this 

equation would be a » .63. In a more realistic case when 
k = .2, 3 = .06, one obtains a = .61. This Illustrates 

p0 

that the value of a when a a Is a maximum is not very 
sensitive to changes in the momentum s, since s is generally 
quite small In actual oases. Using a = .61 and k = .2, 
the value of [a a] Is .55. 

By (2.7), equation (2.31) becomes 


P — CI» Q T 


W 


2/3 


atmosphere-seconds. 


Taking a typical case of a depth of 100 f t» , so that 0^=133, 
we find that 


,, 2/3 

(2.32) I = .37 ~ — atmosphere-seconds. 

R 



it See Part III 




R»®gaiiaiiw0B 


W© have shown that the linimls© carried hy the seoondai’y 
pulse is practically independent of the linear momentum a 
ox T^ixe xs p^iven hy 

ISi© impulse cari’lsd by the primary shock wave is experi- 
mentally measured to be 


w2/3 

^shook “ ""g — atmosphere-seconds o 

A comparison with (3*35?) indicates that the impulse cari’led 
by the secondary pulse is three times the Impulse carried by 
the shook wav?' 

This conclusion can be stated in an approximate manner 
In terms of the durations of the pulses* Using the relation 
between the peak pressures obtained at the end of the last 
section, one can state that the secondary pulse lasts about 
eighteen times as long aa the shook wave* 

This result, and formula (2*38), should not be taken 
too literally but merely aa order of magnitude statements* 

The reason for this is that the In^iulae carried by the secondary 
pulse was ooaiputed between the times when the pressure is zero* 
This includes a relatively long period of time during which 
the pressure is low, and the resulting oontributlon to the 
impulse will not be experimentally noticed and will not have 
much effect In damaging structures* 

I 

ij 

* CSlj P»18 


o 




272 


-S6- 

Part Ilia The Meohanlam o.f Stabilization by 
gravity and the Sea Bed> 

1« The exact equations of motion * 

In this part, the effects of gravity and of a rigid 
v/all upon the motion of the bubble are considered* The 
linear momentuia s produced by these factors in the course 
of one complete pulsation of the bubble la calculated* As 
was shown in part n, it is s which determines the peak 
pressure produced by the bubble at the time of its minimum 
size* An equation for determining the boat location of the 
mine can be obtained by setting the momentum s equal to 
zero* 



>27 


273 


( ■ 



r \ 


\ / 


/ 


Use the same variables A^B as in part II and 1st 
D stand for the (fixed) distar.ee of the fsea bed from a 
point 33 fts above sea level* Tn place of the equations 
(2*2), (2*3), (2.4), wb have* 

(3,1) 2itpA® [d+fg)A^ - ef^^AB + + 


+ lirA^pg(D-B) + a(A) = E, 
3 


where f^ef^^^fg are functions of the ratio A/B which 
represent the influence of the rigid wall on the motion of 
the bubble, ^ir A pg(D-B) represents the gravitational 
potential energy due to the lack of water in the space 
occupied by the bubble, and g is the acceleration due to 
gravity* Explicit expressions for f^, f^, fg are derived 
in [5], and will be reproduced in appendix II of this report* 
Introduce non-dimensional quantities as in equations 
(2*5), (2,6) of part II, sections 3 and 4* The energy 
equation becomes 


(3.2) 


a®[(l+fQ)a^ - 2f^ab + (| + fg)b^] 


d-b 

* 

o 


a 




= 1 , 


where ^ » "^o “ ~ initial distance of 

the bubble from the aea bed* 

The motion of the bubble Is determined from (3*2) 
and one of the> Lagrangean equations (2*12), (2*13)* Using 


* See appendix II for details. 




tne b“®quatlori (2*15), wo obtain 


(3*3) ^ 1 +f._j)b - Ea^f^a] 


•1 * • . '^2 *2 
— ao + ' — b 

at d06 


X 1 „3 

cPS" ^ ^ 
o 


where 


(3.4) 


a 

5E ® 


and the quantities fg are functions of OL only* 

Kie differential equations (3.2), (3.3) are to be 
integrated, subject to the following initial condition* 

• 

at t = 0, a s= a^, a = 0 
b = b^, b - 0, 

where is the smallest root (near of 

a® + = 1. Of course, those initial conditions are not 

exactly realistic, since at the very beginning a shock wave 
is formed by the explosion* But the time interval required 
for lncoH 5 >ressible flow to set in is relatively minute and 
may be ignored* 

The quantity in the brackets on the left-hand aide 
of (3o3) is the linear momentum s of the system* The 
first terra on the right-hand side of (3*3) is due to the 
presence of the rigid wall, and the second terra is due to 
gravi ty* 

By Integrating aquation (3*3) v/e obtain the momentvun 
a at the end of the period of pulsation of the bubbles 

See [4]e 


" - t' - I, 


275 


t. 



r ) 


( 3s 5 ) s s= 


/: 


a 


dK, 


- 2 


df^ 

doT 


• • 

ad 


dfp <1 n 


dt 


+ 


o 



$ 


where the Integration Is extended over the full period 
5 of the pulsation, from the explosion to the time of 


zalnlmum size* 


2* The approximate evaluation or the period and the momentum o 

The results of a numerical Integration of equations 
(3*2), (3*3) are tabulated for special oases In appendix II. 
They serve as a check on the approximations which will now 
be made to evaluate the period u and the momentum s* 

Tlie bubble expands to a maximum size before contracting 
again. Indicate the value of a quantity at the time of maxl“ 
mum size by a subscript 1* Thus t^ Is the time of maximum 
size, is the linear momentum.^ etc* We shall Introduce 
the following approximations which are especially accurate 
when the bubble is In Its balanced position: 

1* The time 5 and linear momentum s at the minimum 
size of the bubble is twice the corresponding quantity at 
the maximum size| 1. e», t = 2t^^, s = 2s^» This assumption 
agrees very closely with the numerical integration of the 
equations* It means that the motion Is approximately 
symmetric about the time t^^ of maxlnaun size* This would 
be exactly correct if the b-coordlnate did not vary* 

2* During the first half of the period of pulsation, 
until time S 1s small and can be taken as zero, so that 

b remains equal to b^* IThis agrees satisfactorily with 
the numerical integration, as well as with experimental 
evidence* 





276 


- 30 ' 


(3,6) 


Substituting b s In (3*2), we get 


aV2 ,/i + f 


The solution of (3«6) is immediately obtained by Integra* 
tlon, with the following result for the half -period t^^: 

/®t^ /* a- 

(3.7) ^ J ^ ^ da 

/l - - lca-V4 

•where a^ is the smallest and the largest root of 

(3.8) 1 - a® - k = 0. 


To evaluate the integral (3.7) it is necessary to 
have a specific value for k. The value of k Is given 
by (2.10) and depends on the distance of the explosion 

from a point 33 ft. above sea level. The following table 
Is calculated from (2.10); 


1^0 

33 

83 

133 

183 

k 

.146 

.183 

.206 

.223 


Selecting a depth of wator of about 100 ft. as typical, 
the value of k is approximately .2« We shall select this 
value of k tliroughout the remaining calculations* A differ- 
ent value for k will change the forimilas slightly. 

For k = .2, the roots a^, of (3.8) are 


(3.9) 


= .118, 


a^ — .924. 




. ^ -'-v 




Tlie last equation (3*9) la noteworthy, since repre- 

sents the non-dimona lonal maximum radius of the bubble. 
The aotual n^xtimim radius Is thus 


A. <= «9S4 L* 


If the Internal energy the gaa were neglected, 

the maximum radius would be exactly L* 

The quantity yjl~^~F^ In the Integral (3*7) Is 
very closely equal to 1 + ~ s 1 + as is shown In 

appendix II« Making this approximation, we can evaluate 
the integral (3*7) numerically* This numerical evaluation 
Is discussed in appendix III, the result being 


•735 + ^ 


The total (non-dimensional) period t of the pulsation is 
therefore 


(3«10) t = ® 1*47 + 


= 1*47 




Pormula (S.IO) shows that If no rigid walls are 
present (b^ ssoo), the actual period T In seconds would 
be 

T = 1«47 C, 


or about 3/2 times c • This provides the physical Inter- 
pretation of the scaling factor 0 mentioned In section 3 
of part II. 

Substltixtlng b = b^j, b = 0 In (3.5), we obtain for 




278 


■32 


the half momentum s 




. », - /‘l ^ ^ I^dt + ^ a^dt 

1 Jc Jo ’ 


or- 


(3.11) 


2b 


r-' 


df. 


o " o 


\/l - a° - Ha -V* » •as 


da 


+ 3=bT 


1 /’“i \/i~^ 


1 /’“i _!i: 


a» - 


da ^ 


using (3.6}« Tho first term on tiie right hand side of 

(3*11) Is the downward momentum due to the rigid wall, and 

the second term Is the upward momentum due to gravity. 

Again taking k = .2, using the approximation 

o, df /dot a 

J1 + f„ ss 1 + a 1 » -jf;- , and evaluating 

° 

these Integrals numerloally, we get the following expression 
for the full momentum s’: 


(3.12) s 


-( 


.03^ . 

Va2 


.019 

TT' 


1 

w 

o 


.704 


^ »148 

TT 

o 


See appendix III for the numsrleal evv.luations. 
3. The stabilized position . 


By the principle of stabilization, the maximum peak 


»3S- 


279 


pressure Is obtained by setting a = 0, with the result 


d 



.704 


► 113 



# 


Expressing d In descending powers of b^, we get 


(3.13) d = 6-g b® + 3.3 b. + .4 . 

A graph of equation (3.13) Is drawn In figure 2. 

More generally, the peak pressure, as a function of 
the distance b^ of the mine from the sea bottom. Is 
obtained by combining equation (2.28), (2.29), figure 6, 
and equation (3. 12). As a typical example of this dependence, 
consider the case discussed in section 3, part I, of a mine 
containing 1500 lbs« of T. N. T. In water of depth 150 ft» 

The scaling factor L, as obtained from figure 1, Is 

L = 27.6 ft. 


The depth D of the sea bed from a point 33 ft. above sea 
level Is 183 ft#, so that 

d s ^ s 6.63. 

The best location, of the mine according to figure 2 is 
bp = .76, or 


= Lb^ = 21 ft. 
o o 


above the sea bed. 


280 


,34- 


If the mine is located at various other distances 
Bq, the pressure factor q can be obtained from figure 6 by- 
first determining s from equation (3»18). The results are 
tabulated below, and the graph is drawn in figure 3» 


■1 

• 6 

B 

.76 

s8 

.9 

1.0 

1*1 

1.2 

s 

-.066 

-.020 

+.003 

.015 

m 

*059 

• 074 

.087 


.51 

m 

1.0 

.96 

.72 

.55 

B 

• 42 


As stated In section 5, part I, figure 3 shows a remark- 
able sharpness in the peak pressure curve as a function of the 
distance from the bottoma 

4. The migration of the bubble * 

A foi*mula for the distance travelled by the bubble in 
the course of Its pulsation Is difficult to obtain because it 
involves a complicated repeated integration. But by combining 
a theoretical argument with the results of numerical integra- 
tion, an empirical formula can be developed* 

It seems reasonable to suppose that the displacement 
** «■ 

A b =s b - b of the bubble la an odd function of a. It 
o 

can therefore be represented by the beginning terms of a Taylor 
exp ansi on j 

_3 

Ab = + OgS , 


•where Cg are appropriate constants* In fact, a theoretical 

juatif ioatlon of this can be given on the basis of the differ- 
ential equations (3*2), (3*3), but this will be omitted here. 

We ean determine the constants c^^, Cg empirically by 
using the results of the numerical Integration tabulated In 

















35 


appendix !• The least squares solution, yields the following 
formula : 


(3.14) 


Ah = 19s (1 - 62a ). 


A graph of (3.14) la dravm In figure 7, and Is oonipared with 
the tabulated values. The forniula (3.14) Is seen to he very 
close to the tabulations. 

5. The correction due to the free surface . 

For the sake of conpleteneas, we shall Include some 
remarks concerning the effect of the free water surface. The 
latter exerts a repellent force on the bubble, aiding the 
downward force of the sea bed. A complete discussion would 
require a knowledge of the potential function ^ and of the 
corresponding values for f^, f^ and fg» A careful examina- 
tion of the previous argument, however, shows that only the 

fli’st terms In the expansions of f. and 2£s In powers of 

® 3 b 

a were used® In appendix II these first terms are obtained by 
successive reflections. The results are as follows: 

If G Is the depth of the bubble below the free surface, 
and X Is set equal to , then the expansion of f begins 


with 




OA O 

and the expansion of begins with 


“ G'(x) 

2b*^ 


where the formulas for P(x) and Q(x) are given in appendix II. 
A short table of their values follows: 


































-37- 


28f' 


f ] 

^ It la to be noted, that x = 1 corresponds to no 

free surface (c = )p while x = 0 corresponds to a 
point midway between the free surface and the rigid bottom. 

An Interesting result Is that P(x) =0 when x ® l/3j 
so that the influence of the free surface on the period of 
the bubble cancels the influence of the rigid bottom. 




284 


-38- 




Appendix I» !Ifhe Mumeyieal Integration of the 
Dlffeg‘entlal Equatlona # 

To test the validity of the approximations made In 

part III, th© exaot differential equations {3,2 ), (3.3) 

were integrated numerically. The coefficients f , f _ , f „ 

o* 1' 8 

and their derivatives with respect to o6 , are tabulated 
In table 4, appendix II. The numerical integration was 
carried out by the Mathematical Tables Project operating 
under the Applied Mathematics Panel.*" 

In this report we have not reproduced the complete 
tables, but have included the graphs drawn in figures 8 - 11, 
which are baaed on these tables. Here we shall sot down only 
the most interesting items in these calculations, namely, the 
behavior at the beginning, at the time of maxlmurj size, and 
at the time of minimum size. The follo^ving table shows the 
results ; 


* la particular. Dr. G. Blanche, Dr. C» Lanczos and 

Dr. A. N. Lowan helped overcome the considerable difficulties 

in the numerical Integration. 



** Details of the computation can be obtained from an extensive 
report by the Mathematical Tables Project prepared for the 
Applied Mathematics Panel. 






































286 " 40 “ 

Uae period i and the momentum a for the cases 
listed above were computed from the fornmlaa developed in 
part III- They are eolleeted in the following table, and 
compared with the numerical values given In the preceding 
table* The agreement is excellent. 


Ti\BLE S. 



1.70 1.70 -.067 -.066 


-41- 


287 


Appendix II« The Velocity Potential for a Pulsating 
Sphere Movln;:; Perpendicularly to a Wall * 

!• Statement of the prohlem » 

An Ideal inoompreasl'ole fluid la bounded by a plane 
infinite rigid wall, and has a pulsating and moving sphere 
immersed in lt« It is required to find the velocity poten- 
tial describing the (irrotatlonal) flow, and to obtain the 
kinetic energy of the liquid* The case of a moving sphere 
is classical and can be found in the standard books on Hydro- 
dynamics o The pulsating sphere, however, does not seem to 
be generally known and has certain features of interest* 

The treatment here follows [5]* 



Let A bo the radius of the sphere at any time T, 
and B the distance of its center from the rigid wall* The 
velocity potential ^ describing the flow of the liquid 
satisfies the potential equation 

^xx ' ^yy zz 


The method used here can also be applied to a spherical 
wall* 


288 


-42- 


and lias the following boundary conditions; 

A* « B* cos 6 on the sphere 

0 on the plane, 

where n is the normal direction pointing into the liquid 
and the '‘prime” denotes a time derivative. 

The oonatructlon of ^ can be decomposed Into two 
simpler problems by setting 



(2) ^ = A' ^0 - B» 

where are potential functions satisfying the 

following boundary conditions; 


(3) 

(4) 


. a 1 on the sphere, 

o n 


3$i 

= ss cos 0 on the sphere, 
on 




on the plane 


on the plane* 


Physically, represents the potential function for a 

sphere of fixed center and radius expanding with unit velocity, 
while function associated with a rigid sphere 

moving away from the wall at unit velocity. The functions 
^ o* ^1 constructed by means of the method of images* 

2* Some theorems on images * 

The theorems that follov; supplement well known theorems 
on Images* See Mllne-Thomson, Theoretical Hydrodynamics, 
Chapter 15* 


,1 





-43- 


289 





We wish to find a potential fvmction <$ ; defined 
outside a given sphere, with a given distribution of sources 
Outside the sphere and vanishing normal derivative, = 0, 
on the houndaryo A standard method for constructing ^ Is 
to place a suitable arrangement of sources inside the sphere, 
called the image of the original distribution* 

The image of a point source and a radial dipole are 
well known,, and will be found In Milne-Thomson , pp, 420, 421* 
Here we consider the Image of a line source* 

Theorem 1 * Consider a sphere with center 0 and radius 


a* The Image of a uniform line source of strength ^ per unit 
length stretching from Qj to Qg Is the following: a uniform 

OQo 

line source of strength U-~ per unit length, stretching 
between the points Inverse to Qg, and a uniform 

line sink of strength per unit length extending from 

0 to Q^* 








290 


■» 44 -* 


Proof * If P ia any point, the Stokes stream 
function 'V (see Milne -Thoms on. Chapter 15, esp. pp. 430, 
421) ia 

Y « - (r-ri) + ^(rg»r^) . 


If P is on the sphere, we have 


ri » 




r. 


a 


» r„ ( a property of inverse points, due to the 

similarity of triangles OPQj^ , OQ^^P), and consequently 

ss m ^ I ifc ft i = const* Therefore =0 on the sphere* 

Ooniblnlng this result with the known image of a point 
source (Milne -Thoms on, pp* 420,421), we obtain the following! 

Theorem 2 * The image of a point source of strength m 
situated at the point Cig, together with a uniform line sink 
of strength ^ per \xnlt length exteiadlng from to Qg, 

where in = (®o that the total strength is zero), is 

the following; a point source of strength - ® 


m 


point Qg 


at the 

and a uniform line sink of strength ^ 


per 


unit length extending between Qj^, Qg* 


5* The construction of 

The function is required to satisfy the boundary 

conditions (3)* If the plane were not present, would be 

a2 2 ° 

^ , which represents a source of strength A placed at 0* 

The presence of the plane, however, causes the boundary condi- 
tion (3) on the plane to be violated. To satisfy it, introduce 
the image of the source relative to the plane, which is a 
source of equal strength at the reflected point Q^. But the 
boundary condition on the sphere is nov/ violated, and to 


11-^- 


- 45 - 


291 


remedy It, introduce the image of the point source at Q. 


relative to the sphere, 
strength ^ 


This image is a point source of 
at the inverse point Q, and a uniform 

line sink of strength ^ per unit length extending from 
0 to this process continues and requires the use of 

theorem 2 for the images relative to the sphorsu The successive 


iinagea fit very neatly together, as indicated in the diagram* 



292 


«* 46 “ 


The successive reflections are described in the 
following table; 


Strength of 
point source 

Situation of 
point source 

Strength per unit 
length of wilform 
line sink 

Situation 
of uniform 
line sink 

A^ 

0 



a2 

^1 




f 

% 

A 

oq’ 



A 

«il^ 

S5i 0«2 


A-jr 

1 t 


etc* 


Set 


( 5 ) 

and 

( 6 ) 


<i t= 


By the Inversion, is also equal to — j- 
expressions for the strengths of the sources are; 


is also equal to — ^ c The general 


^ A point source of strength at and 

(7) / a uniform line sink of strength per unit length 
t^long and 








Introduce the quantity 


OL 


• 


The quantities can be expressed In terms of oc • 

We have 


i ( 


Thus 


" 2 B - ■55; “ X - ^ 


•^1 “ ^+1 - 1 ^ » *•• f 

S" 


so that the d*a are convergents to a continued fraction. 

The quantities can be obtained as fimctions of 

oi from the definition (6). They can also be obtained by 
solving a second order linear difference equation with 
constant coefficients for the quantity ^ , obtained from 
the rooursion formula (9). ^ 

It Is convenient to expand In the neighborhood 

of the 8'urfaoe of the sphere In terms of spherical harmonics. 
The expansion of a unit point soToroe lying outside the sphere 
at a distance X from the center is 




^ -r Sg F|j^(003 8) + Pg(c08 9 ) + e«. f 


while a point source lying Inside the sphere lias an expansion 


^ ”3 Pg(c 08 9-) •}■ *9s ^ 

R R 


u ' 


294 


- 48 - 


wliere {Pj^(coa 9)^ are the Legendre polynomials of cos 0* 

(See diagram on page4i^« Integration with respect to X 
yields the expansion of a uniform line source. The ex- 
pansion of Q la then easily obtained by using the distri- 
bution of sources and sinks given in (7). 

Per the present purpose it suffices to know imrely 

the value of ^ - v f #^dS. the mean value of over the 

4nA^''K ° ° 

sui'face K of the sphere. The integral of a unit point 

source over the surface of the sphere la, by (10) and (11), 

(12) ^ ^ » 


according as the source Ilea outside or inside the sphere. 
The first of these is to be expected from the mean value 
theorem for potential functions, while the second Is a 
constant, independent of the position of the sources Inside 
the sphere. 


The value of 


itfl® 




4> i3S can now be obtained by 


using the distribution of sources and line sinks (7). The 


contribution of the distribution inside the sphere reduces 
to A x <3ue to the source of strength A at the center. 


This follows from the second result In (12) and the fact 
that the successive Images, each consisting of a source and 
a line sink, have a total strength zex’o. The contribution 
due to the som’oo A^D^^ at line siiik of 

strength per unit length along equation 

^n«l 

(7) ) is 

A^D - jA. log 

or 

A ^+1 


°Qn+l 


A D. 


h+1 


■"n-1 


log 


“3T 


f 


295 




f , 

V. / 


-49- 

"by (5) and (6)# Thup v/e obtain tha final result 


/ 


(13) /#edS = 4trA' 

k 


1 + D 


a * 2 ( Vi " 3^ V) 


l*h.e term Is due to the point source A® at 


4* The construction of ^ 


l! 


The function ^ la required to satisfy the boundary 
conditions (4), If the plane were not present, would 

be the velocity potential due to a dipole of strength ^ 


placed at 


0, namely, ^ " ’ Ig 


The boundary cox^dltlon on the 


plane, however, is violated and to satisfy it, introduce the 
imago radial dipole of strength - ^ situated at the reflected 
point Ql (see the diagram on page 45)* To remedy this boundary 
condition on the sphere, which has now been violated, introduce 
the image of this radial dipole relative to the ^here* This 
image is another radial dipole of strength at the 

inverse point (See Miln©-Themson,p* 421 )eStc» 

The successive reflections are described in the following 


tables 


Strength of 

Situation of 

Dipole 

Dipole 

A» 

0 

2 


% 


- A*' 

Ql 

A® A® 

o' 



. aS a® 



Q'2 

A® A® A® 



s 


etc* 




ill 

5e 


«•: 

‘■,l; 

■ •ssn 


r ' 
; Ji 


1 :.. 
V I 


V'"i 

mI 

1 


4E 


im 


w. 

t?n 

t}i 

I 

r ' 

?~T 


296 


-50- 


The general expressions for the strengths of the dipoles 
are: 


a3 
A 


at , there is a radial dipole of strength "g“ f 


(14) 


at 


Qn+i, n «« VI It 


A® 3 

T • 


Ihs expansion of a unit radial dipole lying outside 
a sphere at a distance X from the center is 

(16) «» Pj^(cos 0) — ■ Pg(coa 6) ■»»«»» g 


while a radial dipole inside the sphere has an expansion 


(16) 


igP^(003 0) ■¥ •••• 


These are obtained from (10), (11) by differentiation with 
respect to X. See Milne -Thoms on, pp. 442,443* 

For the present purpose we are interested in the 
expansion of only up to terms involving P^ (cos 9)* 

XTslng (14), (15), (16), we obtain 

(IV) 4^ = ^ 2] ^ Z ^ » h"- ®> 


n-0 OQn+x 


•i. ^ y i — » — i — + terms In hi^er 

* ^ Legendre polynomials! 


Pj^(cos 9) 


n« o 




It is convenient to evaluate Jl^ ^^dS and U^coa 9dS • 
using the orthogonality property of Legendre polynomials. 






I— 


-61- 


and (17), (5). (6), we have 


n>o 


oo 

J $ ^coa dds - |iT A®(1 + 3^D®) 


5* GJhe kliletlo energy of the water» 


^e kinetic energy of the water la 


by Qreen*s theorem* By (2), and the Green’s theorem 


os = Ai^ os. 

7”= - 1 OS - as * 


Bie boundary conditions (3), (4) yield 

(20) *j'= dS - 2AbJ ^ ^dS •¥ cos 9dsj . 

Ihe Integrals appearing In (20) have been calculated 
In the* preceding sections* By (13), (18), (19), 

(21) ^ = 2fr^A® (1+Xq)A^ Bfj^AB + (^+fg)B^l , 



-52- 


whers the ooefflolents fg are the following 

fuiiotlons of the quantity ^ ^ ^ s 


= D, + 
o 1 




(22) / fi=|2]D®a5^i 


^2 = 2 


CO 


The oxpraaslona for d^, as functions of at, are 
given In (9) and (6)* 

The series for f^^, fg converge for oc s 1/2 

(which means A s B)* They are tah-alated in table 4 below. 
Expansions in powers of oo begin as follows; 


f© « oc + ^ 


f — i ^ j. ^ j. 

_ -OC + -oc ^ ... 


■P — a. j. 

^2 “2“- *5“^ * *•' 


df /d«. 


The exnanslons for 1 / 1 + f -==== 

° /r+ 

In part III, begin as follows: 


required 


fi + f„ = 1 + 


■ 7 ■”• ~~ = 1 - s ®^ + 

/rr?; ^ 


■ la?;' . , 












wall and a free aurfftoe * 

Consider a pulsating bubble at a distance b from 
a rigid wall and a distance c from a free surface® For 
convenience, the free surface will appear vertical In the 
diagram® 



If 0 la the original position of the center of the 
sphere, the successive Images with respect to the rigid wall 
and free surface will be at P^f P£* Qgl ^3* Q3I ©to® 
Their distances from 0 will be given by the following 








66 - 


30 i 


I 



formulas j 

~ Sm(b+c) if n = 2in 

= 2m('b+c) + 2b If n s gm 4. i 

= 2m(b+o) If n = 2m 

= 2m(b+o) + 2c If n = 2m + 1 


The signs of the Images are given in the diagram* 

Images with respect to the sphere will be 
neglected* 

The first term in the expansion of f^ in powers 
of a can be obtained by finding the potential at 0 of 
the infinite aeries of images Fg, Qg, etc* Some 

single algebraic transformations lead to the foimmla 


^o = 



where x = 




fix) 


1-x* 


9 -x*^ 26 


It is to be noted that 


0 

3 b 


— [f(A) + ^f*(x)3 
(b+c)^ 



To facilitate comparison with previous results, 


302 


■=56 


3fo 

and caii be written as follows* 

o D 

^0 “ ^ (l-x)(2xf(x) - log 2) = I^F(x) 

3f 

Tb ~ " ^ 2(l-x)^Cf(x) + xf*(x)l = - a(x) 

2 d ■' 2'b‘‘ 

A short table for P(x) and G(x) la given on page 36 • 


irtJZ. - ; 




” .. vv 




Appendix III» Mtunerioal Evaluation of Som e 
Eoflnlte Integrals 

la part III, the problem of evaluating the definite 
Integrals in equations (3#7), (3* 11) arose. These Integrals 
have square root singularities at the limits of integration. 
In this appendix we shall develop a quadrature formula, which 
was used for this evaluation but does not seem to be as well- 
known as it should. It is based on Tchebycheff polynomials. 

The nth Tchebycheff polynomial Tj^(x), -1 ^ x S 1, is 
defined by 

Tj^(x) = oos(n arc cos x). 


or in other terms, Tj^(x) = cos nS , where x - cos e. 

The Tchebycheff polynomials are orthogonal with respect to 


the weight function 


1. e. , 


y. 


dx = 0 


if n m. 


This follows Immediately when we make the transformation 
X — oOS S. 

By virtue of the orthogonality property, the follow- 
ing theorem can be proved? 

Theorem . Let x^, Xg,..., x^^ be the a zeros of 
Tjj^(x) in the range -1 ^ x n f (x) be any polynomial 


of degree at most 2n-l. Then 


J Vi-«® 


;•» ••• 
U'r- 


304 


58 - 


The proof is similar to the proof of the Oauss 
quadratAire fonmila* Ifhls theorem Indioates an exact 
evaluation of the integral involving 2n parameters 
(a pcljn^omlal of degree 2n*»l) in teimis of n specially 
selected points* It la exactly analogous to the more well 
known (Jauss quadrature forarala, but it is simpler In two 
ways; tile zeros of Tj^(x) are easy to obtain, 

since they are merely the zeros of cos n 0 where x = ooa 9f 
and the weight factors inultlplylng f(xj.) are all equal to 
1/n* 

If f(x) is not exactly equal to a polynomial of 
degree 2n-l but can be approximated by one, we oan write 
the approximate quadrature formula 



The accuracy of this approximate formula depends on the 
closeness with which f (x) oan be approximated by polynomials 
of degree 2n-l« 

The Integrals in question In part III are of the form 



where v Is some exponent, and a^, are the two zeros of 
the expression under the square root sign* This can be 
written as 

a) (a - 



da 


^1 - a'' - 


) 



) 


W' 




■89 


305 


where the function 


- a)(a - a^) 


/ 


1 - a - 


has no 


singularities and la rather smooth* The Integral can now 
be evaluated by using the Tohebycheff quadrature formula 
(changing the range of Integration by a linear transformation 
from a^, a, to -1* 1)* The integrals were evaluated 

S seros of Tg(x)j with the re- 
The accuracy depends on the close- 
\/(a3^-a)(a-a^) 


'o* “1 

in this way by using the 
suits stated In part III* 


ness with which the remaining factor 


a^. 




^3 k 


can be approximated by a 9th degree polynomial* It la fortunate 
that the Integrand need be computed only for 5 intermediate 
points* (Also, the values obtained by using the three zeros 
of Tj(x) agree within 1 percent with the values quoted*) 

It is of interest to see how the momentum a depends 
on the parameter k* If the explosion takes place near the 
surface, say for a model experiment, the value of k would 
be approximately #16* For k = *16, the computations yield 


s 





• 760 



In place of (3sl2)s All the constants have Increased, but 
the alternate additions and substractlons tend to cancel 
these increases* Thus, the new equation for the stabilized 
position, where s = 0, is 


d = 6b? + 3.32b„+ *4 
o o 


which is practically identical with (3* 15)* The total period 
for this case turns out to be 


t s 1.4811 + 



-60- 

«hloh Is practically identical with (3«10)* 

Finally, If the internal energy la neglected, l.e., 
if k as 0, the IntegrRla 'become Beta fnnctions and 



The stabilised position where s «= 0 la 

d a 5«lb^ + 3*lb^ + .4 • 
o o 

These deviate somewhat from (3*12) and (3* 13)* 




Iri 





oeiir »«0E oooz OOOIC06 go8 eoz ao9 o«s oo^ oos ooz 










































ggy ort oat oe' o»' ol' aa' os' ot-' os' o'er 





■ BO .70 





'iSi. 



0^7 oei 037 on act oe oa 













J 
{ 

I 

i 

I ' 

. i i 


318 


— 83 “ 

STTR5PJARY OP FORMULAS 


1» Scaling factors (for T* K« T»)i 
L a 13.2 ft. 


C = 2.85 


W 


1/3 


D 


5/6 


sec. 


where W = weight of T<* W« T. In lbs., = distance. In feet, 
of the explosion from a point 33 ft. above sea level. 

2. Period of pulsation: 

wV3 

T = 1.470 = 4.19 sec. 


in the absence of outside Influences j If the bubble is at a 
distance b^ from a rigid wall, this Is to be multiplied by 


(i 


'O / 


3. Maximum radius; 


If the internal energy of the gas Is neglected, the 
maximum radius of the bubble is L. If the internal energy is 
allowed for, the maximum radius is a^L, where a^^ is the 
largest root of a® + = 1, and k is given by the formula 

1/4 ^ ' 

k = .0607 dJ'' • 

4. Non-dimensional momentum at time of mlnlmam size; 


The momentum due to gravity and a rigid bottom is 





where R Is the distance In feet from the center of the 
bubble and q is a factor whose graph Is drawn In figure 6* 







320 


85 


BIBLIOORAPHy 


[1] G« S* Hudson •= "Early and UltlTnate Damage due 

to an Underwater Explosion against 
10-lnch Diaphragms". Taylor Model 
Basin Report 509, Aug* 1943* 

[2] D. E. J. Of ford 


[3] 0. I« Taylor 


[4] C* Herring 


[5] M. Shlffman 


[6] E* H« Kennard 


[7] J* 0. Kirkwood 


[8] M. Shlffman 


C9] E* P. Willis 

and M* I* Willis 


Admiralty 

Naval Construction Dept*, WA-669-7, 
UndQX 23, May 1943* 

- "Vertical Motion of a Spherical Eutble 

and the Pressure Surrounding It". 

Taylor Model Basin Report 510, 

Pet* 1943* 

- "Theory of the Pulsations of the Gas 

Bubble Produced by an Underwater 
Explosion”* Division C, NDRC Report 
C4-sr20“010, OSRD 236, Oct. 1941. 


AMP Memo 37* Ij AMG-NYU, No. 7, 

Jhly 1943* 

- "Migration of Underwater Gas Globes 

due to Gravity and Neighboring 
Surfaces". Taylor Model Basin 
Report R-182, Dec. 1943. 

- "Memorandum on the Plastic Deform- 

ation of Marine Structures by an 
Underwater Explosion vyave"- 
OSRD No. 78S, Aug. 1942. 

- "The Effect of Non-Spherloal Shape 

on the Motion of a Rising Under- 
water Bubble". AMP Memo 37*5; 
AMG-NYU, No. 23, Sept* 1943* 


- "Underwater Explosions Near the 
Sea-Bed"* HMA/SEE Internal Report 
No. 142; OSRD No* WA-1078-4, 

Sept* 1943* 



I 




i i 


THE BEHAVIOUR OF AN UNDERWATER EXPLOSION BUBBLE 
FURTHER APPROXIMATIONS 


A. R. Bryant 

Road Research Laboratory, London 


j 

i 

/ 


British Contribution 




March 1945 




ii 



321 


THE BEHAVIOUR OF AN UNDERWATER EXPLOSION BUBBLE 
FURTHER APPROXIMATIONS 

A. R. Bryant 
March 194S 


.femory. 


* nutiber of approxiiiate formulae relating to the o-.havlour of an imderwator explosion 
bubble are presenteo nere as a supplement to tho report ’The behaviour of an unPerwater 
explosion oubble*, noreaft-or calteO Report «, Taylor's non-dimensional units are used 
throughout . 

The equations and graphs given make passible the calculation of the displacement and 
momentum of the bubble tn«ards a number of rigid surfaces, viz. an Infinite o’ane, an infinite 
cylinder, a sphere and a disc. In the case of the infinite plane the ^qua-ions arc based on 
the work of Schiffme-.n, and on O.S.R.C, Report no, 38**i, and arc valio right up to tho case where 
the bubble touches the plane it its itaximum radrus. 

for completeness two equations, based on graphs given in Bureau of .'nips (b.S. Navy) 

Report 1944 - l, are included wnereby the minimum radius and tho peak prousuro in the pulse 
emitted by the collepsing bubbi >. can be calculated in the case where oil surfaces are absent. 

I ntroduction . 

This note Is an extension of Report 4 in which equations and graphs were given enabling 
some of the principal quantities associated with the underwater explosion bubble to be determined 
approximately. The equations wer;. based on S.l. Taylor's theory of the motion of the bubble 
together with Conyers H-Tring's theory of tne influence of plane free or rigid surfaces. 

Tne equations of motion of the buoiile near an Infinite rigid plane nave bean extended by 
Schlffman(l), beyond tho approximation given by Herring, rijht .ip tn the case where the bubble 
touches the plane at its maximum radius. Tnc integration of nis equations to give an approximate 
formula for the momentum and displacement, and a cemparisdn with some exact integrations of tho 
equations iKive been given in a recent puperCc), In tha present note these approximations nava 
been converted Into the more familiar non-dimensional units given by Taylor, and slightly modified 
to make them of more general application. 

A number of other approximate formulae relating to the motion of the explosion bubble 
are added here, some extracted from, the report "A simplified theory of the effect of surfaces 
on the motion of an explosion oubPU", hereafter called R-;prjrt 3, in ord.r to bring tn;- 
collection of approximations in Report A up to date. as in that note, the equations are 
collected together in Part I for .,asc of reference, with their derivations omitted. Their 
oerlvations are given in Part II. 


notation . 


The notation employed is tnat given in the previous paper.* Taylor noiv-dimensional unit 
are usee throughout and defioted by small letters, i.-. all lengths in feet are divid.d by tho 

length scale factor t = ton*, wh -re M is tnt weight in lb. of T.N.T. h.-,vihg tn, samt total ensrg 

as the bubbls under consideration. All times ^r. divided by tnu tint factor (t/g)*, wherr* g 

Is the acceleration flue to gravity. As before, ill non-dimensional i-qu-itions will oc labelled 

as such. 


322 


PART^ \ » 

Su/nmary of Approximaticn Formulae 

Tht mom An turn co >> <?t ant m in the presence of an infintl3rit*id surface ,. 

If the rigid plane Is at a distance d beloM the bubble and is horUontal, the mome'^tum 
constant m (pi^sltlve up^ros) Is 

0. 12 2 0.0107 (z l)* 

m s “"“T — TT75 — (fton-dlmenslonal) (2) 


This equation is valid right up to iht; point where the Dubote touches the plane at Its 
naximum radius; the equation diff^^rs rroo the r.'sjlts of full nu/noricai integration by, at most, 
54 of the larger values of m, ovur ih',- range 2 ^ * l.S to 16. 


0.C107 (2 - 

If the plane Is not horiiontal the momentum ■ ■■■ oirected towards tne 

0 122 ^ 

plane, is to be added vectorially to tne momentum ’~'yT 76 gravity. 

^0 


The Momentum Constant m in the presence of other ri p id su r face s * 


m 


(attraction coefficient of surface) 


(l.U 




or m 


(noR-3imcnsioniil) 


(attraction coefficient of surface) 



(non-dimensional) 


(“) 


(**a) 


The attraction coefficient is a jeo'octrical factor taOulateO in deport B for a number 
of surfaces end plotted in fijure 2 for the cylinder, disc and sphere. 

Equations (s) ana (ta) >a'o only valid when the maximum OubOle radius is small compared 
to the distance Oetween Dubole centre and surface. 


Relation hetuien motrentum constant m and d i splacement of t'ubbl s a t end of 
ft rst oscil lotion , 

Oisplacenent h^ and momentum censtant n have the same direction and are related numerically 
by 

= 3.57 (m^ - 0.008) (non-dimensional) (3) 

or more accurately by the curve in fijura t. In quation (3) n^ and m are the numerical values 
without regard to sign. 

tfinimum radiu-^ l o/ien no surfaces an, present. 

,a^ 0.446 tl/4 ^ 0.190 c (non-dimensional) (5) 

This equation agrees witn 3 or 4 per cent with the values given in figure 3 oT report *. 

If surfaces are present then equation (7) of that paprr nxist be used. 

Peak Pressure in Pulse emitted by Col t apsinff bubb le. 


When no surfaces are present 


250 m' 


,4/9 




(«) 


where . .. * 


- 3 - 


323 


Where Is the peas pressure in Ih/square inches, R the distance of the bubble centre In feet 
and .<■ the equivalent weljht of T.s.t. in lb.; Is the no^imonslonal depth of the charge. 
«hen surfaces ar- present equations i7) and (8) ,-.nd Report A must be used. 


ihe Attraction of an Bxplo sion Bubble to an 
[n finite Ri^id Pl ane 

lntearalM^'n^'“'’"?n®^t“°"’ f ® ® ®een 

bubble - I (, tnnV ■ ' sp-.cial case where the wall is horizontal and below the 

a rijit' u" n ase considered is that qf a CiV.ne at a dlsuncc o non-dimensional units above 

tSir f T , Is 'ouhd to aja>e very closely with the results of 

their full nuinerieal Integration, and, when put in Taylor units, gives 


m * ^ I - -2:^ f a £oi|^ » 

1 . 0 0 *0 


(non-dimensional) (l) 

minim „ a*!, ™ «"'ich at any Instant near to the occurrence of the 

minimum bubble radius, equals a\ , wnere a Is the radius of the puboTe and v Its linear velocity,. 

fonsula blob's'" ’’‘ ""‘R - l.d. when d » oi, the 


-rveti 


(norwdinens ional) 


Integration given in previous 

l/d2 fnr^‘’“' '* accordingly values of m were plotted against 

r:. r;.: “ “““ »■ - - 


« °.1R2 
TTU6 
o 


0,0 107 (z - i) 


(non-dimensional) 


Mere the momentum is positive upwards - i.s. away from the plane. 


0.7 a ^ 


0.62 aj z 
0) o 


{nw'v-w i fiic.'ns I ona I } 


z . 2 ^ t° l>0 irue within about -•* over the rang- 

^0 2 to ~ $, fnsortific, tMs in the nbove equation jives 


Oa.0138 2 


TTT/6 ~ -- 7 - irya - 


(non-dlmens ional) 


from this It appears that the simple first order theory valid -trinti., » 
of d large compared to the maximuni bubble radius, is In fact a fair aourix' , T 

point Where the bubble touches the plahu -at H. Jnlll IZll ‘Ion right up to the 

If the plane Is not horizontal then It is .seeps-ary to add the momentum —7 jX" 
directed towards the plane, vectorial ly to the mcmentum upw.ards due to gravUy? 


The 


324 


- u - 


The Relation between the Momentum Constant m and the Displacement at the 


End 0 f the First Oscil laiion. 


In orJer to calculate the displacement of a BuODle it Is dosKaDle to have some relation 
between dIspU cement at the end of the first period and the momentum constant m. The method 
adopted by Conyers Herrinj, and In Reports A and was to assume that in any given case the 

displacement with the surface present bore the same ratio to the free displacement under gravity 
of a bubble at the same depth (with surfaces absent) as the momentum bore to that acquired freely 
under gravity with surfaces absent. This in fact presupposes a linear relation between displacement 
and momentum at constant depth Zg< 

The values of m and -f is placement h^ at the end of the first period, as determined from all 
available full Integrations of the equations of motion of the bubble have been collected together 
in the following table. 

























- 5 - 


325 


acgulred, since points represent Inp cases In wrileh jravlty ulone acts lie on the same curve as 
those In which a tree or rljlo surface contributes appreciably to the total momentumi 

Fro“ Figure l It follows that the relation 

• 3.57 - O.OOe) (non-dimensional) (3) 

agrees very closely with the results of numerical Integration down to values of h^ of O.OtS. 

In equation (3) onV i the numerical value of h^ and m are to bo taken; clearly the displacement 
and moemturo will always have the sane direction. (3). 


T/ie Attraction of the Bubble to Various Surfaces, 

It Is seen from the foregoing that the general prccedure In calculating the displacement 
of a bubble at the end of Its first oscillation falls Into two parts, vozi (I) the calculation of 
the momentum acquired by the bubble towards or away from the surface [to which must be added 
vectorlally the momentuh due to gravity), and (il) the determination of the displacement h* from 
equatloij (3) or from Figure 1. 


In the neighbourhood of an Infinite rigid plans the momentun nay be calculated from 
equations (l) and (2), while if tue surface is a free plarft surface, equaticns (S) and (Sa) of 
Report A must be used. Similar 'approximate* equations have been obtained for a number of surfaces, 
in particular the sphere(U), Infinite Cy1inder(<l) and disc (5). In Report G It was shown that the 
momentum acquired at the end of the first oscillation towards a rigid surface can be written 


m « (attract Ion coefficient of surface) 


(*) 


where a^ la the noi>-uimenslonal maxlmuni oucbie radius. This may be simplified still further by 
using the approximation ” O.lM, discussed above, giving 


m " (attraction coefficient of surface) 



(la) 


This attraction coefficient, which Is a puHily geometrical factor, has been multiplied by 
U X (distance)^, and plotted In Figure 2, for the sphere, cylinder and disc, In the case of the 
cylinder the attraction coefficient used Is only an approximate expression for a certain Integral, 
and Is not valid when the distance- of iho OubOie from the cylinder axis approaches the value of the 
radius of the cylinder, over this region the curve Is drawn with a brokon line; It has been 
put In by eye since It Is known that the curve must tend to the value unity when the distance of 
the bubble from the cylinder's surface Is very smal'I. 


when the iiioxiinuni bubble radius approaches the value of the distance between bubble and 
surface these equations are no longer strictly valid. Since there are a$ yet no equatlona for 
thees surfaces In which still higher order terms are eonsidereo, In such cases the approxlimte 
equations will be the beSt estimate. In this connection It Is worth noticing that In the one 
case Where such an “exact" theory is available - vli, the Infinite rigid plane as treated by 
Schlffmann - the approximate theory Is not much In error even up to the point where the bubble 
touches the piarwj. 


Formulae for Minimum Radius and Peak Pressure in Pressure 
Pulse due to Bubble Collajise when Sur-faeet are Absent 

In ifeport A oqiiatlons were given for Sj, the minimum bubble raolus, and the peak 
pressure In the pulse emitted by the collapsing bubble. The equations were transcendental and 
required graphical solution. Two alternative equations have been put forward by the U.S. Bureau 
of Shlps(6) to represent the results of solving these transcendental equations. These alternative 
formulae are thus more convenient for nunvirlcal work, and are given hero for cqmpleteness. 


0.44S z 


-U/» 


+ C.198C 


( 


(non-dimensional) 


(5) 


- £■ - 


326 

In equation (si the coefficients jiven by Bureau of Ships have been Increased by about 
I, 19 per cent to bring them Into better nceord with the graphs of Report t ahd with the data 

obtained froin the full numerical Intepntlon of the equations of motion. 

’ The Bureau of Shjps formula for P*» Is 

nt 

. < RP„ - 2b0 {&) 

j where the quantities In the brockets are non-dimensional and where R Is the distance from the 

i charge in feet, and M the weight In 1b. of T.N.T. having the sanie bubble energy ei the actual 

I charqs: P_ Is in Ib/sauore Inch. Eouatlons (s) and l6] eoree within about B oer cant with the 

S 'HI . ~ 

I curves }n Report A and with the results of the fuM numerical Integration of the eqiMtloht of 

i motion. it Is to be noted that since both RP^ and a^ are functions of the mowentuiil m equations 

{ (S) and (6) can only be used when surfaces are absent. It any surfaces are pfesent which cause 

I an appreciable alteration of the fflamentum m then the original equations In Report A must be used. 

I ' References , 

i (l) The Effect of a Rigid Wall on the Motion of an Underwater Gas Bubbis. 

I H. Schlffman. t.M.P. HemoJTtl; AWt-RYU, No. 7. July/ 19«B. 

I (2) On the Best Location of a Mlnu Near the Sea Bed. AiH.p. Rep. 37iiR. 

\ AkG-NYU. No. 49. O.S.R.D. 3S41. May. 1944. 

' , (?) In O.S.R.O. 3841 a relation between the o isplacement and momenlum Is given 

\ ‘ Which becomes, In Taylor units, 

I " 11.23 I 1 - 774 

This relation agrees quite well with the points obtained ffoCi their 
i i' nunsrical Integration but deviates markedly for higher values of from 

I J the data in Table I, Equation ( 3 ) above Is therefore to be preforrod. 

^ i 

, (4) The results are collected In Report B. These equations are called *approi<Iimte* 

i since, In contrast to Schlffmann's treatment of tha*lnflnlte rigid plane, they 

are obtained by neglecting terms of the order of and higher powers, d 

I being the distance of the bubble centre from the surface# 

^ (S) “The Attraction of an Underwater Explosion Bubble to a Rigid Disc', 

I (d) Guships 1944-1. 


! ) 
I 


i I 

- ii 

^ P 

i ^ 

I 




Fig r RELATION BETWEEN BUBBLE DISPLACEMENT AFTER ONE OSCILLATION AND MOMENTUM 

constant” m 














THEORY OF UNDERWATER EXPLOSION BUBBLES 


B. Friedman 

Institute for Mathematics and Mechanics 
New York University 


American Contribution 


September 1947 


329 


TABLE OP OONTMTS 

Page 

Preface ii 

Introduction 1 

Outline 2 

Section I: suinmar;^ of Foriaulas 6 

section II: Analysle of Easperimental Data ... 13 

Table 1: First Bubble Periods versus Depth . 14 

Table 2: versus 555®^® 15 

section III: Derivation of Fonoolas 18 

Uomentua of the Bubble 21 

Migration of the Bubble 23 

Radius of the Bubble ........ 24 

Peak Pressure ........... 24 

Seotion IV: The Elec trosta tie Problem , , , . 27 

Potential Fvuxctlon 31 

Appends jc: 

Tables of f(x) and ^(x) 33 

Graphs 36-63 


330 


THEORY OP TJMDEHVifATEH EXPL03I03S BTOBLES 


latoodmetloa 

The theory of the behavior of a gas bubble produced 
by an underwater explosion has reached such a state of 
completeness that It seems desirable to give a unified 
account of some of the main results. \jVhen an e:^losive 
is detonated under water^ a shook wave Is first emitted 
and then the gaseous products of the explosion expand 
under the influence of their high Internal pressure, Be» 
cause of its Inertia^ the gas bubble overe:^ands to a 
very low pressiire and then the hydrostatic pressure of 
the water recompressee the bubble to nearly its original 
size. At this stage the bubble starts esQsanding again 
and a pressure pulse is emitted. This process of expan- 
sion and contraction may occur five or six times before 
the bubble breaks up and dissolves or escapes from the 
water. 

The behavior of the bubble is affected by the pres- 
ence of surfaces, such as the bottom of the water, the 
walls of a tank, the target or the air-water surface. 
Normally, the bubble would tend to move upward since it 
is lighter than the surrounding water. However, the free 
siu?face of the water repels the bubble while a rigid s\a»- 
face of any kind such as a wall or a bottom attracts it. 
In some eases these effects may be so strong that the 
bubble actually moves downward. 

Besides influencing the motion of the bubble, the 
surfaces also change the period of oscillation and modify 
the size of the peak presswes produced. The theory 


331 


presented here explains these effects and gives a quanti~ 
tative estimate of their size. The values predicted for 
the period show excellent agreement ^th those obtained 
experimentally, while the values predicted for the peak 
presstare and the distance moved by the bubble show only 
fair agreement. 

An Important application of the theory is the deter- 
mination of the amount of energy left in the bubble after 
the shook wave has passed. Also tiie energy In successive 
bubble oscillations can be found. There is a large amount 
of energy dissipated in the transition from shock wave 
stage to bubble znotion and also between successive bubble 
oscillations, which cannot be explained on the basis of 
the energy radiated by the pressure pulse. The explana- 
tion of this large energy dissipation is still unknown. 

Outline 

As developed in this paper the theory is an exten- 
sion of that given in AMP Report 37. IR, Studies on the 
Gas Bubble Resultiiig from Underwater Exploelona : On the 

Best Location of a Mine gear the Sea Bed . There, the 
motion of the bubble in the presence of a rigid bottom 
was investigated and it was shown that the exact theory 
could be successfully approximated by the addition of a 
term to the kinetic energy. In this paper we show tiiat 
the effects of euiface, bottom, walls, targets, etc., can 
all be approximated In the same way by the addition of a 
suitable term to the kinetic energy. The evaluation of 
tills term depends upon the solution of an ”electrostatic 
problem." In Section IV wo work out in detail the case 
of a bubble between a free surface and a bottom. Other 
oases can be treated in the same way. 

Section I presents a collection of formulas and a 
summary of metiiods which can be used to predict the pe- 
riod of oscillation of the bubble, the distance its 


332 


06 iite« moves during the first oscillation, the majcimujn 
and Hiinimuni radius of the bubble and, finally, the peak 
pressure emitted by the bubble. The formulas are given 
in terms of certain integrals which can be evaluated by 
the method given in Report 37. IR. Numerical integration 
of the differential equations of motion of the bubble is 
completely unnecessary, por convenience Figures 1-6 con- 
tain graphs of these integrals for tho most frequently 
occurring values of the parameters. 

Section III contains a discussion of these formulas 
and a short indication of their proof. A careful analy- 
slf, is made of the dependence of the parameters in the 
bubble motion upon the properties of the explosive. It 
is shown that, by a study of the periods of bubbles 
placed at different depths, it is possible to determine 
the amount of energy left in the bubble after the shock 
wave has passed and also to determine the exponent in the 
equation for the adiabatic ejqjanslon of the explosive. 
This seems to be one of the very few methods by which 
this exponent can be found. 

A similar procediire can b© used to determine the 
amoxmt of energy left in the second bubble oscillation. 
The experiments indicate that only about 16 percent of 
the original energy of the explosive remains. Calcula- 
tions show that the energy radiated by the pressure pulse 
emitted by the bubble at minimvun size is not large enough 
to explain the energy loss . The explanation of this high 
dissipation of energy is one of the major xmsolved prob- 
lems of the theory. 

As was mentioned before. Section IV contains a solu- 
tion of the "electrostatic problem" equivalent to the 
problem of a bubble placed between a rigid bottom and a 
free aiirface. 

In Section II the theory is applied to the analysis 
of some experimental data obtained at Woods Hole by Arons 


333 


and. his co-'^orkarg . Three hiaidrsd grams of TetryJi -^sre 
det-onated at varying depths below the surface in water- 
23.5 feet deep. The period of oscillation, the peak 
pressure and the distance the bubble moved were measured. 
The agreement between theory and experiment as regards 
periods is excellent; as regards pressure and distance 
the bubble moves, the agreement is only fair. 

Since the agreement as regards periods is one of the 
outstanding successes of the theory, it seans worthwhile 
elaborating on it. Let f be the period of oscillation of 
the bubble at a depth H feet beneath the surface. In 
section II it is shown that f(H + varies linearly 

with 33) ' where is a complicated function of 

H, so that we have 

T(H + 33)^"^® = a + ^ 3S)"^/® 

where ct and ^ are constants independent of H. It is also 
shown that a = and p » where E is the 

amo\int of energy left in the bubble after the shock wave 
has passed and C-j^ and are two constants depending upon 
the exponent in the equation for the adiabatic expansion 
of the ex^Jloslve. 

Prom the experimental data, T(H 33)^^^ and 

+ 33)“^'^® are calculated and then the constants oc, and 
p are determined by the method of least sqviares . Prom the 
values of oi and p the amotait of energy left in the bubble 
and the adiabatic constant can be determined. In the par- 
ticular case considered in section II it is found that 
about 48 percent of the original energy of the explosive 
is left in the bubble. 

Assumptions 

As in Report 37. IR we shall idealize the problem by 
making the following assumptions: 


334 


?h@ T?ater Is an ideal inconroressibla riuld., 

2« The bubbae reinalns spherical in shape. 

3. The gas inside the bubble expands adlabatioally. 


335 





i: 


( 


Seotion I 

Summary of Formulas 
units of Length and Time 

Let A be the radius of the bubble at time T* B the 
distance of Its center from the bottom, and H the dis- 
tance from the surface. We put Z = H + Z*, where Z^^ Is 
the height of the water column whose pressure equals at- 
mospheric pressoro, so that the hydrostatic prossurs at 


"I 
. ! 


, 1 


1 

i 


i 


0 i ? 

r, I 



h 

P 



i 


bodom 


the center of the bubble is pgZ. 

Let E be the amount of energy left In the bubble 
after the shock wave has passed and let M be the mass of 
the explosive. We write E = rQM so that rQ la the amount 
of energy per unit mass of explosive left In the bubble 
after the shock wave has passed. 

All the formTilas will be given In dimensionless 
terms. The unit of length Is 

(1.1) L = (5E/4TrpgZ^)^/® 

where Zq is the value of Z at the time of detonation. 

The mit of time Is 


336 


^-1 


n » / « ~ ' —i/S 

C = lavegzi-zo; ' . 


Note that the tinit of velocity is proportional to the 
sqiiare root of Z^, for 

(1=S) L/C = (2gZQ/3)^/®. 

Parameters in the Bubble Motion 

If the presaxire and volume of a unit mass of the gas 
formed by the explosive are connected by the adiabatic 
relation; 


( 1 . 4 ) 


PV'*' = K, 


then the internal energy of the bubble when expressed in 
non-dimensional terns will depend upon a parameter k 
which is defined as follows; 

(1.5) k = K(E/M)'‘>'(pgZQ)'^'"V(Y - 1). 

Since V - 1 is very close to zero, the value of k will 
hardly change when Z^ varies slightly. In most cases, 
therefore, we shall be able to treat k as constant inde- 
pendent of z^. 

The effects of free surface, gravity and bottom are 
expressed with the help of a f^lnction tp^ , y/e put 

(1.6) -[f(x) + log 2]/(H + B) 

s -^(x)/h 

where 

ti-7) ^ = 

and 

( 1 . 8 ) 


00 

f / -s: ■> = {?X 

O 




n 


(£}n + 1)~ - 


337 


and 


(1.9) fefu) + (1 - *)ts: — s +~|-^]. 

^ ^ 1 (2n -H 1)^ - ^ 

Tables and graphs of the ftsnotlons f (x) and (x) are 
given in the appendix (see Figures 13, 14 and 16 and 
Tables 3 and .4) . Note that IT the '!?ater iis infinitely 
deep, B = 00 so that froi& (1.7), (1*9} and (1.6) we have 

(1.10) X = 1, ^(x) = 1/2, - -1/2H. 


Bubble Quantities at Time of Maximum Size 

Small letters, a, b, h, etc., will be used to repr@° 
sent the non-dimensional values of the quantities repre- 
sented by the corresponding capital letters. The sub- 
script zero attached to these quantities will indicate 
their value at the beginning of the bubble motion, the 
subscript one their value at the time of maximum size, 
and a bar attached to them will Indicate their value at 
the time of recompression when the bubble size is a mini- 
mum. 

The non-dimensional time from the beginning of the 
bubble motion to the stage of maximum size is 


(1.11) 

where 

( 1 . 12 ) 

and 




= r 


l/x". ka-"®'(V=3n 


_ r*l da 


(1.13) 


338 


I 


Heye? Sjj and denote the smallest and largest roots, 
respectively, of the denominator . Graphs of and Ig 
for the most frequently occurring values of k and y 
given in Figures 1 and 2. 

The vertical momontum at the time of maximum siae is 
(1.14) ,^ = L ^(Ij - I Lp^l^) * 4 ^ Lpj^Ig) 


where 

(1.15) 

(1.16) 

(1.17) 

(1.18) 



riaT'^T-xr 

a^/2^ 

I ■■ r;i i r, I i ii |i j ii " »w wff i t- i r- . mu ■■ n i ■ 

■;s’"ta-srr-iT 

da 

.5 . to-S(T-l)‘ ’ 


da. 


da. 


The maximum radius of the bubble, a^, is the root 
near 1 of the eqiiation: 

(1.19) a^ + + Ssj/aa® = 1. 

A good approximation to is given by the formula; 

/ *t r>/^ ^ ... k /Y*l,l \ ^,2 3 ^2 

(1.20) a^ - 1 - -g - v-h-j — + -g)k - -g 

Bubble Quantities at Tims of Mlnimxmi Size 

The time, t, from the beginning of the bubble motion 
to the stage of minimum size is twice the time to the 
stage of maximum size, that is, 

t ^ 




t- 

- 


■t 


e 


( 1 . 21 ) 


339 


I 5Jhe vertical momonttan, a, at the time of miniraim 

i size Is twice liie vertical momentum at the time of maxi- 

I mum size, l^t is, 

1 

i ( 1 . 22 ) i = 283 ^. 

f The minimum radius of the bubble, a, ia the root 

5 near zero of the equation: 

(1.25) a® + ka‘^^Y-1) + 3|2/2^3 « 

If we put 

(1.24) 5 = g 38^25 

! 

i and 

;■ (1.25) = fu, 

! then u la the root of the equation: 

5 (1.26) u « + s» 

V 

I 

A graph of u for various values of e and f is given in 

I Figure .7 . 

I The non»dlmeneional distance the bubble moves domi-* 

j 

) ward is given approximately by the fomula: 

If- T 

I ( 1 . 27 ) Afa = 35(1,7 + I LjPj^Ig) 

I where 



340 


The non-diai©nBloaal peak ppessxrpe at the time of min- 
imum size is 

(1.30) (a®a)- ® u^-Y][i + aLfh]“^ 

where 

(1*31) n « 

Aotaal Yalwee of Bubble Qnantities 

If 9 Is the actual period from the beginning of the 
bubble motion to the time of lain'iwmTn eise« then 

§ m g£ 

or using (1.21), (1.11), (1.1) and (1.2) we find that 


( 1 . 38 ) 

there 

(1.33) 

(1.34) 
Note that 

(1.35) 


>ttZ 


-5/6 ^ 




06 = 2l3^(2g/3)'®/^ (2irp)“^/® 

P * l2(2g/3)“'^^® (2frp)“^/® E^/®. 


fz®/® 


,-•1/2 


0® + P^l^o 

so that is a linear fuaetion of In a later 

section we shall use this formula to find cz and p and then 
the value of E. 

The ”•»« and minimum radii are found from formulas 
(1.80) and (1.23) after multiplication by the unit of 
length, L, given in (1.1). 

The peak pressure of the bubble at a distance H from 
the center of the bubble is 




341 


(1.36) P = 2 

where the value of (a^a)‘ is given by foTinula (1.30). 

At the time of maximum size the pressure in the bub- 



342 


section II 

Ayxaljals of Experimental Data 


In this section some experimental data obtained by 
A» Ai*ons and his co-workers at Woods Hole Oceanographic 
Institution will be analyzed. It will be shown that the 
theoretical values of the period agree very well with the 
experimental data while the values for pressure and mi- 
gration do not agree as well. 

The experiments consisted of exploding three hundred 
grams Tetryl at various depths beneath tlie siarface in 
water 23.5 feet deep. Table 1 shows the values observed 
for the period of bubble oscillation. 

Before those data can be compared with the theoretical 
results, the value of E must be known. We shall find E 
by fitting a straight line to the points obtained when 
T2^^^ is plotted against * Table 2 gives the val- 

ues of and the corresponding values of 9 ^-^^ ' • 

A straight line is fitted to these values by the 
method of least squares. The eqxiatlon obtained is 

(2,1) TZg-^® = 3.876 - 16.96 

Figure 8 shows how closely the straight line fits 
the data. This closeness of fit is a confirmation of the 
theory. 

When equation (2.1) Is compared with (1.35), we find 

that 

« = 3,876, 


( 2 . 2 ) 


p = 16.96. 


CO o 

<0 lO 

^ Xj) 

H H 


oHt-a3ioo>t~-^inrto>w 
in to '>1* •a* '!' <• ■'S’ '<? sH xj' Xji U5 


•p 

m o 
o 


lOOODHOltOC'CO 


«0 Ex 
rH H 


CO CO Cfe 

H f— I rH 


0) Oi o> o 0> 

H i~8 H H H 













10 

H 














H 

0> 














50 

10 














H 















• 

• 













i 

% 







o 






H 

lO 


o 






H 






lO 

xt* 

H 

t- 






(0 






«o 

to 

50 

lO 






H 






r*l 

r-| 


IH 






• 






• 

• 

• 

• 



















P% 









•% 

•» 






CM 

CM 

so 

x? 

rH 

o 



o 

o> 

50 

'.O 



a 

CO 

o 


CM 

xi» 

to 

in 


■<# 

o 

50 

xj» 


£-• 

o 

in 

CO 

10 

CO 


50 

tn 

50 

<o 

50 

to 

lO 


o 

H 

H 

(H 

H 

iH 

iH 

iH 

r-^ 

H 

rH 

rH 

H 



s> 

• 

m 

• 

• 

• 

• 

• 

• 

• 

» 

« 

• 



n 
















t*- 

m 

Q 



co' 

2 

CO 

co' 



xji' 




> 

<n 

O 

01 

10 

CM 


xi< 

so 

o 

cO 

lO 




in 

in 

(0 

(0 

50 

50 

50 

to 

50 

50 

lO 

lO 




H 

• 

H 

• 

H 

• 

H 

• 

H 

• 

H 

• 

rS 

• 

H 

• 

iH 

• 

r— 1 

9 

H 

• 

iH 

• 





O 

«% 

in 

% 

0) 

•« 

CM 

lO 

CO 

% 

o> 

CM 

o 

H 

pH 



Cx 

O) 

O 

H 

to 

CM 


01 

iH 

O 

50 

lO 

r*t 



in 

in 

(C 

«) 

to 

50 

50 

50 

50 

JO 

in 

in 




H 

• 

f— 1 

• 

H 

• 

r4 

• 

• 

H 

tM 

• 

H 

• 

r— 1 

• 

H 

• 

H 

• 

H 

« 

f"4 

• 
















m 

-p 

to 

o 

in 

o 

10 

O 

in 

O 

o 

O 

o 

o 


0) 

• 

m 

9 

9 

t 

• 

• 

* 

c 

• 

• 

• 

• 

(0 

to 

xt* 

xj' 

to 

lO 

to 

50 


00 

D 

01 

xa- 

lO 












r-1 



H 






















344 


i 

1 


H 



H 

3,5 

3.158 

3.160 

3.170 

.0488 

7.0 

4.0 

3.223 

.0372 



3.239 


8.0 


3.247 




3.263 



4.5 

3.290 

.0352 



3.296 


10.0 


3.307 



5.0 

3.353 

.0298 



5,. 357 




3.363 


12.0 

5.3 

3.419 

.0262 



3.430 




3.438 



6.0 

3.442 

.0238 



3.448 


14.0 


3.452 



6.5 

3.516 

3. S 27 

3.631 

.0216 



Table i 


TZq/® 

3.524 

3.554 

S .560 

3.571 

3.575 

3.617 

3.621 

3.634 

3.680 

3.634 

3.696 

3.707 

3.710 

3.722 

3.732 

3.736 

3.746 

3.796 

3.788 

3.795 

3.817 




.0198 


.0166 


.0121 


.00710 


.00385 


H 



15.4 

3.830 

.000620 

16.4 

3.895 

-.00210 

17.3 

3.969 

-.00525 

18.5 

3.938 

-.0108 

18.8 

3.971 

-.0122 

19.0 

3.915 

-.0140 


3.985 


19.1 

4.032 

-.0148 

19.2 

3.980 

- .0157 

19.6 

3 . 923 

-.0188 

19.7 

3.963 

-.0197 


3.996 



4.167 


19.8 

3.917 

-.0204 


3.953 



5.969 


20.2 

4.091 

-.0264 

20.4 

4.174 

-.0288 i 






345 


FCx^UUIkS (1.33) axiCi (1.54) show th&b k aiici p uspdjau 
on the value ol* E and of and Ig. If we consider the 
ratio a“/P have 

(2.3) = 4l2lg^(2g/3)"^/2 or ^ = 1.155 ^ 

p 

which la Independent of E. A graph of I^/l2 different 
values of k and y is given in Figure 3. 

Three physical q\aantitles are needed to fully describe 
the bubble motion — the values of K> Y S. Since we have 
only two constants (& and ^ determined, we must assume the 
value of one of the three quantities . We shall assume that 
Y = 1.25, the value proposed by Jones for T.N.T.» and then 
By the use of Figure 9 we find that k = ,23. From this 
value of k, using Figures 1 and 2 and formulas (1.33) and 
(1.34), we find that rQ s 500 calorie s/gram. Since the 
detonation energy of Tetryl is 1060 calorios/gram, this 
value for rQ is another confirmation of the theory. 

Since we know the value of E ® rQJM, we can use (1.1) 
and get 

(2.4) L = 13.88 (w/Zq)^’/® 

where L is in feet, W in pounds Euad in feet, and 

C = 2.997 

since k varies with depth as , the value, k =s .23, 
is really an average value over the range = 33 to 
Zq = 56.5. If we assume that this value of k corresponds 
to a depth halfway between the bottom and the surface of 
the water, we find that k varies from .213 to .244, This 
change in k can be neglected in the calculation of the 
period, momentum and migration. However, in the calcula- 
tion of the peak pressure it must be taken into account 

-2/3 (y-1) 

since the pressure varies as k ' ' ' ' , 


346 


Fig'^'« shows how the perlodsi calculated iTom the 
equation (2.1) compare with the observed results. The 
agreement is excellent. 

Using k = ,2S and i'orraulas (1.27) and (1.28) the nii- 
gration was round by nuiaerloal integration of for each 
value Zq . The caicvilated results are compared v/ith the 
observed values in Figure 11. The agreement seems to be 
fair 

The peak pressitre is calculated from formulas (1.06) 
and (l.SO) . If we use the values of k and y determined 
by the period measurements, the calculated pressui-es are 
about one-half ttie observed pressure. Theoretically, it 
would be possible to use one experimentally determined 
value of pressxire with the two constants a and p to solve 
for the three quantities rQ, k and Y- This was done but 
it did not lead to consistent results. It seems likely 
that the value of y at the time of minimum size may be 
different from the value of y dviring the time of expan- 
sion and contraction. 

Despite this difficulty with the magnitude of the 
pressure, the variation of pressure with depth Is given 
approximately by formula (1.50). If we multiply the cal- 
culated pressures by a constant chosen so that the calcu- 
lated presstire agrees with that experimentally observed 
at the depth of 18.5 feet, we get Figure 12. 



347 


Section III 

Derivation of Formulas 


Consider the motion of a gas bubble in water of fi- 
nite depth, taking into account the effects of the bottom, 
the free surface and gravity. As before, we let a be the 
radius of the bubble at the time T, B the distance of its 
center from the bottom, and H the distance from the sur- 
face. We put z ^ H + z’“* where is the height of the 



water column whose pressure equals atmospheric pressure, 
so that the pressure at the center of the bubble is pgZ. 

Just as in Report 37. IR we find that if we use coor- 
dinates R and 9 to describe the motion of the water, the 
velocity potential describing the flow is 

0 . 1 ) 

R 

where is the “image" potential necessary to satisfy 
the boundary conditions on the bottom and on the axirface. 
The primes Indicate time derivatives. 

By classical hydrodynamics we can show <see Report 
37. IR) that the kinetic energy of tho water is given by 


348 


(3. a) STrpA'^LA’^d -f- kp^) - + I b'^(1 + k%^)] 


where 91?^, y?g ana <p,^ are fionctlons of a> B arid H which 
represent the effect of the surface and the bottom. Note 
that ®g, 9?g ai'*e of tiie dimensions (length)"^, 

-2 -3 

(length) , (length) respectively. 


!Eh0 potential energy of the displaced water is equal 
to the volume of the bubble multiplied by the hydrostatic 
pressure at the center of the bubble, that is, to 
4'nA®pgZ/3 . 

Assume the pressxxre and volume of one unit mass of 
the gas in the bubisle under an adiabatic change are con- 
nected by the relation 


(3.5) 


PV'*' = K. 


If H is the mass of the explosive, then for the actual 
bubble 


PV^ « KM*^ 

and the internal energy of the gas when the bubble has 
radius A is 

G(A) = f PdV = lOlV-'VtY - 1 ) 

= KT>I^(-5 ttA®)^"V(Y - 1). 

By our assumption the total energy, E, loft in the 
bubble after the shock wave has passed is equal to the 
sum of the kinetic energy of the water and the potential 
energy of the displaced water and the internal energy of 
the gas. V/e have, then 

2TrpA^[A'^(l + Ajp^) - SA^jrgA'B* + ^ B'^’(1 + A^^^)] 
+ § TTA^pgZ + G(A) = E. 


(3.5) 


If we use L and C defined in (1.1) and (1.2) as 
units j equation (3.5) can be wrltton In terms of non= 
dimensional quantities as follows: 


a®[a^(l + aLpj^) - 2aba^L^902 * a^L^^g)] 

‘®*®’ * =. 1 , 

where ^ is the internal energy of the gas ex- 

pressed In non-'dimensional quantities, k is defined in 
equation (1.5). 

It was shown in Report 37. IR that the period can be 
found by assuming- b is constant so that b = 0. Equation 

(3.6) reduces to 

(3.7) a^a~(l + aL^?^^) + a® + ka“®^*^'^^ = 1. 

If we approximate (1 + by 1 + eqmtion 

(3.7) can be solved for a and we get 


a®/^(l + aLf^/2)da 


where a^ and are respectively the smallest and large 
zeros of the denominator. We find then that the time 
from beginning to maximum is 


^1 " ^1 S 



350 


which is formula (1.11) ♦ The t-otal period from beginning 
to minimum will be twice 


Momentum of the Bubble 

The vertical momentum, s, of the bubble is equal to 
a^^/3‘ Using the Lagrangian equation associated with the 
b coordinate, we find that 

(3.13) ^(a®6/3) = 

Integration gives the value of s at the maximum aa 


Sj^t = 2[ /h a^A ^ dt a- z;^ /h a=dt] . 

‘o 

Using equations (3.7) and (3.8) and the fact that 

adt = da 


we express s^ as integrals over a. We have 

,1 = h ^ p da 


(3.13) 


. pa., a®/^(l + aW2)da 

• 7 g . .. - 

o 


where 


351 


{ 



These equations eire the sane as (1.14)-(1.18) . 

The momentum can be used to give us an approximation 
to the energy equation (3.6) which takes into account the 
fact that b does change. We shall neglect the terms 

which are of second and third order* 

respectively, as compared to the first order term 
Now, since 

b = 3s/a^, 



equation (3.6) becomes 

(3.15) a®a^(l + aLfj^) + a® + ka”®^^"^^ + 5s^/2a® = 1. 

If s^were known as a fiaaction of a* this could be inte- 
grated s ^ 


(5.15) J ■ 


g^3/2(x + a.Lp^/ S)d& 

ri - a® - - 3 s72? 


s t. 


The vertical distance the bubble moves oaaa be found 
from this formula since 


= 5s/a® 



352 


so tliat 


i.l7) Ab = 3 J' adt/a^ ~ ^ f 7“ 


4- aL^/8)da 


SiJac© a is given as the indefinite integral of the right 
side of (3.1j^), Ab Is really a double Integral but w© can 
change It to a single Integral by the follov/ing arguraent: 
During the first part of the bubble motion, that Is, 
until the bubble reaches its maximum size, s increases 
from zero to a^. During the second part of the motion, 
from the bubble maximum to the bubble lolnimum, s increases 
from 3 2 ^ to i = 28^^. The momentum at any time during the 
second half of the motion is i minus the momentimi devel- 
oped as the bubble contracts from its given size to its 
mlnimnin size. Because of the symmetry of the bubble mo- 
tion, this minus momentum is the same as the momentum de- 
veloped when the bubble increases from its initial size 
to the given size. We may then, conclude that the dis- 
tance moved during the bubble expansion Just balances the 
distance which wotzld be produced by the minus momentiam so 
that the total distance moved is given by one term 

pa + aL^,/2)da 

Ab = Si / ■ ■ =:=,;: : ,c=. x:r i --T-r-T-r:;— 

(5.18) - 5iW 


= 3s ( ly + ^ ^ ’ 


(3.19) Ir; = f 


(3.20) Ig = 


a, VI 


■fir: 


da 

a-^/^ da 

a® - 3 s2/2»3^ 


3sV2a'' 


This proves (1,27). 


353 


Radius of fee Bubble 

I?:xe maxlsaum radius of the bu'Dbla is found by putting 
a = 0 in equation (3.15) and solving 

(3.21) a® + + 3s^/2a® = 1 

for the root near one. If we e:^and a In powers of k and 
solve (3.21) we find that 

(8.22) = 1 - § - (^4-1 . I)k2 - !!i . 

^is is the same as formula (1.20) . 

To find the minlm'um radius of the bubble wo put 
a = 0 in equation (3,15) and consider it at the time of 
minimum size. Tlie eqmtion reduces to 

(3.23) a® + + 3s®/2a® = 1 

and the root near zero is the minimum radius. Note that 

(3.23) is the same as (3.21) except for the value of the 
momentum. The equation simplifies if we put 

(3.24) a^=^u, e = 3s 

g 

and neglect that a term, since a is near zero. We have 

+ eu-^ 1 


or 

(3.25) u » u^'“^ + 6. 


Peak Pressure 

The pressure in the water is given by Bernoulli’s 
Law as 



354 


Ihe second term can bo neglected compared to the first. 
Using (3.1) w© find that 

P _ (A®a' ) ' (A^b')' eo» 6 . 

p - T- * • 

For moderate distances from the bubble only the first 
term is important, so that 


(3.26) 


P _ 1® (A^al 


P -JS— T- • 


The value of (a*'a) ' can be found by multiplying 
(3.15) by a and then differentiating. This gives 

2a^a(a^a)‘(l + aLf^^) + a^a^aLf-^ + 4a®a 

+ k(4 - 3Y)a^“®'''a - 3s\"®a = a. 

Since the peak pressure ^111 be found at the time of min- 

0 S ♦ • 

Imum size, we may put a = 0 ©nd solve for (a a) . The 
result is 


/ S * \ ** ^61 

(a a) = 2,^ 






(3.27) 


9s'" . 3k(Y - 1) z1-3y 

^ ^ a 

- 4a 

1 + aLjo^ 

if we make use of equation (3.23). Using the notation of 
(3,24) equation (3.27) becomes 


(3.28) 


a 


5(y - 1] 

u!S/ 


+ ul-f] . 



355 


Notice that since u is near one the bracket does not 
change lEueh as k changes. 

We may write formula (3,26) for the presstare in a 
more convenient form as follows: 


P = 


2 

2 ^(a a) - ■g pgZjj • g{a a) 
c 


_ 2 


L,_2. 


(3.29) 


(r - l)pgZ, 




by formulas (1.3) and (3.28). 

Replace L by its value given in (1.1); then 


(Y - l)pgZ 


R 




This formula can be combined with the formulas (1.33) and 
(1.34) for a and p to solve for the three parameters of 
the bubble motion: rQ., K and y. 

suppose, for example, that we know P when the momen- 
tian, s, is zero. This means tiiat e = 0 and u = 1. Con- 
sider the ratio P/ct. We have 


(3.31) 


P 


(Y - l)pg(^)^/^ 

T!% 


2Il H 5^ 


and the right aide depends only on k and y. combining 

O 

this with the formula for et /p, which also depends only 
on k and y> will be able to solve for these quantities. 


356 


section IV 

The Electrostatic Problem 


In section III functions and <Pr,^ were intro- 

duced into the energy equation to take account of the 
presence of free and rigid surfaces. However, the re- 
sults obtained depended only on the function 
function evaluated, to a first approximation, 

by finding the kinetic energy of water motion due to a 
fixed expending bubble. Oonsidor the velocity potential 
for a fixed expanding bubble, assumed spherical, lo(sated 
at a distance H beneath the svirface In water whose depth 
is D feet. If we take spherical coordinates, R, 6, ^ 



with pole at the center of the bubble, the problem can be 
formulated mathematically as follows s 

Find a potential function f(R,e) such that 


(4.1) 

M = A 

8R * 

on 

the sphere 

(4.2) 

o 

II 

•ciS 

on 

bottom. 

(4.3) 

t = 0 

on 

s\irf ace . 



Tha last condition la obtained by taking Bernoulli* a 
equation on the surface, assuming that gravity can be 
neglected, and than neglecting square terms. 

Equation (4.1) is satisfied by taking 


o. 

This is equivalent to assigning a source of strength A A at 
the center of the sphere. We can then satisfy equations 

(4.2) and (4.3) by reflecting this sowca in the bottom 

and in the free surface. It turns out that the image with 
respect to one boundary must be reflected in the other 
boxmdary and this process carried out on the successive 
images leads to an infinite sequence of images: Let 

(4.4) f = A^A(^ + f’) 

where $' is the potential due to tlae images. It can be 
evaluated by a method similar to that in Report 37. IR. 

After § has been found we get the kinetic energy of 
the water by evaluating 

I P /* M ^ 

over all boundaries. Because of conditions (4.2) and 

(4.3) this reduces to the integral over the sphere. 

Using equation (4.1) we have that 

(4.5) E=-gp J i|^dS=A J = A^A J #dco 

sphere 

2 

where dco is the element of solid angle so that ds =* A dco. 
By the Mean value theorem 

i dw = $’(0) 

-.11 ^ 

where $'(0) is the value of the potential f' at the center 




358 


of the sphere. Prom equation (4.4) and this result we 
find that 


(4.6) 


E = 2ttpA^A^(“ + ?’{0)) 

— SiTpA^A^Ci + A$ (0) ) 


SO that ~ (0) • 

Solution of the Electrostatic Problem 


[Chis shows that we need the value of $ only at the 
center of the sphere. It will be found by eonslderlng 
the infinite set of Images produced by reflecting the 
original source at the center and its images ij[i the bot- 
tom and free surface. 

For convenience of representation we shall draw the 
stirfaces vertically. The situation Is as follows: 



Here, is the source of strength one at the center 
of the bubble. S^i Sg» S_g» ••• denote the Imago 

sotirces and sinks. They are obtained as follows; 


359 



St Is the reflection of in the bottom 

S_j_ is the reflection of in the surface 

5„.i is the reflection of S ^ in the bottom 

n+i ~n 

S_jj is the reflection of in the surface. 

Note tlmt a reflection in the bottom, to satisfy equation 
(4.S), gives an image of the same strength, while a reflection 
in the free surface to satisfy equation (4. '3) gives an image 
of the same strength v/lth the opposite sign. (A source of 
negative strength is to be considered a slnJi:.) V.'e find that 

the strength of is (-)“ at a distance 2nD from 

the strength of (“) at a distance 2nD"SB from 

the strength of S_g^^ Is (-)'^ at a distance 2nD from 

„ , -n 

the strength of S__gy^__^ la (-) at a distance 2nD+2B from s^. 

$'(0), the potential at the center of the bubble due to 

this collection of sources and slnlcs can be obtained by com- 
bining the effects of and S 

The potential at due to Sg^ and S_gjj is 

(-P 

nD 


while that due to and S_y ^_2 is 


(4 7) ( )nr 1 1 •[- j(-)“ _(^-gP) 

' ' ' ' *• '2n+2)D-2B 2nD+2B'‘ [ (2n+l)D+D-2B] [ (2n+l)D- (D-2B) ] 


The total potential due to all the image sources and sinks 
is , therefore , 


(4.8) Z ^ 


0 (2n + ir "• (1 - 




360 


or, if we let - 1 = x, this can be v/rltten as 


(4.9) 


^ ^0 (2n + i)^ - 


The series on the right can be expressed In terms of 
tabulated functions as follows: 


(4.10) 


fix) = 2 z: — — s 

0 (2n + 1)*^ - x‘ 


- 2 ^Clog ni-JJS) + r(i-J-2)] - I tan f 


and we have 


i' (0) = - i(f (x) + log 2) 


(4.11) 


Note, that if D --5» 00 , $’(0) = - . By differentiation 

wo find that 


(4.12) -W " " 5 H ^ ■ A • 




Table 3 


Table 4 


m 

f(x) 

0 

0 

.05 

.09184 

.10 

.1852 

.15 

.2816 

.20 

.3829 

.25 

.4910 

.30 

.6081 

.35 

.7379 

.40 

.8840 

.45 

1.051 

.50 

1.246 

.55 

1.481 

.60 

1.769 

.65 

2.156 

nr \ 

• r 

2.619 

.75 

3.292 

.80 

4 eSQG 

.85 

5.968 

.90 

9.305 

.95 

19.31 

1.00 

CO 


X 


X 

0 '{ x ) 

0 

.3466 



.05 

.3729 

-.05 

.5157 

e 

H 

O 

.3952 

-.10 

.2794 

.15 

.4143 

-.15 

.2366 

.20 

.4504 

-.20 

.1862 

.25 

.4440 

-.25 

.1264 

.30 

.4554 

-.30 

.05528 

. 3 o 

.4651 

-.35 

-.03021 

.40 

.4731 

-.40 

-.1356 

.46 

.4796 

-.45 

-.2593 

.50 

.4849 

-.50 

-.4150 

.55 

.4891 

-.55 

-.6104 

.60 

.4924 

-.60 

-.8608 

.65 

.4950 

-.65 

- 1.190 

,70 

.4969 

-.70 

- 1.637 

.75 

.4982 

-.75 

- 2.274 

CO 

o 

.4991 

-.80 

- 3.244 

* 85 

.4996 

-.86 

- 4.880 

• 90 

1 

.4999 

-.90 

- 8.181 

.95 

.49998 

-.95 

- 18.15 

1.00 

.5 

- 1.00 

-00 









362 



1 













365 




































MIGRATION OF UNDERWATER GAS GLOBES DUE TO 
GRAVITY AND NEIGHBORING SURFACES 


E. H. Kciinard 

David W. Taylor Model Basin 


American Contribution 


December 1943 


377 



NOTATION 



•f 


t 


ji 

{ 


^'1 


5 j 



D 

D’ 


9 

9x< 9yt 9z 

H 

I 

■^ 0.2 


M, N^, Ny, N, 

P 

Po 

Pa 

Pc 

Q 

R 

Ro 

«2 


S 

Ti 

Tio 

t 

W 

X, r, z 
5 ^ 0 . Z 
*0 


*»> 

f 


Cosines of the angles between the upward vertical and the X, Y, 

Z axes, respectively 

Depth of water 

Hydrostatic pressure, including atmospheric pressure, expressed 
as an equivalent depth of water 

Acceleration due to gravity 

Components of the acceleration due to gravity in the directions 
of the X, Yf Z axes 9 x ~ ^ 9 ^x> 9 y ~ “ 9^yt 9 z ~ 9^ z 

Vertical component of Q 

Impulse per unit area or fpdi 

Impulse per unit area due to the peak of pressure Included be- 
tween the Instants at which p has one-fifth of its Intervening 
maximum value 

Functions of the distances from certain surfaces, varying from 
case to case 

Pressure in the water 

Hydrostatic pressure, Including atmospheric pressure, at the 
level at which a charge is detonated 

Hydrostatic pressure p, expressed in atmospheres 

Pressure of the gas in a gas globe 

Linear displacement of the center of the gas globe from the 
time of detonation until the time of the first peak 
recompresslon 

Radius of the gas globe 

Radius of gas globe when its pressure equals hydrostatic pres- 
sure at the level of its center 

Maximum radius of the gas globe 

Distance from the center of the gas globe to a point in the 
water 

Horizontal component of Q 

Period of first oscillation, up to first peak recompresslon 

Value of Ti at the same level if no bounding surface Is near 

Time 

Weight of charge 

Coordinates of the center of the gas globe 

Values of X, Y, Z at instant of detonation of charge 

A dimensionless coefficient referring to depth below the sur- 
face and occurring only in Figure 5 

Ratio of the specific heat of the gas in the gas globe at con- 
stant pressure to its specific heat at constant volume 


( 


378 


DIGEST* 

The gas globe formed by an underwater explosion not only pulsates 
in size but also usually changes position as each pulsation occurs (1).** 
This migration may be of importance because the first recompression or con- 
traction of the globe thereby comes to be centered at a point different from 
that of the initial detonationj and the location of the point of rocoflipros- 
Bion influences the damage that may be done by the associated secondary pulse 
of pressure. 

Accurate solution of the general equations of the motion of the gas 
globe set up by Herring (4) in this country and by G.I. Taylor and others (5) 
(6) (7) (8) in England is, in general, possible only by numerical solution of 
the differential equations; it has not yet been found poodibie to embark on 
such an enterprise In this country because of the labor involved. For pur- 
poses of analysis, calculations of the motion of the gas globe have been made 
by an approximate method. The assumptions tinderlylng this method are set 
forth In the first part of the report, 
together with a discussion of the ap- 
parently conflicting and paradoxical 
motions of the ga.s globe under various 
conditions. 

For example, in free watei? of 
unlimited extent the gas globe rises be- 
cause of its buoyancy. Near a rigid 
boundary such as a vertical wall, the 
globe is attracted toward the boundary, 
as shown in Figure 2. Near the surface 
of the water, the gas globe Is repelled 
from the free surface. However, although 
the action of gravity is always present 
to cause the globe to rise through the 
water by virtue of its buoyancy, the at- 
traction of a rigid bottom in shallow 
water, or the repulsion from the free 
surface of the water, may decrease the 



Figure 2 - Curves of Size and 
Position of a Gas Globe 
near a Vertical Wall 


The charge krb fired 9 inches froa a rigid 
vertical wall. Note that the velocity of 
the center of the globe Is greatest during; 
the compressicn phases. 


* This digest is a condensation of the text of the report, containing a description of all essential 

features and giving the principal results. It Is prepared and Included for the benefit of those who 
cannot spare the time to read the whole report. 

** Nuabers in parentheses indicate references on page 53 of tfala report. 



379 


rise due to gravity; In certain cases, they may actually produce a downward 
motion of the globe. 

The results found by acceptance of the assumptions given and by use 
of the approximations are summarized In a aeries of formulas and curves. Many 
of the results are stated in terms of the radius of the globe at maximum 
expansion. For tetryl or TNT, this is estimated from observation as 


R, 



[18 j 


where .B* is in feet, 

W is the charge weight In pounds, and 

is the total pressure, including atmospheric, expressed in 
atmospheres . 

Figure 4 presents this information graphically. 



Figure 4 - Maximum Radius on First Expansion of the Gas Globe 
Produced by a Charge of W Pounds of Tetryl or TNT 

Tha center at the globe la d feet below the surface of the sea; lines sre drawn for 
eevsral values of d; see Equation [18] at top of page. 


Vertical migration due to gravity alone Is a special case treated 
first by the approximate method. By neglecting the action of the gas in the 
globe, It becomes possible to express the solution In dimensionless terms, 
and this Is done for four depths in Figure 5 pri page 13, and In Table 1 on 
page 13. The minimum depth below the surface at which recompresslon will 


380 


occur is assuined to be equal to 
the sum of the maximum radius 
and "he rise to the time of the 
first peak. The results are 
shown in Figure 6. 

The migration caused 
by the proximity of a surface, 
in the absence of gravity, is 
shown in Figure 7 on page l6, 
expressed as a fraction of R 2 - 
The combined effect of gravity 
and a vertical wall for a sin- 
gle small charge is shown in 
Figures 9 and 10. Curves for 
estimating the rise of the gas globe under a fi-ee surface and above a rigid 
horizontal bottom are given in Figure 11 and 12 respectively, page 20, for a 
wide range of charges. 

In Figures 15 and l6 on page 26, the two components of migration, 
vertical and horizontal, are given for the combined effect of gravitation, a 
free surface, and a vertical wall. As the v/eight of the charge increases, 
the gravitational effect is shown to predominate; thus the downward motion of 
the globe from a small charge is reversed with an increasing charge in the 



Figure 6 - Cui've Giving a Rough Estimate of 
the Minimum Depth Dm below the Surface 
at which a Charge W may be Detonated 
without Blowing through the Surface 
before Undergoing Recompresslon 


1.4 

1.2 

1.0 

c 

fc 0.8 

•» 

£ 

SO.6 

G 

a 


\ 

1 

\ 




Free 

5 



♦ 15 

♦ 1.0 

■ 

i 

■ 




I 

B 

□ 

□ 

— 

\ 

\ 




r 



i 

i 

H 

m 

B 

■ 

■ 


B 

Bl 







\ 

\ 

V 






i 

■ 


■ 

■ 

i 


B 

B 


OispI 

1 



\ 

\ 



aeement upwart 


■ 

n 

B 






1 

n 1 
\ 


— 1 — 1 — 

Free Surfoce 


/ 



'horizontal Displocament 
toward rigid vertical well 


i 

1 

B 


B 

j 


0.2 












■ 




B 


5WW 







- 



— 

■ 

■ 

B 

■ 

B 





LJ 

0 


r 


19 6 7 

1 

3 < 



10 12 14 16 18 2‘ 


Distane* from Wall Z in f««t Haiqhf abov* Bottom in foal 


Figure 9 - Upward and Horizontal 
Components of Displacement of 
the Gas Globe from 3A Ounce 
of Tetryl or TNT 

The charge Is assumed to be detoiated 10 feet heloa 
tlM surface of the eater, far above the bottoin, and 
Z feet from a rigid vertical rail. The curves show 
the displacement from the point of detonatirai up to 
the point of greatest rscompression, according to 
approximate calculations. 


Figure 10 - Displacement of the Center 
of the Gas Globe from 3A Ounce of 
Tetryl or TNT, up to the Instant 
of Maximum Recompresslon 

Detonation Is assumed to occur at the height shown 
above a rigid horizontal bottom in water 20 feet 
deep. Positive ordinates represent a rise, nsgative 
ones a descent. The curve is based on approximate 
calculations. 





•. - -i 




381 

oases studied, and the motion la upward for all charges over l/2 pound. The 
horizontal motion diminishes with an Increasing size of charge In the cases 
studied at all charges over 1/2 pound. 

Finally, a rough eatlmate is made of the effect of the migration 
upon the pressure that la generated In the water by the recompresslon of the 
gas globe; curves are shovm In Figure 17 on page 28. A migratory displace- 
ment equal to one-third of the maximum radius R 2 decreases the peak pressure 
by about one-third, whereas a displacement equal to reduces the peak pres- 
sure to one-tenth of Its value for zero migration. The Impulse, however, is 
much less affected by the migration. 

Important eases remain to be taken up, in particular those In which 
the distance from the wall Is less than ZR^, and In which the charge lies be- 
tween two rigid surfaces, like the ground and a ship’s bottom. 

The conditions for exact similitude with respect to migration can 
not be reconciled with those governing the flow of destructive energy from a 
charge to a target, as applied In the nominal theory of TMB Report 492 (11). 
The application of model tests of migration effects to the prediction of full- 
scale phenomena is therefore subject to correction for scale effect and any 
direct expansion from a very small scale to full scale, without full know- 
ledge of the scale effects, may lead to erroneous conclusions. The exact 
formulation of the scale effect corrections will form the subject of further 
work. 


382 


MIORATICJ! OF UNDERWATER GAS GLOBES DUE TO 
GRAVITY AND NEIGHBORING SURFACES 


ABSTRACT 

Approximate formulas are assembled, and illustrated by curves, for 
the migration of gas glebes under water due to the action of gravity and of 
neighboring surfaces. In addition to the effect of a single surface, rigid 
or free, consideration is given to the combination of a free surface with a 
rigid bottom or a vertical wall. The general' analytical procedure by which 
the formulas were obtained is described but most of the details are omitted. 

INTRODUCTION 

The sas 2lo^'“ hu an imrte'mirnl-.eT' flirnl ftsi on not onlv DUl sates 

in size but also usually changes position as each pulsation occurs (1).* 

This migration may be of importance because the first recompression or con- 
traction of the globe thei-eby conies to be centered at a point different from 
that of the initial detonation, and the location of the point of recompres- 
sion Influences the damage that may be done by the associated secondary pulse 
of pressure. Measurements of the migration will be reported separately but a 
number of analytical results have been obtained, and these results will be 
assembled here for convenience of reference. Deductions of the formulas may 
be found elsewhere (2). 

The motion of the water around the pulsating gas globe is suffi- 
ciently slow so that compression of the water can be neglected. Furthermore, 
good experimental support exists for the assumption that the globe remains 
approximately spherical during the larger part of the first cycle at least. 
For these reasons certain aspects of the migration are adequately covered by 
old investigations on the motion of spheres, which are summarized in Lamb's 
Hydrodynamics, Section 100 (5). 

A thorough survey of the problem has been given recently by Herring 
(4) and numerical studies have been made, especially of the gravitational 
displacement, by Taylor and others in England (5) (6) (7) (8). Calculations 
by an approximate method have been made under the author's supervision at 
the David Taylor Model Basin. More e;<tended calculations are in progress 
under the direction of Professor R. Courant of New York University; these 
will be described in a later report. 


Nunbers in pmanthosSE itidicato references on page oi wilo 


GENERAL FEATURES OP THE MIGRATION OF A GAS GLOBE 

The migration results from the action of gravity and from the ef- 
fects of bounding surfaces such as a rigid wall or bottom or the free surface 
of the water; see Figure 1. Theae various actions are not simply superposed 
upon each other, because the extent of the migration is greatly increased by 
the periodic compression of the gas globe and the degree of compression is 
Itself materially decreased when the rapidity of the migration becomes large. 
The migratory motion implies the existence of kinetic energy of translation 
in the surrounding water; this energy is abstracted from the energy of the 
radial motion, with the result that the inward motion of contraction ceases 
sooner than it would in the absence of the migration. 

If the gas globe were fixed in size and far removed from all bound- 
aries, it would Pimply rise with an aceeleratlon of 2g, or twice the ordinary 
acceleration due to gravity; for the water surrounding the gas globe is acted 
on by a buoyant force equal to the weight of the displaced water, and the ef- 
fective mass of the water is only half of the mass of the displaced water for 
tl»e type of motion that results in the upward displacement of the globe (5). 

The effects of a bounding surface or wall in the neighborhood of 
the globe can be regarded as arising in the following manner: While the gas 
globe is compressed, the pressure in the water is positive, and this pressure 
Is increased owing to the blocking effect of the wall. The pressure increase 
due to the wall is greatest between the gas globe and the wall, and the in- 
equality of pressure thus produced has somewhat the same effect as if it were 
due to a gravitational field acting toward the wall. The gas globe then 
floats away from the wall in accelerated motion. During the expanded phase, 
on the other hand, the pressure is less than hydrostatic, and the deficit of 
pressure is greatsi- on the side toward the wall, as it is relieved less on 
that side by the inflow of water. Thus during the expansion phase motion of 
the water is developed that acts to carry the gas glob© toward the wall. The 
action on the globe throughout one cycle can be regardf*. as equivalent to a 
sort of buoyant force acting alternately away from the wall and toward it. 

The action during the expansion phase predominates, however, both 
because this phase lasts longer than the compression phase, and because the 
buoyant action on a large globe is much greater than that on a small one. 
Since the first phase after detonation is one of positive pressure, the ini- 
tial effect will be a slight displacement of the globe away from the wall; 
but thereafter the distribution of momentum in the water will always be such 
that the gas globe moves toward the wall, the momentum increasing in magni- 
tude during each expansion phase and losing only a little during each com- 
pression. The actual velocity of the gas globe, however, will be greatest 











385 





during the compression phase, when the 
momentum becomes concentrated into a 
comparatively small volume of water sur- 
rounding the globe. Thus the center of 
the gas globe moves continually toward 
the wall but advances chiefly In spurts 
during the compression phases, as shown 
in Figure 2. 

The effect of a free surface, 
when the gas globe is not near enough to 
the surface to break through, is approxl- 
0 o.oi osa 0.03 0.04 0.08 mately opposite to that of a rigid sur- 

Figure 2 - Curves of Size and expansion of the gas 

Position of a Qas Globe globe accelerates the water above it up<» 

near a Vertical Wall . ,,,,, 

wai’d and that below It downwara. While 

Ths shsrge -m firsd 9 Inehan flfcw a rigid , , ... ^ 

vertical sail. Hete tiiat th^ velocity of expanded, the presBure near 

th« canter of the globe is greatest during It, is very lOW, and this deficit Of 

the coaprasslon phases. 

pressure acts so as to check and then 
reverse the radial motion. Because the 
pressure remains constant on the free surface, the deficit is less, or the 
pressure is greater, near the surface than it would be if there were addi- 
tional yiater instead of air above the surface; and because of this relative 
excess of pressure, the water lying either above or below the gas globe is 
given an excess of momentum downward. During the next compression phase this 
momentum becomes concentrated in a much smaller volume of water and, provided 
the effect la not canceled ty the simultaneous and opposed action of gravity, 
the globe is carried dovmward. Arguments from momentum in the water are dan- 
gerous, however; some further remarks on this subject are introduced at the 
end of this report. 

By this action, uhe gas globe is attracted toward a rigid boundary 
but repelled from the free surface of the water. Although the action of 
gravity is always present, to eause the globe to rise through the water by 
virtue of its buoyant nature, attraction by the bottom in shallow water or 
repulsion from the free surface of the water may merely decrease the rise due 
to gravity, or it may actually produce a downward displacement of the globe. 

The effect of a boundary should increase as the gas globe approaches 
the boundary. Under certain circumstances, however, observation shows that 
marked departures from sphericity of the globe may occur. Near a free sur- 
face, for example, part of the gas may blow through the surface while the re- 
mainder migrates down into the water; see Figure 5. Furthermore, a gas globe 







Figure 3 - High-Speed Photographs of a Gas Globe from 
a Charge Fired Just under the Free Water Surface 

The charge was fired 1 Inch under the surface of the eater. In Photograph 2 part of the gas 
has vented through the water surface. In Photographs 4 to y the lower part of the glo'uu la 
Boving downward. Hore gas appears to be venting through the surface on the second expansion 
shown In Photograph 7. 













38? 


midway between two rigid boundaries may break in two, one half migrating each 
way (1 ). Such features can only be handled by a more complete analysis in 
which the ohaps of the globe is not limited by assumption but is left to the 
control of the hydrodynamic action. 


DESCRIPTION OP THE MATHEMATICAL METHOD FOR DETEP>MINING MIGRATION 

Provided the gas globe remains spherical, the effect of gravity 
alone is easily found. The analysis can be extended to include the effect of 
plane boundaries by using the method of Images that Is familiar In electro- 
static theory, and by expanding in negative powers of the distances from the 
surfaces, as in Herring's report (4) The entire analysis, including a spe- 
cial method of approximation for the first oscillation, has been completed 
(2) but is not given here. 

If the solution is required to satisfy the boundary conditions only 
as far* as the Inverse second powers of the distances from the boundaries, dif- 
ferential equations of the following type are obtained: 




-W (I' + 111 

+ i rH {n^X + Ny¥ + N^z) - I (JT - Xo) + gy(Y -Y^+ g^iZ » £„)]| 


X -jNjcR^R -■^^j^R*R^dt-^J^RUt [2] 

Y^jNyR*R-j^f^R*RUt-^J^R*dt [5l 

Z ’■‘~NgR‘R - j^l^R*R^dt~^‘-J^RUt [4] 


Here R is the x’adlus of the bubble, 

X, Y, Z are the cartesian coordinates of its center, 
t Is the time, 

R, X, Y, Z stand for dR/dt , dX/di, dY/di, dZ/dt, respectively, 

X^, y«, Zg denote vali’.es at t « 0, 

p is the density of the water, 

Pg is the total hydrostatic pressure including atmospheric 
pressure, at the point Ag, Yg, Z^, 

B. is the pressure of the gas in the globe, supposed to be a 
known function of K, 


388 


9x< 9y> components In the X, Y, Z directions of the gravita- 

tional acceleration g, 

C is a constant depending on the initial conditions, and 

Af, Nx, Ny, stand for simple functions of X, Y, and Z, depending 

upon the choice of axes and upon the nature and location 
of the boundaries. 

An accurate solution of these equations can be effected only by 
numerical integration. This method has the disadvantage that many repeti- 
tions of the entire calculation are required to obtain results covering a 
wide variety of conditions. 

For the first oscillation of the globe, on the other hand, formulas 
can be obtained which, although less accurate, are widely applicable. These 
formulas give the position of the gas globe at the peak of the first recom- 
preaslon, which is particularly important because it is the point of origin 
of the first secondary pulse of pressure. 

The method of approximation is based upon the observation that, as 
the time t advances, the integrals in Equations [2], [3], and [4] grow chief- 
ly while R is large^ whereas, because of the factor 1/J2* preceding the inte- 
grals, they are effective in causing displacement of the gas globe chiefly 
while R Is small. Approximate values for the displacement during the first 
recompresslon can be obtained, therefore, by substituting for these integrals 
in Equations [2], [3h and [4] constants equal to the values of the integrals 
at the instant of greatest compression. In calculating these values, on the 
other hand, an approximate value of dR/dt, obtained by neglecting certain 
terms in Equation [1], is sufficiently accurate. The same expression for 
dR/dt leads to a corrected value of the period. 

The first period Is thus found to be 

r, = Tiod + O. 2 OMR 2 ; [5] 

where T’„ is the period when no bounding surfaces are near. Here M is the 
coefficient that occurs in Equation [l], and iZg Is the maximum radius of the 
gas globe during the first expansion; see Equation (l8]. 

The approximate formulas obtained for the displacement of the cen- 
ter of the gas globe from the position of detonation X^, Y^, Z© to the point 
Xi, y., Zj at which the next minimum radius occurs, may be wTitten 

X^ - Xq Ve B^RzU 
y, - Fo = ByR,U 

Z, - = n V 


[6a] 

[6b] 

[6c] 






s-‘j^ ^.jk vv-’or 


where 


L 




’ “?t(^ ?i'0«2 + 


B* - 




Here Ny, and are to be evaluated at the point X^, Y^ , Z^, and P, Q, 
and £/ represent the three integrals defined as follows: 

Q. I ,.(l+ ,5] 

" - 1 (‘ + 5 [<1'» • - T^i (f*)’' » ‘ no) 


In which 


B* “ By* + By* + 5,® 


Bo is the radius of the gas globe when the pressure of the gas equals the hy- 
drostatic pressure at the level of Its center, and C’and C" have such values 
that the roots of the quantity in square brackets are in each case the saae 
as the limits of Integration, that is 1 and or y^. The gas is assumed to 
behave as an adiabatic ideal gas, in which the ratio of Its specific heats 
is y. 

For gas globes due to underwater explosions, B, /Bo exceeds 2.5 and 
the tern In y has only a small Influence upon the values of P and Q: this 
term represents the effect of the gas upon the motion during the expansion 
piiasSt If tills term Is dropped, P and Q are easily obtained as series in 
powers of MB,; 


P * 0.182 (1 - 0.18 MRz • • • • ) 


[12a] 


Q - 0.467<l + 0.28JlfB2 } [12b] 

A curve for the integral U, defined in Equation [10], as a func- 
tion of B* has been constructed by numerical Integration, on the simplifying 


390 


a!?sumptlon that y * 4/3. As 5* Increases, U decreases; the conversion of the 
energy of oscillation into translatory kinetic energy, as a result of gravity 
01* of the presence of boundaries, checks the inward motion' and thereby dimin- 
lahes the extent of the compression, with a resulting deereese in V. For 
Si/Ro ® 2.65, which seems to be within reason as an estimate for actual gas 
globes, the curve is represented closely by the formula 


U = 



1.Q6 

0.009 

1 + 4000B^ 


H3] 


With the introduction of these approximate values of the Integrals, 
Equations [6a, b, c], [7a, b, c], and [11} become, for sea water of specific 
gravity p = 1 .026, if W la in pounds, 


where 


X.-X. = FBx, Y^-Y^ 


FBy, Z^-Z.^-FB, [I4a, b. c] 


F = 




2.60 Ri 


0.009 


1 4- 4000 52 
B = + /bJ^ + By^ + Bz^ 


[15} 

[16} 


Bx = 0.0846(1 + 0.23 MKz) C;^ ^ - 0^223 (1 - 0.18 MBg) [1?} 

1>A 

with two other sets of equations similar to Equations [15l» [16}, and [17]» 
in which X Is changed to Y or to Z, respectively. Here in the products MR^, 
NxR>^, NyR^, it is sufficient to use the same unit of length in both 

factors, but elsewhere iJj has been assumed to bs expressed in feet; is the 
hydrostatic pressure at the initial level of the center of the gas globe 
measured In atmospheres; and Cx, Cy and are the cosines of the angles be- 
tween the upward vertical and the X, Y and Z axes, respectively, so that gx - 
~ gCx> 9y ~ 8z ~ ° 9^z‘ 

Here Bg, the maximum radius, may be assumed to vary as (W/p^)*. For 
tetryl or TNT a fair estimate seems to be 

= 4 .l(-^)^ feet [18] 

where W Is the weight of the charge in pounds and p^ is the total pressure in 
atmospheres. For tetryl the best experimental evidence available would re- 
place 4.1 by 4.2, whereas for TNT, Figure 2 in Reference (7) gives 3-95. Equa- 
tion 118} is plotted for several values of in Figure 4 ; p^ is specified by 


100 . 



Figure 4 - Maximum Radius on First Expansion of the Gas Globe 
Produced by a Charge of W Pounds of Tetryl or TNT 

Tte canter of tha globs is d fsat bslon the surface of the sea; lines wa dram for 
asTsral values of d; see Equation [18]. 


giving the equivalent depth d in feet below the surface of the sea, so that 
Pvi * ■> + ‘*/33. 

With this value of iZg and with Af, N^, Ny, now expressed in terms 
of feet, Equations [15] and [1J] become 



or, if the small term in M Is dropped, apprcslmately, 

Bx - 0.142 3 . 7 s .V^ (^)« [21 ] 

with similar equations in Y and Z. 

The accuracy of these approximate formulas Is hard to estimate. 
Serious doubts arise as to their validity when the center of the gas globe 
comes closer to any bounding surface than ZR^ or twice its maximum radius. 
No correction has been made for the change in hydrostatic pressure as the 



392 


I 



gas globe rises or sinks. Furthermore, effects due to compressibility of 
the vfater, associated with the emission of acoustic radiation, have been 
neglected. 


MIGRATION DUS TO QRAVITY ALONE 

If X is taken vertically upward, &?uatlons [1] and [2] become, tdien 
no boundary is near, 

“ F - I + I ff (JT ~ [22} 



Hers g is the acceleration due to gravity. 

Numerical integrations of these equations have been given by Taylor 
(3) and others (6) (7) (8) The effect of the gas, as represented by the 
occurrence of p, in Equation [22], is usually not large; its smallness arises 
from the fact that, in practical cases of motion due to gravity alone, the in- 
ward motion of the water during each compression phase is arrested chiefly 
not by the gas but as a consequence of the conversion of radial kinetic ener- 
gy of the water into kinetic energy of translational motion. Tliat is, when 
the center of the, gas globe is nearly stationary, the radial kinetic energy 
of the inrushlng water becomes converted, at the Instant of peak compression, 
entirely into energy of compression of the gas; but if translational motion 
of the gas globe occurs, part of the kinetic energy remains in the water in 
association with the translational notion. For this reason the Inward radial 
motion is checked at a larger radius than when the globe is stationary. In 
migration due to gravity alone, nearly all of the energy usually thus remains 
in the water, and the motion during the compression phase is nearly the same 
as if no gas at all wore present. Because of this conversion of the energy, 
the radial oscillations gradually die out, as the velocity of rise increases, 
especially if the hydrostatic pressure is very low or If the gas globe was 
produced by a large charge. 

In UNDEX 10 (7), Plguires 1 and 8, two plots are given, based upon 
numerical integrations, from which estimates of the rise due to gravity can 
be made for a wide rar-se of charge weights aiid depths. These estimates agree 
within 8 per cent with values calculated from the convenient approximate 
fomula 





[24] 


393 


where Is the weight of the charge In pounds, 

la the total hydrostatic pressure, Including atmospheric pressure, 
sxprsEssd in atmospheres, and 

H is the rise in feet from the point of detonation to the location of 
the next peak compression. 


The total rise during the first compression and the re-expansion should be 
about 2H. The charge may vary from 1 ounce to 1000 pounds, and the depth may 
be as much as 300 feet for the larger charges. 

In using this formula it must be remembered that if the point of 
detonation is too close to the surface, the geses will blow through the sur- 
face, at least in large part, and no typical recorapresslon can occur. This 
ought almost certainly to be the case if the point of detonation is closer to 

olic ouiiauc oiitui oiic luaA^iituiii i'auj.uo aovajLii^u ujf i>iio ^ao ^.luuc .til ai/o iii’au 


expansion . 

The formula for H as given by the approximate calculations of the 
present report may be found by putting By = Bg = 0, = 1, ~ 0 and drop- 

ping the term in B* in Equation [19]« which turns out to be negligible in all 
interesting cases of purely gravitational action. Then Equations [l4a], [l6], 
[19]. and [2T] give H = Xi - Xq - 4.0 VW/p^, in exact agreement with Equa- 
tion [24], 

It will be noted that the rise H up to the first peak compression 
increases as the weight of the charge Increases, and also as the hydrostatic 
pressure decreases. The increase results partly from the greater buoyant 
force on a larger gas globe and partly from an Increase in the time occupied 
by the oscillation. 

An Interesting plot given in Reference (8) is reproduced in Figure 
5- It shows the radius of the gas globe and the upward displacement of its 
center as functions of the time, for four different values of the Initial 
hydrostatic pressure, as found by numerical integration. The plot applies 
approximately to any charge at suitable depths; small errors will remain 
owing to the fatft that In the calculations no allowance was made for the gas 
pressure. It is interesting that at the smallest depth, ?o = 1 , the gravita- 
tional effect is BO large that the radial motion is almost deadbeat, so that 
no succeeding pulses occur. 

The scales in Figure 5 vary with W, tne weight of the charge in 
pounds. Unity on the axis of ordinates represents L feet, and unity on the 
axis of abscissas represents T seconds, where L and T are given by the 
formulas 

L = 10 ^ feet 

T = 0.56 17^ seconds 


















/ 

/ 


4 ' 



/ 

/ 

f. 



















— Rediu! Curvit 

— Ri«* Curv** 


Figure 5 “ Curves, Obtained by Numerical Integration, Showing 
Variation with Time of the Radius of the Gas Globe 
and the Rise of the Globe toward the Suri’ace 

For notation and unlta, see the text> This figure is copied froa Reference (8). 


TABLE 1 

Values of the Coordinate Units for Figure 5 


Charge 

L 

r 


*< 

L* 


pounds 

feet 

seconds 

«* •= 1 

*0 “ 2 

*0 “ 3 

zj - 4 

1/1 6 

5 

0.39 

5 

10 

'5 

20 

1 

10 

0.55 

10 

20 

30 

40 

50 

26.6 

0.90 

27 

53 

80 

106 

100 

31.6 

0.98 

32 

63 

95 

126 

300 

4 l ,6 

1 .12 

U2 

83 

125 

166 

1000 

56.2 

1.30 

56 

112 

168 

225 


* Her« is total hy^ostatie preeaure in equivalent feet 
of 8oa miter> where 33 feet ~ \ atooaphoree 


The four pairs of 
curves refer to an Initial 
hydrostatic pressure, in- 
cluding atmospheric pressure, 
at the center of the globe 
equivalent to z^h feet of 
sea water, or to L, 2L, 3i, 
and 41, feet, respectively. 
Some values are given In 
Table 1 . 

A question of In- 
terest In practice concerns 
the depth at which the globe 
from a given charge may be 




expected to execute a complete 
oscillation and emit a second- 
ary pressure pulse during a 
first phase of recompresslon. 

If the charge Is too near the 
surface, at least part of the 
gas will blov; through the sur- 
face and no recompresslon of 
the full globe can occur, as 
illustrated in Figure 3 on page 
5. The proper criterion Is un- 
certain. It may be assumed 
tentatively as plausible, and 
as supported somewhat by exper- 
iment, that recompresslon will occur only when the depth exceeds both the 
maximum radius as calculated for a spherical gas globe and the calculated 
gravitational rise to the first peak compression.'* The minimum depth Unde- 
termined in this way is plotted on a basis of charge weight in Figure 6. 

If the depth of detonation exceeds D^, recompresslon of the gas globe should 
occur, although the emitted pulse of pressure may not be very effective if, 
because of the gravitational rise, the recompresslon occurs very near to the 
surface of the water. 

GENERAL EFFECT OP SURFACES 

A nearby surface limiting the body of water attracts or repels the 
gas globe. In a rough way this effect is superposed upon that of gravity, as 
is evident from the linear combination of the two terms in By, By, as in 

Equations [7®- ’1> [^71* [20], [21]. Some interaction of the two effects 

arises, however, from the fact that the Integral £/ in Equation [6a, b, c], or 
the quantity B defined by Equation [l6], depends upon both effects. 

Comparisons for charges of different weights are most easily made 
at distances from the limiting surface In proportion to the maximum radius JZj* 
Then at corresponding distances the factors Ny, Ng in Equations [7a, b, 
c], [1?1» [20], [21] actually vary as ^/R2, so that th’fe corresponding surface 
terns in 5 ;^, By, By do not vary at all, whereas under fixed hydrostatic pres- 
sure the gravity terms vary as or as wi, where W is the weight of the 

charge. For this reason it turns out that, at distances of the order of 2R2 


'* For saall charges, where ttie rise is decreased or even nude negative 'an affect of the free siarface, 
detemlned the maxlmuB radius. 



Figure 6 - Curve Giving a Rough Estimate of 
the Minimum Depth D* below the Surface 
at which a Charge W may be Detonated 
without Blowing through the Surface 
before Undergoing Recompresslon 


396 


from the surface and at ordinary depths below the surface of the water, the 
effect of a bounding surface should predominate for small charges like deto- 
nators; for charges of a few ounces the two effects should be comparable In 
magnitude, and for charges of 100 pounds or more the gravity effect should 
usually predominate. 


MIGRATION DUE TO A SURFACE IN THE ABSENCE OP GRAVITY 

If Z denotes the distance from the surface, = A^j.= 0, and it is 

found that 


M=±i 


[25a, b] 


where the upper sign refers to a rigid surface and the lower sign to a free 
surface. 


The period of the first oscillation will be approximately, from 


Equation [5] 


= r,o(i ± 0.20^) 


In terms of the period 2’jj for 2 = « and the first maximum radius Bg- Thus a 
rigid boundary lengthens the period, a free surface decreases it. For Z - 
2 R 2 ) however, the chiange In the period is only 10 per cent. 

If the gravitational effect l.s neglected, as is justifiable for the 
gas globe produced by a detonator under ordinary pressures, from Equations 
[I4c], [15]» [l6], and the Z analog of Equation [17J» 


Z,-Z„ 


« W ^ 


2.60 B 
0.009 

1 + 4000 5 * 


^ , Ro\/Ro\^ 


Here - Z„ represents the di-splacement of the gas globe, from the point of 
detonation up to the first peak recompression, measured positively away from 
the surface. 

For R 2 /Z near-'l/2, approximately 


= + 1.23^ 

XV 2 ^ 


whereas at very small R 2 /Z, approximately 


Z,~ Z, -r 


+ g.i “V 


[28a] 


[28 b] 


1 .. 






i 


397 




Figure 7 “ Effect of a Single Surface on a Qas Globe 
when Gravity is Neglected 

\Z^-Zq\ denotes the dlsplaeeaent of the center during the interval mtU the first 
coopression, the distance of the point of detonation from the surface, 
aj the maxinua radius during the first expansion. 


peak 


Thus the effect of the surface should fall off with increasing distance Z at 
first nearly as ^/Z, then more rapidly and ultimately as 1/Z®. 

The sign indicates that a rigid surface (upper signs) should at- 
tract the gas globe, whereas a free surface (lower signs) should repel it, in 
agreement with observation. The two effects are nearly equal in magnitude, 
but the repulsion Is a little greater. ’ 

Equation [27] is plotted in Figure 7* In using these formulas it 
must be remembered that ifg varies with the hydrostatic pressure, as indicated 
in Equation [l8]. The formulas probably become unreliable when Z < ZHg; the 
^ ) corresponding parts of the curves are shown broken in Figure 7. 

MIGRATION DUE TO GRAVITY AND A SI’’*"' -or- 

Assume that the surface 
the gas globe, make an angle ^ wi' 
tance of the center of the gac> gif 
coordinate of the center measured 
upward in a vertical plane; see Pi 
are Equations [l4a, c], [15], [l6j 
sin 0, Oy =0, Cg = cos $ i and the 







S98 




Figure 8 - Diagram Illustrating a Gas Globe under the Simultaneous 
Influence of Gravity and a Neighboring Rigid Surface S 

The circle represents & globe of gas surrounded b/ water; the nass of water is 
bounded on one side by a rigid wall. 


while = Ny= 0. Hence, for the displacement of the globe to the first 
peak recompresslon 




F = 


2.60 Rt 


]/b4- 

Y 1 + < 


009 


4009 
B = /^+ V 


Bx = 0.0346 

[l-f- 0.23-^ 

^ sine 
Pa 

fl + 0.23^1 

^ cose - 0.223 Fl - 0.18 

L X J 

Pa 

L * J' 


[30a, b] 

[31 a] 
[31b] 
[32] 


or, to a good approximation. 


= 0.0346 ^ sin 0 [3^a] 

Pa 

= 0.0346 cosO - C.223 [34b] 

Here Xj - Xo and X, - Zq represent components of the displacement measured 
positively in the direction of increasing X or X. 

The first period, from Equation [5]* is 

r, = r,c[l + 0.20 [351 





3S9 




/ \ 

\ ’ 


In the formulas, values .of W are In pounds and all values of X and 
Z in feet; is the total hydrostatic pressure in atmospheres. 

The case, © 0, applies to a rigid horizontal bottom. The formu- 

las for $ m n, on the other hand, are found to apply to a free surface, with 
Z measured away from the surface and hence downward, provided changes are 
made corresponding to the assumption that, for the free surface, 

[36a, b] 

in place of Equations [29a, b]. Hence for a rigid horizontal bottom and for 
the free surface of the water, the formulas can conveniently be written 
together as follows: 


Z\ Zn 


2.60 


+ Ti 


A~ AA/V 
V«VV9 


[37] 


4000 V 




= ±|o.0346[l ± 0.23 - 0.223 [1 + 0.18 [38] 


or, to a good approximation 


Bj; = ± [o.0346 I*- - 0.223 (^)*j [38a] 

The upper sign refers to the rigid bottom and the lower sign to the free sur- 
face. The symbol IB^i denotes the numerical value of Bg taken without regard 
to sign; and Zj - Zo represents in each case the displacement measured posi- 
tively away from the surface. 

The first period Is ■ 

r, = r,o[i ± 0.20 (39] 

These formulas probably become unreliable when 

Z < 2J?g = 8(g)^ 

Because of the negative sign between the two parts rf Bg, the grav- 
itational effect and the effect of the surface oppose each other in the case 
of a free surface or a rigid surface below the charge, whereas the two ef- 
fects are in the same direction when a rigid surface lies above the charge. 
The gas globe from a small charge near the surface of the water sinks Instead 
of rising. 


J/Tkl'V 


l.4| 

1.2 

« 

■S 1.0 
.5 

^0.8 

I 

SO.6 

o 

a 

.s 0.4 

o 

0.2 

0 



\ 








\ 

\ 








\ 







V 

\ 

> 



Oispioe6m«nt upwarc 


— 1 1 



y 


\ 

'Horizontal Oisplacemani 
toward rigid vartical wall 















1 











i 

[ i 

i i 


r « 

> 

r 


10 


Olstonc* from Wall Z in f«it 



Figure 9 “ Upward and Horizontal 
Components of Displacement of 
the Gas Globe from 3/4 Ounce 

I, rPdt*y»XFl 

the charge Is asstassd to be detonated 10 feet belo« 
the Burfece of the water, far abovo the bottom, and 
Z feet froD a rigid vertical The curvea show 

the dleplacement from the point of detonation up to 
the point of greatest raconpression, according to 
approximate calculations. 


Figure 10 - Displacement of the Center 
of the Gas Globe from 3A Ounce of 
Tetryl or TNT, up to the Instant 
of Maximum xecompresslon 

Detonation Is assumed to occur at the height shown 
above a rigid horizontal bottom In water 20 feot 
deep. Positive ordinates represent a rise, negative 
ones a descant. The curve Is based on approximate 
calculations . 


The effects due to gravity and to the surface are almost additive 
but not entirely so, because of the occurrence of B or \B^\ in F, When grav- 
ity and the surface produce opposite effects, as In the case of a rigid bot- 
tom or a free surface, the net displacement is a little greater than the 
numerical difference of the values that the two displace.ients would have if 
they occurred singly. In such cases the gas globe contracts to a smaller 
radius than it would if only one effect occurred, and this decrease in the 
minimum radius increases the displacement. Otherwise, as In the case of a 
rigid wall located to one side of the globe or the rigid bottom of a boat 
above It, the two displacements are slightly decreased by their coexistence. 

The formulas are Illustrated in Figures 9 arid TO, which refer to a 
3/4-ounce charge of tetryl or TNT detonated at Z feet from a surface. In 
Figure 9 the surface Is assumed to be a rigid vertical wall, the charge is 
detonated 10 feet below the surface of the w.ater, and the bottom is assumed 
to lie much deeper. The vertical rise of the center of the gas globe and its 
horizontal displacement toward the wall, up to the point of maximum compres- 
sion, are shown by curves. In Figure 10 curves are shown for the same charge 
Z feet above a horizontal rigid bottom in water 20 feet deep. It will be 
noted that the gas globe descends If formed loss than about 3.5 feet from 
either the bottom or the free surface. 

The formulas for a free surface and for a rigid bottom are plotted 
in general terms in Figures 11 and 12, as explained under the figures. 




‘I 

I 




Ti 

u 





I 

i 

' i -i 

i 

i 








Figure 1 1 - Rise H of the Oas Qlobe Formed by W Pounds of Tetryl or Ttn* 
Detonated at a Depth Z below the Surface of the Sea of Infinite Depth 

it la the rlM of the center of the globe froa the tine of detonation up to the tine of the 
flrat peak reeanpresslon; both H and Z are escpreased in terns of iig, the intervening maxlmm 
radius. Curves are shown for three values of Z. For JSj see Figure 4 or . Equation [18]. 



Figure 1 2 - Curves for Estimating the Rise H of a Oas Globe at a 
Distance Z above the Bottom of the Sea 

The rise is fron the time of detonation iqp to the tine of the next peak recumpresslon. ^2 
is the intervening maximum radius In feet; is the total hydrostatic pressure In atmos- 
pheres or 1 + d/33 where d is the depth of water in feet at the point of detonation. Curves 
are shown for four values of z/R^’ ^2 Figure 4 or Equation [18]. The curves are 

drawn on the assumption of Infinite depth; they are fairly accurate If the gas globe Is at 
least ^R2 below the surface. 


I.,. JL 



402 



figure. 13 - Plot of the Critical Depth Do for Migration 
near the Bottom of the Sea 

In aaa water of depth leae thim <?« Seat, the gas globe should rise during the first rseoapresslon If 
detamtion osours at a dlatssce g * atMve tbe bottoa, where Zg ~ ^2 ih plotted in tsrws of a 

larger scale shoen at the right. In water of depth greater than the globe should sink toward the 

bottoai If ^ = gg. For fair acouracy the gas globe should be at least 5Ag below the surface of tbs sea. 


From Figure n it Is seen that migration downward can be produced 
by a free Surface only if the charge is less than 0.2 pound, provided deto- 
nation occurs at a depth at least as great as 2/Jg below the surface. In the 
absence of more exact calculations, it may reasonably be surmised that the 
globe from 1 pound or more of TNT or tetryl should ralgrate upward, however 
close to the surface it may be formed. 

The effect of the bottom is more complicated because the total hy- 
drostatic pressure, as influenced by the depth of the water, enters as a new 
variable. In order to illustrate more concretely the implications of Figux>e 
12, there is plotted on a basis of W in Figure 15 the depth of sea water 2>o 
at which Sg/p^ •= 1.33; the value of Z when Z « 2Bg in water of this depth is 
snown, on a different scale, as 

The formulas for Dp and Zg are 



In water shallower than Dp, the gas globe should rise if for;»ed a& a distance 
Zg or greater above the bottom; in water deeper than Dp, it should sink when 
It is formed at a distance equal to Zg, and also at progrc.'slvely greater dls- 
tiuioes as the depth of water is increased. 

Unfortunately the approximate formulas become unreliable at those 
short distances which are of greatest practical interest; they should be 
fairly accurate if Z ^ 2Bg, 




403 


It is particularly unfortunate that calculations do not exist for 
charges detonated on the bottom. They are made difficult by the Inevitable 
distortion of the gas globe. As Z Is diminished below ZR^. the attractive 
efi'ect of the bottom should probably increase and then decrease again. This 
conclusion is based on the following ideal case. The water flow around a 
hemispherical charge lying with its flat face on a rigid bottom and detonated 
at its center should resemble half of the flow around a spherical charge of 
the same radius detonated in open water; gravity should, therefore, cause the 
gas globe from the hemisphere to rise. Prom the analytical results it may 
reasonably be surmised that the gaa globe from 10 pounds or over, detonated 
on the bottom under any depth of water of practical interest, will probably 
rise during the first recorapression. 


MIORATION OP A GAS GLOBE IN SHALLOW WATER 

The combined effect of the free surface and of a parallel rigid 
bottom can be obtained by extending the method of Images. If Z is taken to 
stand for the distance of the center of the gas globe above the bottom, it is 
found that only those changes need to be made in the formulas, as obtained 
for a rigid bottom alone, which correspond to the assumption, instead of 
[ 29 a, b] 

1.39 


ilf = r, - 


2? 


n, = s. 


[40a, b] 


where D is the total depth of the water, 1.39 represents 2 log 2, and T\ and 
S 2 stand for the series 




Z 


1 


1 , 


1 


D-Z D+Z 2D-Z 2D+Z 

„ _ X . „__1 _ 1 . . . 

F (D - Z)^ (D i2D~Z)^ (ZdTW 


ZD -Z 

1 


Hence the displacement of the gas globe .messured upward, during the first ex- 
pansion and recorapression, is 




2.60 BgRz 




[41] 


B- = 0.0.346 [1 + 0.23 (r, - .d ^ 
- 0.223 [1 -0.18(r,- 


[42] 


404 


or, very nearly 


« 0. 0346 ^ - 0.223 t^2a] 

^ A 

The first period, from Equation I5]* Is 


T, = r,o[i + 0.20 (r, - R, 


[433 


Here distances are to be measured throughout In feet. The formulas 
become questionable if either bovmdary la closer than 2^2 or 8{W/p^)t 

These formulas are the same as those obtained for the bottom alone 
except that \/Z Is replaced by T, - 1.39/i^and l/Z* is replaced by S,. If 
the small term containing and D is omitted, it is clear that the displace- 
ment is the same as that due to a single surface at a distance Z, such that 
Si or 



In Si the effects of the bottom and of the free surface are added 
In a sort of quadratic fashion. If Z = D/2, so that the charge lies midway 
between the surface and the bottom, 


Thus 


Tj = 0, Si 



Z, = 0.74 Z 


SO that the displacement is roughly the same as that when either surface 
alone is present at about three-fourths of the actual distance to elthpr top 
or bottom. As the charge is moved toward either surface, however, the effect 
of the other surface rapidly decreases. Thus if Z = 0.35 i? or 0.65 D, the 
effect is about the same as that due to the nearer surface acting alone at a 
distance 0.91 times its actual distance. 

The effect of the free surface on the pex’iod somewhat exceeds that 
of the bottom. Hence when the charge is detonated midway between the two the 
period is shortened. The first period is 

ri = r,„(l- 0 . 28 ^) 


EFFECT OP PROXIMITY TO A FREE SURFACE AND A VERTICAL WALL 

The wall is supposed to be plane and rigid and to extend from the 
surface to a great depth. Let X denote the distance of the center of the 






405 







Figure 14 - Diagram Illustrating 
a Gas Globe near the Surface of 
the Water and also near a 
Vertical Rigid Wall 


globe below the surface, measured down- 
wsrdy snd ^ j.ts dist-9no6 f*z*oni thi© w 9 .ll \ 
see Figure 14. Then, In Equations [l4a, 
c], [ 15 ], [l6], [ 17 ] and its Z analog. 


clearly = - 1 , Cy - = 0. 

The analy' 

sis gives 


af -I _ i _i 
^ Z X L 

[44a] 

+ 

1 

II 

[44b] 


[44c] 


where L = Vx^ + Z*, and JVy = 0. Thus, for the displacement from the point of 
detonation to the point of peak compression. 




2.60(-g)*i?yi?2 


V 1 +4000B* 


Zi Zo 




x/b + — — 

y 1 + 4 


009 

40005 * 


B = + b/ 


[451 


[46] 

[47] 


- 0=0346 [l +0 = 2.? MRz + 0.223 (l + - 0.18 AfJZ,] [48] 

J ' i/'L 


5^ - - 0.223 [l - 0.18 (l - [491 

or, very nearly, 

Sx = -- 0.0346 ^ + 0.223(1 + [501 

5^ « - 0 . 223(1 - [ 51 ] 

Here X, - X^ is the upward component of the displacement, while Zj - Zj 1? 
the horizontal component measured positively away from the wall. 

The first period, from Equation [ 5 ], is 


[ 52 ] 


406 


The formulas are probably unreliable when either JIT or ^ Is less 
than Sfta or 8{W/p^)i 

here the additional terras containing L represent the prinoipal ef- 
fect of the interaction between the surfaces. Crudely speaking, the repul 
slve effect of the surface is increased by a factor 1 + X*/L^, while the 
attractive action of the wall la decreased by a factor 1 - Z*/L*, as compared 
to what these effects would be if the other surface were not present. The 
interaction between the two effects Is greatest when X = Z. Then L ■ VS’Xand 
the repulsion from the surface is increased In the ratio 1.35, while the at- 
traction toward the wall is decreased in the ratio O.65. These numbers will 
be somewhat modified, however, by the concomitant change in B. 

On the period, the surface effect again predominates and results in 
a shortening; the first period is 

T, = [1 - 0.20 |a] 

The most interesting feature in this case la the variation of the 
displacement with weight of charge. As the weight increases, the gravita- 
tional effect comes to predominate. In order to illustrate this fact, Fig- 
ures 15 and 16 show curves of vertical displacement H and the horizontal 
displacement S toward the wall, for a charge detonated at several distances 
Z from the wall In combination with several distances X below the surface of 
the water, plotted against the charge weight W. These figures also serve to 
indicate qualitatively the relative magnitudes of the two displacements at 
shorter distances from the surface, where the numerical formulas become un- 
reliable. 

This case has some resemblance to that of a floating mine exploding 
near a ship. For the relatively slow motion Involved in the production of 
migration a ship should function as a rigid obstacle. The ship extends down- 
ward, however, only to a limited depth. For this reason the attraction to- 
ward the ship should be considerably less, and the rise a little greater, 
than in the ideal case here considered. 

PRESSURE IN THE WATER AS INFLUENCED BY THE MiaRATIOH 

The pressure generated in the water by the recompression of the gas 
globe may be greatly altered by the migration. The general effect is compli- 
cated, as is illustrated by G.I. Taylor ( 5 ). The pressure will probably be 
further modified, however, in consequence of depai-tures from spherical sym- 
metry, so that calculations based upon the assumption of symmetry possess in 
most cases only a limited Interest. For this reason the following rough meth- 
od of estimating the pressure as irodlfled by the occurrence of migration may 
be of interest. 








407 


( ' 



Figure 1 5 Vertical Rise H of a Ras Globe near the Surface 
of the Sea and near a Vertical Wall 

The rise La fron the tine of detomtlon to the tiae of tto next peak reconpresslon. Xj la the 
interveniu* naximsia radius. Curves are shoim for 4 positions i (A) X = Z = 2fi^ (B) X = SSg, 
X - 3-5*25 (C) X - 3.5*2) * ■ 2*25 W X ~ 3.5*2> ^ ® 3.5*21 X is the dietahoe of the point 

of detonation belojt the surface and Z the distaaoe of this point from the wall. tKis the weight 
of the ctierge in pounds. For *2 see Figure 4 or Equation [1S]. 




(: 


Figure ^6 - Horizontal Component of Displacement S 
of the Gas Globe of Figure 15 


408 


The motion of the water can be resolved Into three parts superposed 
upon each other, a spherically aymmetrloal part associated with the radial 
oscillations of the gas globe, a part caused by any bounding surfaces that 
may be present, and a part associated with the motion of migration. The 
pre5sm*e can then be divided into three corresponding parts, provided the 
Bernoulli terra pe®/2 is omitted, so that the pressure is simply proportional 
to the rate of change of the velocity potential. The part of the pressure 
that is associated directly with the migratory motion is then essentially of 
dipole character and hence falls off relatively rapidly with distance; it may 
therefore be dropped in a rough calculation, except near the gas globe. The 
part due to a bounding surface, if any, Is simply the pressure due to the im- 
age of the gas globe in the surface and is easily allowed for on this basis. 
There remains then the part of the pressure that is associated with the rad- 
ial motion. This part is altered by the migration because the radial motion 
is altered. 

The radial part of the pressure is given by Equations [6] or [ 7 ] on 
page 45 of TMB Report 480 ( 10 ) with the omission of h*; it may not be cor- 
rectly given by Equation [8] of that report, however, in which the term in w/ 
is not negligible and is Influenced by the migratory motion. The pressure p 
at a point distant r from the center of the gas globe is thus 


P 




[53] 


where p Is the density of the water and denotes the total hydrostatic 
pressure at the level of the gas globe. Only the phase of intense compres- 
sion is of Interest, hence ^uation [1] of the present report can be simpli- 
fied as before, and even the email termMI^2 can be omitted for the present 
purpose. Thus from Equation [1] 




pR' 


jip^-p,)RUR- 1543 


TTne approximate values employed previously for X, f, Z can then be Inserted, 
and they may conveniently be expressed in terms of the linear displacement 
of the gas globe from the instant of detonation to the Instant of peak recom- 
preasion, which is 

Q = /(X, - Xo)* + (Fi - Xo)* + (^1 - ■Zo)* “ /h* + [551 


The pressure as thus estimated 5.s found to depend on the ratio Q/Rm, 
where Rf is che first maximum radius, and to be proportional to PfR^r, The 
impulse I <=f(p- is proportional to S^Vpi^/r. A single graph applicable 


409 



Figure 17 - Curves and Ponnulas for Estimating Roughly the 
Effect of Migration on the Pressure in the Water 

is the naxloun pressure to the aster at the instant of greatest rsconpresslon of the gas globe 
in pounds per square inch, is the total hydrostatic pressure in poimde per square inch and 
is ti« sane pressure expressed in atnospheres, is the peak inpulse defined as /pdt integrated 
between points before and after the peak at which - pg is nne^flfth of its value at the peak, 
r denotes distance In feet fron the center of the gas globe et the instant of naxinun recoapres- 
slon, ^2 first maxlmua radius of the gas glebe in feet, 0 is the linear displacenent of 

the gas globs from the point of detonation until the instant of greatest recompresslon, in feet. 


to all migratory cases can he constructed, therefore, by plotting against 
Q/Rb values of 


Pm*^ 
Pa^2 ' 


T 

r I 



where Is the total hydrostatic pressure measured in atmospheres and 
is the maximum excess of pressure above p^. This Is done in Figure 17- The 
Impulse Iq z Is taken between the two Instants at which the excess of pres- 
sure p ~ p„ is l/5 of its Intervening maximum value. The values of Iq g thus 
serve to give an idea of the estimated width of the pressure peak. 

To use Figure 1 7. the value of the displacement Q is first esti- 
mated by use of formulas given previously; then values of y and z are read 
off the curves and p„ and Igg are calculated from the formulas 

Pm “ ?o “ ^ PaV pounds per square inch 


[56a] 


410 


I 


0.2 





pound-seconds per square inch 


[56b] 


In which r and are In feet. 0^ in pounds per square inch and /0.2 in pound- 
seconds per square inch. 

If there is a plane bounding surface in the neighborhood, a correc- 
tion is then to be added representing the effect of a phantom gas globe lo- 
cated at the mirror image in the surface of the actual one. The pressure and 
impulse due to the image are calculated from the same formulas as those due 
to the actual gas globe, with v made equal te the distance from the image. 

The total pressure and impulse are then the sum of these respective quantl- 
tlerj for the actual globe and the image if the surface is rigid, or the dif- 
ference if the surface is a free one. At the surface Itself, the effect of 
the image is to double the excess 01 pressure above hydrostatic pressure on a 
rigid surface, or to keep the pressure at the hydrostatic level on a free 
surface. 

Prom Figure 17 it is seen that, as the migration Increases, the 
peak pressure decreases, but the width of the peak increases so that the im- 
pulse due to it is about constant. At Q =< J? 2 / 3 , the peak pressure has de- 
creased by about a third; the rate of decrease then becomes greater, so that 
at Q = 1 the peak pressure is only a tenth of what It would be In the absence 
of migration. 

The increase In /0.2 large values of Q/Rz in the figure arises 
from the fact that the total range of pressure becomes small and the peak, 
defined as extending from one-fifth of the maximum to one-fifth on the other 
side, comes to Include almost the entire range of positive pressure. The 
ratio of /0.2 to the total positive Impulse increases from about l /5 at $ =» 

0 to 4/5 at Q = R2. 


MIQRATION OP A GAS GLOBE IN A TROUGH OR BOX 

The necessary formulas have been written out for a charge detonated 
Inside a rectangular box partly filled with water; this obviously includes as 
special cases a deep we"'! of rectangular cross section or a trough with par- 
allel vertical sides. The series obtained in these cases are complicated, 
however, and for this reason no results will be cited here. 


MIGRATION AND SIMILITUDE 

In considering similitude as it applies to phenomena of migration, 
it should be noted that physical processes of four different types are in- 
volved and each process imposes its own characteristic requirements for si- 
militude. These differing requirements are not entirely reconcilable. Thus, 
in order that similar motions may occur on different scales: 




/ 



411 




0 


O 


a. The laws of the non-coapresslve motion of water require 
that pressure differences and the squares of velocities shall vary 
In the sane ratio, or 

Ap St V* 

b. The intervention of gravity requires that all pressure 
differences shall vary, as do gravity heads, in proportion to the 
linear dimensions, or 

Ap » L 

where L Is any convenient linear dimension. 

c. If a confined mass of gas Is present, its pressure must 
usually remain unchanged with scale, on the assumption that the 
mass of gas present is varied in proportion to the cube of the 
linear dimensions. Hence, all pressures must remain unchanged. 

d. Certain boundary conditions, such as exposure to the at- 
mosphere or the presence of cavitation, nay fix the actual value 
of the pressure at certain points. 

The hydrodynamical requirement a. Is consistent with many types of 
similitude. The addition of the gravity requirement b, restricts the choice 
to one In which «* « L, as In ship model testing. Then, also, Ap « 

If the migration effect on a gas globe is large, the gas has little 
effect on the radial and translatory motion, so that the pressure in the gas 
globe may be assumed to be zero. Ihe pressure at certain points Is then 
fixed, as in d. To keep Ap « L, the atmospheric pressure must then be adjust- 
ed in proportion to L. Furthermore, since the kinetic energy In the water is 
proportional to or to L*, and since this energy ihay be assumed to be a 
f^ed fraction of the energy released by the charge, and since the latter 
energy is proportional to the weight of the charge W, It follows that W must 
be varied in proportion to L*. Thus linear dimensions and pressures vary in 
proportion to and, since v® « L, velocities and times vary In proportion 
to^^. The atmospheric pressure must also be varied In proportion and 

the depth of the water must be varied in the same ratio if the depth is sig- 
nificant. In corresponding positions, the maximum radius and the migra- 
tory displacement of the gas globe likewise vary as 

When the migratory displacement Is small relative to R^, the simil- 
itude is not exact, because the motion is then appreciably influenced by the 
pressure of the gas. The partial failure of similitude in this case is not 
apparent from the approximate equations written In this report, because these 
equations are based upon a fixed value of Rt/Rft, whereas In reality this ratio 
will vary somewhat with the hydrostatic pressure. 






412 


In any case, similitude of the type described does not extend to 
the dimensions of the charge nor to the shock wave. These features can be 
Included only if gravitational effects are neglected. If that is done, re- 
quirements a, c, and d can be met by keeping all pressures and velocities the 
same at corresponding points, while linear dimensions and times are changed 
in proportion to W^. The effects of the gas pressure are then correctly 
covered; and the similitude will hold for migration due solely to the pres- 
ence of bounding surfaces. This Is the type of similitude that Is familiar 
in the discussion of underwater explosions with neglect of all gravitational 
effects. 

The various conditions requiring study thus lead to different cri- 
teria for similitude, a situation which occurs also In other applications of 
the ship model testing method. Thus in model tests of ship propulsion it has 
long been the accepted practice to break down the modal r-eslstancs into two 
parts which are stepped up to full-scale values by the use of different laws 
of similitude. 

It Is hoped that a similar procedure can eventually be established 
in the present case,- so that migration effects observed on small scale can be 
made the basis for a correction of the results of direct scaling according to 
the nominal theory based on the solid angle subtended at the charge by the 
target, as explained In TMB Report 492 (11). 

However, such a procedure is not yet possible. A study of migra- 
tion from that point of view is being made and the results will be communi- 
cated in a later report. 

NOTE OK MOMENTUM IN THE WATER 

In thinking about the motion of gas globes it is often natural to 
resort io reasoning based upon the principle of momentum. Much greater care 
must be used, however, in applying the principle of momentum to the noncora- 
pressive motion of liquids than In applying the principle of energy. It is 
very easy to go astray and arrive at the wrong conclusion. The fundamental 
I’eason lies in the fact that the transmission of momentum involves only the 
pressure itself, whereas the transmission of energy depends upon both pres- 
sure and particle velocity; because of this difference, momentum in an in- 
compressible liquid is more readily transmitted to great distances than is 
energy. 

To Illustrate the care that must be used in considerations of mo- 
mentum, consider a sphere of the same mean density as water, so that it will 
remain suspended without rising or sinking. Let an upward force be applied 
to it, causing it to rise in accelerated motion. The sphere is thereby 


413 


caused to press upward against the water in order to accelerate it; the water 
loads the sphere, in fact, with an equivalent mass equal to half the mass M 
of the displaced water, and an upward force F » Ma/Z must act on the water 
where a is the acceleration of the sphere. This force imparts upward momen- 
tum jFdi to the water. 

Yet if the total amount of upward momentum is calculated from the 
usual formulas, for the water lying within any given distance r of the center 
of the globe, the result la zero. The water around the sides of the gas 
globe moves downward as that above and below it moves upward, and, as re- 
gards the water Inside of any spherical surface concentric with the gas globe, 
the downward momentum on the sides just cancels the upward momentum above and 
below the sphere. The question thus arises, what has become of the upward 
momentian Imparted to the water by the upward force F? 

The paradox is redoubled by the following consideration. Since 
the sphere moves upward, water must on the whole move downward. The total 
momentum In the water must, therefore, be directed downward, not upward. It 
is easily shown that this downward momentum is, in fact, of magnitude zjFdt, 

The solution of the paradox is found upon careful consideration to 
lie in the occurrence of a decrease in the pressure near the bottom. As a 
result of the motion of the sphere, the upward force of the bottom on the 
water is decreased by 3F. One-third of this decrease compensates for the 
lifting effect of the sphere caused by its upward motion and thereby absorbs 
the upward momentum given by it to the water; the remaining two-thirds of the 
decrease allows part of the downward force due to gravity to develop in the 
water downward momentum of magnitude zjFdt. 

These considerations are based, of course, upon the assumption of 
incompressible water. If the action is so rapid, or the body of water so 
large, that non-cpmpreaslve theory is not adequate to describe the motion 
throughout, then part or all of the upward momentum given to the viater will 
remain in it, although perhaps not in the neighborhood of the sphere. 

The motion of the water around a moving spherical gas globe of 
fixed radius Is exactly the same as around a sphere of equal size moving at 
the same rate, hence the same considerations apply to the motion of the gas 
globe. The statements made ift this report have been carefully worded so as 
to be correct as they stand; caution must be used if the references to mo- 
mentum are altered or extended. 


REFERENCES 


(1 ) "Motion of a Pulsating Gas Globe Under Water - A Photographic 
Study,” by Lt, D.C. Campbell, USNR, T® Report 512, May 1943. 

(2) Deductions of the formulas are available for- inspection in 
manuscript form at the David Taylor Model Basin, 

(5) "Hydrodynamics," by Horace Lamb, M.A., LLD., Sc.D,, P.R.S., 
University Press, Cambridge, 1924. 

(4) "Theory of the Pulsations of. the Gas Bubble Produced by an 
Underwater Explosion, " by Conyers Herring, Volume II of this compendium. 

(5) "The Vertical Motion of a Spherical Bubble and the Pressure 
Surrounding It (S.W. 19)," by Professor Q.I, Taylor, P.R.S., Volume II of this 
conpendiiim. 

(6) "Behaviour of the Gas Bubble from 1 Ounce of TNT Detonated at 
Depthis of 3 and 6 Feet, Including an Approximate Computation of the Effect 

of the Surface on the Vertical Movement," Department of Scientific and Indus- 
trial Research, Road Research Laboratory, Report to the Admiralty, UNDEX 7* 

Note ADM/86/AJH, December 1942. 

(7) "The Behaviour of an Underwater Explosion Bubble - Approxima- 
tions Based on the Theory of Professor G.I, Taylor," Department of Scientific 
and Industrial Research, Road Research Laboratory, Report to the Admiralty, 
UNDEX 10, Note ADH/91/ARB, December 1942, 

(8) "Calculations in Connection with S.W. 19" (See Reference 5). 
by L.J. Comrie and H.O. Hartley. Prepared by the Scientific Computing Ser- 
vice, S.W. 26. 

(9) "The Pressure Pulses Produced by the Oscillation of the Under- 
water Explosion Bubble from 1 Ounce of Polar Ammon Gelegnlte," Department of 
Scientific and Industrial Research, Road Research laboratory, Report to the 

Admiralty, UNDEX 16, Note ADM/99/ARB, February 1943. 

• 

(10) "Report on Underwater Explosions," by Professor E.H. Kennard, 
Volume I of this eomqaendium 

(11) "The Design of Ship Structures to Resist Underwater Explosion - 
Nominal Theory," by Captain W.P. Roop, USN, TMB Report 492, August 1943. 




OF RISE OF LARGE VOLUMES OF GAS IN WATER 


G. I. Taylor and R. M. Davies 
Cambridge University 


British Contribution 


June 1944 



415 


THE RATK OF RISE OF LARQE VOLUMES OF 6AS IN WATER 
Q. I. Taylor amd R. M, Davies 
Tune 1944 


Summary . 

The work described in this report is concerned with the rate of rise of bubbles, ranging in 
volume from l,5 to lOO ec., produced by the non-explosive release of a volume of air in nitrobenzene 
or in water. 


Measurements of photographs of bubbles formed in nitrobenzene showed that the central portion 
of the upper surface was spherical in form. A theoretical discussion, based on the assumption that 
the pressure over the front of the bubble is the same as that in ideal hydrodynamic flow round a sphere, 
shows that the velocity of rise, U, should sc- related to the radius of curvature, R, jn tho region of 
the vertex, by tho equation u = Z/j/gR: the agreement between this relationship and the experimental 

results is excellent. 


For geometrically similar hubbies of such a diameter that tho drag coefficient should be 
independent of Beynolds number, it would be expected that u would be proportional to the sixth root of 
the volume, V; measurements, of about ninety bubbies -show considerable scatter in the values of 
although there is no systematic variation in the value of this ratio with the volume. |f U is expressed 
in em./sec. and v In c.es., the experiments give the formula u = ZU.B if this frimuln is applied 

to the volune of gas given off by the explosion of jeo lbs. of Amatol, tha calculated rate cf rise is 
about 17 ft. /sec. Although no direct measurements cf this velocity have been made, this estimate is 
net Inconsistent with measurements of tho time between the appearance oif the dome and tho plume when 
depth charges are exploded at grebt depths. 


Introduction ami Sxperimntal Meth od . 

The rise of gas hubbies in liquids has been studied by several workers*, but In all the work 
so far published, the bubbles hive been so small that the results are not applicable to the study of the 
rise of large volumes of gas, such as those produced in submarine explosions. In the experiments here 
described, bubbles ranging in volume from 1.5 to 3» c.cs. were formed In nitrobenzene contained in a 
tank, 2 feet x 2 feet x 2 feet 6 inches filled to a depth of about 2 feet with the liquid. The bubbles 
were photographed by spark photogr.iphy at intervals of about lO roll 1 iseconds, using a revolving drum 
c-amera in the manner previously described w. in some further experiments, bubbies covering a range of 
volume from 4.5 to 200 c.cs. wore formed in a cylindrical tank, 2 feet e inches diameter, filled with 
water to a depth of 3 feet s inches, end their mean velocity of rise over a measured distance was 
determined. In both sets of experiments, the air volunc was datarmlned by collecting the bubble in a 
graduated glass cylinder, 

considerable difficulty was found In producing single, large bubbles of gas, and the method 
finally adopted wa-s to pivot an inverted beaker containing air, which was then tilted so that the 
air was released. in general, the air is released from the beaker in a stream of bubbles of varying 
sizes, but by adjusting the rate of tilting, it was found possible to arrange that the air was spilled 
into a single bubble. 

Two successive photo.-niphs of a typical bubble formed in this say in nitrobenzene are shown 
in Figure i, the time-interval between tho two photographs being 10.3 millisecs. |n addition to the 

bubble 

• Allen, M.S., Phil. Mag., vol. 50 pp.323 and 519 (i900). koefer, k. v.d.I., voI.st, p.1174 (t9i3) 

Miysgl, 0, Tohoku Imperial university. Technological Reports, vol. 5, p.l35 (1925): Vol.8,p.587 (l929) 

Phil. Nog., Vol, 50, p.112, (1925). 

M Taylor, 0.1. and Davies, R.M. ’The Motion end shape of the hollow produced by an explosion in 
a I iquid'. 


I 


bubble, the photoyraphs show a steel Dal), i Inch dianeter, soldered at the lower end of a vertical 
rod tmnersed In the liquid; this e'’rangeinant was used to find the scale or the photographs and to 
give a reference marh from which the vertical displacement of the bubble could be measured. 


In the original photographs, a region of turbulence is clearly shown behind the larga bubble. 
This is due no doubt to some anisotropic optical property of nitrobenzene when subjected to viscous 
stresses. That such stresses exist could be inferred from the fact that the largest of the small 
bubbles in the wahe of the large one Is not spherical and Is rapidly changing In shape. This bubble 
has a dismeter of about « mm, still smaller bubbles are less distorted and one of diameter about 
2 irni., seen to the left side of the 6 mm, bubble is distorted so that Its length/diameter ratio Is 
about t.i. 


The rate of shear which might be expected to produce a distortion of this amount has been 
calculated i. in the field of flow represented by the equations 

u • Cx, V a -Cy. w a 0 ... ... (l.l) 

an atr^bubble uf mean radius a would be pulled out sc that 


where L and 3 are the length and oreadth of the bubble and /z and T the viscosity and the surface 
tension of the liquid. for nitrobenzene, m a o.oie poises, t a 43.9 dynes/cm., so that for the 
2 mm, bubbles, a * 0.1 cm., l '/8 » 1.1 and (L - 8)/(t r B) • 0.05, giving 

0.05 X 4J.9 

^ * 2 X 0.018 x“o7i ■* sec 


The rate 1 of dissipation of energy per c.c. In the flow represented Oy equation (l.l) is 

//Ckl i'^VX 2 4 

/ J “ ° X 10* ergs/e. c. /sec. 

lf the rate of dissipation were constant through the wake and if wake extends over the whole 
of the region which appears disturbed In figure 1, nwiely through a diaireter of 5.9 cm,, the total 
rate of dissipation in the wake is 

l, 3 » X 10** X (volume of wake) 

•= i. 3 k X 10" X (5.9)- X 3.8 
4 

= 1.4 X 10* urys/sec. 


The total rate of dissipation would be known if the drag co-efficient, Cp of th* large bubble 
were Known, since the density p of eitroeonzono is i.z gm/c.c, whilst the velocity of rise, U, Of 
the large bubble In this experiment was 36,7 cm/s«c. and its maximum transverse dimension, 2A, was 
5.1 cm., the total rate uf dissipation was 


C- X * /O 7T X u 


Cp Ji X 1.20 X 


/5.1 

36.73 X 77 x^— j- 1 ? 


• 6.1 X 10® Cp ergs/sec. 

Since Cp is of the order 1.0 (see Table 1) it will be seen that the rate of dissipation which 
would distort the ouooies by the observed amount is of the same order as that deduced from the rate 
of rise. 


i G. I. Taylor •formation of emulsions indefinable fields f flow" 
proc. Soy. Soc. Vol. 146 p.501, (l934) 




- 3 - 


427 




For the l^irgpst of thp srroll bubbles in the wake, viscous stresses would produce such a 
distortion that the* formula (i,2) would not Do expected to apply. 


Thff uniformity of thr velocity of rls.^ of th»' ouooles, c-^nri the order of m:»gnltude of th»* 
('xpcrimental error in tne mc.^surement of the velocity may be jud.vd from Figure 2, in which« time t, 
and the vertical oisplaC',T.H.ru, x, of thj. two ^juddIcs ire plottid as 'hsclssne and ordinates, respectively. 
Thr actu:*! measureo values of X and t for the hubbl" c? Figur i e.rc i??dicatcd by .the circular dots 
in Figure 2* - nd those fora second. Verger bubble t.y crossrs; thu straight linos of closest fit drawn 
through tho obsi^rv.d points ere rt.-noted by *a* and *8* respect ivt-ly. it will be scon that the scatter 
of the experimental points is not excessive and that the velocity of rls'? jf the two bubbles is r&asonably 
constant over the interval mccisurod. 


Thh' shape of the profile of tho bubbles was found by measuring the films « n a travelling 
micrjscvpe fittoo with tw-- independent irwtions at right -».nglcs t. -^ne .••.nothrr. The results f-.r the 
U'wer photjgraprt of Figure \ are shown graphicAlly in figure 3 whore the circular dots represent points 
on the central, regular portion of the* profile of the bubble. dr*ducod from the microscope readings. 

In Figure 3* the vertical -and horlyontal axes are parallel to the corresponding axes in tho tanx and the 
origin is taken at tho uppermost point on the bubble; the dimensions given in Figure 3 refer to th^: 
actual size of the bubble, Thu crosses with vertical axes and with axes at 45® to the vertical in 
Figure 3 represent points on the profile of the same bubble, obtained from measurements of photographs 
taken 105,7 and 132.6 milliseconds earlier than the lower photogr^iph of Figure i. The agreeront between 
the three sets of points shows that tho shape uf the cap uf the bubble underaoes very little varlatun 
••ver the range ■ f tlnie covered by the three photogr^piis. 

The Curve in figure 3 is nn arc of circle of radius 3. 01 cm,, drawn to oass throuah the origin 
and since the scatter of the observed points around this curve is within the limits of the errors toOo in 
measuring the film, the upper port of the bubble is e portion of .1 sphere within the experimental error. 
It is worth noticing that the angle suotendud at the centre- of the circle by the arc In Figure 3 is about 
75°, Whilst the angular width of the whole buObl,-, In Figure 2 (referred to the centre) is about bO®. 



a. The vertical motion of a gas bubble uith a sbherieal eap, 

Measureeients of the pressure over the front pert of the surface of a spher.i exposed to a wind 
shows that the distribution of pressure is very similar to that calculated assuming Ideal hydrodynamic 
flow. Similar measurements over a solid spherical cap set with its vertex facing the wind show that 
the removal of the roar part of the sphere does not greatly affect the pressure distribution over the 
front except near tne rim of the cap. 

according to the hydrodynamic theory of Ideal fluids, the pressure p at angle rp from the 
stagnatTon point of vertex of a sphere (see Inset In Figure 3) moving with unlfonn velocity u In a 
fluid of density p is 

j 9 5 

P ' P. ♦ * (1- j- sln^ifl (2,1) 

where p^ Is the pressure at the depth i, of the point concerned, at a great horizontal distance from 
the sphere. 

Thus 



pj. “ Po ° ept (2.2) 

where Pg Is the atmospheric pressure, and, at anglegct for a sphere of radius R, 

z « d -f R (I - eos<p) (2.3) 

Where d is the depth of the vertex. 

From equations (2.l), (2,2) and {2^3) 

P • Pq + gpp ♦ gpB(i-cos(^) ♦ ipu^d--^ Sin 2 <^) ... (2.4) 




* f 


41S 


- yi - 



I 

i 

i 

I 


{ 



t 

I 

t 

I 



' i 


:.S I 

" i i 

: . I 

-i I 


1 j ■ 



nl 

-HI 


The condition which must be satisfied over the surface cf the bubble is that, apart from 
surface tension effects Kfhich arc small when the diameter of the bubble is greater than 2 cm., the 
pressure inside must be equal to the pressure outside. At the vertes, the pressure is 
(P(j ♦ 9 pf + i and this must therefore be the pressure of the air inside the bubble. The 

expression (2.«) for the pressure Is therefore satisfied alt over the sphdrical cap if 

9 2 2 

j" U sin (T = 3 R ( I cos <j>) (2,5) 


This equation is satisfied for small values of ilf 


u‘ 4 
“gR ” “ 

or u • I (2.4) 

for, in tlBt case, 1 im A - cos q. 

hJ"* o\sin^ oj 



To see how far the theoretical relationship (2,6) is verified experimental!/ and to obtain further 
data concerning the rise of bubbles, fourteen bubbles, rising in n i t robenzene, were photographed. 

The results of the measurement of the films are summarized in Tabic i, where the first three columns 
Qtve the volume* v of ihe bubbies« the rflblus of curvAture« ft. n^Ar th^ shd the observed 

velocity 0^ rise* u* The fourth column in the table gives the maximum transverse dimension* 2A of the 
bubble and the fifth column the angle =*sln ^ A/fi* The sixth column gives the drag co-efficient. Cq» 

of the bubbles calculated from the equation 

CjjXTTA^Xipu^ • gpi (2.7) 

The last column in the table gives the value of Reynolds number* Re, referred to the Tadlus, a» 
of the maximum transverse section of the bubble* l.e,* 

UA 

gg a **, (2.9) 

Where v is the kinematic viscosity of the liquid* Tor nitrobenzene* the viscosity is O.difi poises at 
14®C., and the density is 1,20 gm*/c,c., hence v «= 0 015 cm^/sec. 


Table k 


bubbles in ,Vitrobemene » 


Volume 
V (c,c) 

Radius of 
curvature 
R 

Veloc it/ 
U (cm,/ 
sec.) 

Ma<ifflum 

transverse 

dimension 

2A 

5ln"* a/R 

Drag 

co-ef- 

ficient 

C*. 

V 

Re/nolds 

number 

Re 


(cm.) 


(cm.) 



• 

1.48 

2.41 

29.2 

2.66 

38.4° 

0,53 

2780 

3.50 

2.09 

29.6- 

3.14 

48,6® 

1.02 

3090 

4*06 

2.0V 

28.9 

3.48 

58.3® 

1.00 

3360 

tt.31 

2.17 

28,0 

3.1S 

46.8* 

1.37 

2950 

6.40 

2.78 

36.5 

4.98 

63.6° 

0.52 

5900 

7,30 

2. 65 

34.2 

4.40 

58.1* 

0.80 

5010 

8.02 

2.67 

34.0 

4.23 

52.5° 

0,97 

4790 

“ 

3.01 

36.7 

- 

-. 


«• 

3 . so 

3,17 

37-Z 

5.26 

56.1® 

0.56 

6520 

9.18 

2-77 

33.0 

4,53 

65,8' 

1.00 

5020 

18 .40 

3.30 

37.3 

s.fo 

30.7° 

1.27 

6340 

21.25 

3. Si 

38.1 

5.85 

56,5° 

1.07 

7440 

28.1 

V.16 

43.0 

- 




33.9 

4.27 

42.1 

6.19 

46.5° 

1.25 

8700 


V.S4 

48.2 

■ 

• 

~ 

r 


) 







I 





- 5 - 


419 


The securacy oT the relationship (j.6) can bo judged from Figure U, In which the vsJues of u 
given In table l are plotted against the values oTv^ it will bo seen that the experimental points 
are, on the whole, reasonably well represented by the straight line given by the relationship u o 2/3 
i/g?. 


The values at Cp given In table I arc very variable and they do not appear to Show any consistent 
variation with respect to either of the variables or Se. At an early stage In the experiments it 
was thought that might be a function of <t^ and, for comparison, a series of experiments was carried 
out in a wind-tunnel in order to find how the drag co-efficient of a rigid spherical cap varied with the 
semi-vertical angle Thu cap was supported so as to face the wind and Its leeward side was closed 

with a plane metal disc; using pressure orifices connected in turn to a .manaTreter, the pressures at 
various points on the curv'wd and flat surfaces of th(> cap were determined and the drag co-efficient 
calculated In the usual way'. Four oodles, with a radius of i Inch and wIlhgEi^ equal to 90°, 75°, 

5B° and gc° respectively vfore used In th--. experiments and the values cf Cp are given in the second column 
of table II, In the expcirlroents, the wind speed was kept constant at 1500 cm./sec., and the values of 
the Reynolds numbers, defined by equation (2.8) are given In the third column of the table. 


TABLE II . 


SiliA. 


_ i r. : r ~ 

.VI*,- V a y. 



=0 

Reynolds Number 

90° 

O.U63 

25400 

75° 

0.521 

24600 

55° 

0.6X7 

20000 

30° 

0.862 

12400 


These values of Cq plotted against are shown In Figure 5 by the circular dots; this diagram 
also shows the values of Cp given in table 1 for bubbles rising in nitrobenzene. 

It Is clear that, whereas the results of the wind-tunnel experiments lie on a smooth curve, the 
results of tne nitrobenzene experiments show a rather large scatter, and. In addition, thn general trend 
of the curve of closest fit drawn through the points for the bubbles (indicated by the broken line In 
Figure 5) differs considerably from the curve given by the wind-tunnel experiments. 

It is difficult to be certain of the reason for these effects. They may, for example, Oe due 
to the variation in Reynolds numoer, although this is unlikely since the values of Cp for the bubble show 
no systematic variation with Reynolds number. it is more likely that the effects In question are due to 
variations in the shape of the bubbles. in thic connection, it must bo reraentcred that our photographs 
show only the projection of a bubble on a vertical plane, so that the lower surface of a huhhle is 
invisible unless it is convex downards. Visual observation shows that the lower surface is. In fact, 
usually concave downards, its curvature being less than that of the upper surface, and the difference 
between the .observed (Cp, curves for the oubble and for the wind-tunnel experiment may be caused by 
the difference in the wake in the two experiments, due to the difference in the geometrical forms of the 
bubble and the flat-bottomed spherical cap. 

Similarly, in the case of the bubbles themselves, the scatter of the experimental values of Cp 
for a given value of may bo caused by differences in the value of the ratio of the curvatures of the 
upper and lower surfaces when is constant 


■ See for example, R, Jones, Phil, Trans. A vol. 226 p.231, (1927). 


■■ -r 



’^MataefcliSS^ 



420 - 6 - 

rAa relationship betueen the volvms and the rate of ri^e of a bubble. 



If all the bubbles were geometrically similar and If the drag co-Bfflelent were independent of 
Reynolds number, it would be expected that the velocity of rise would be independent of the density 
of the liquid and would be proportional to (volume) Figures « and 7 show the results of experlnants. 
Involving 13 bubbles rising in nitrobenzene and about 75 bubbles rising in water, which were carried out 
to tost the truth of this prediction. In Figure 6 , the velocity of rise, U, (In cm./sec.) Is plotted 
against the volume, V, (in c.c.) whilst. In Figure 7, the ratio is plotted as a function of 

In the diagrams, the results for bubbles in water and In nitrobenzene are plotted os circular dots and 
crosses surrounded by circles rospectively: the two diagrams also show the curves derived from the 
experimental results of H>yagl and of Koefer, 

Figure 7 shows a considerable scatter of the experimental points around the horizontal straight 
line of ordinate 2U.8 which represents the Man value of derived from the observations; in 

the same way, the experimental points are widely scattered around the curve u » 2».g In Figure 6 , 

It Is worth noticing that In Figure 7, there is no systematic variation of with V, and that the 

points for bubbles in water and in nitrobenzene can be represented by the same curve; In all 
probability, the scatter of the points is^ue to the reason already put forward to account for the 
discrepancies in Figure S, namely the lack of geometrical similarity In the bubbles. 

4, tine of rise of gas from a deej> submarint extlonon, ■ 

Photographs of bubbles produced by a spark* show that SuCh bubbles pulsate violently and rise 
at a very variable speed during the first few pulsations. The amplitude of these pulsations dies down 
very greatly after three or four cycles and the rate of rise becomes more nearly constant, and, at the 
same time, the bubbles assume a mushroom-like form which is rather similar to that shown in Figure 1 for 
air released non-explosively. 

Measurements of the time-interval between the appearance of the spray dome and that of the plume 
have been made for depth charges filled with 300 lbs. of amatol, fired at different depths. The results 
show considerable scatter. The mean curve representing these experimental results, Is shown in Figure 
0 of the present report. It wilt bo seen that the mean rate of rise of ges from the depth charge Is 
very rapid (of order us ft. /sec.) when the depth of the charge Is less than about 90 feet; when the 
charge Is deeper than so feet, the mean rate of rise rapidly oecreases. Fran the sound-ranging 
observations it appears that the plume comes through the surface during the fourth oscillation If the 
depth is about 90 feet and that bubble oscillations can hardly be distinguished beyond this point. 

It Is not possible from these observations to deduce the rate of rise of gas after the bubble 
oscillations have ceased. |f, however, the rate of riee during the first four pulsations Is assumed 
to be Independent of depth, the slope of the curve of Figure a for times greeter than zi seconds would 
give the rate of rise after the fourth pulsation, with this assumption, the time for the gas to rise 
from 90 feet above the charge to 134 feet above It would be 3.0 seconds, corresponding with a mean 
velocity of 44/3.0 = 14.7 ft. /sec. if the u.me from no to 134 feet above the charge were taken from 
Figure a as 2.2 seconds, the corresponding mean velocity is 24 / 2.2 » 10.9 ft. /sec. 

Taking the amount of gas released from 1 gm. of amatol, after the water vapour has been 
condensed, as 550 c.c., the volume released from a 300 lb. depth charge is 8.8 x 10^ c.c. The rormuia. 
u » 24.8 deduced from the present experiments wuuld therefore predict that the rate of rise of gas 

from a 300 1b, depth charge would be 

U • 24,8 X (8.8 X 10^)^^* = 526 un./sec. » J7.2 ft./sec. 

For comparison with the experimental results, a straight line whose 5 l>>pe corresponds with this velocity 
Is shown In Figure 8 - it win- be seen that the actual vertical velocity of gas from an explosion, when 


? r 5 


pulsation 









pulsation has died a«ay, Is of the ssine order ns that predicted from msasuremcnts with htihhUs whose 
volumes are cf order one teh-mill I. nth tc one millionth of that ;it the explosion products. 

Though the worh here discussed is cgncarned with single DuPbles, it tray be renarked that if the 
gas separates into two buboles, the velocity of each would only be reduced by J 2 i per cent it the 
bubbles rose independently. if it separates Into 6u <,-qua! bubbles and each risos independently, the 

rate of rise would be reduced by 50 per cent, but It they co-operate so that each bubble moves in a 
rising current produced by the others, the rate of rise will not be reduced so much. 




. . - 








rtTTT- 




■\ -■■ 




c<- ' -- 








TT 


J" I-.. 


U-._- 


1— b 


-(■; . 


?1f- 1* Successive spark photographs of ar 

air-filled hubble rising in nitro'ben 2 sene 


Time-interval between photographs s 10.3 '.lilllsecs 


Velocitv of rise of bubble 


36.7 cm. /sec. 


Diameter of steel ball shown in 
unner part of photograph 


= ^ inch . 















































































t n 




r't 


( ' 


A SIMPUFIED THEORY OP THE EFFECT OF SURFACES 
ON THE MOTION OF AN EXPLOSION BUBBLE 


A. R. Bryant 

Road Research Laboratory, London 


British Contribution 


October 1944 


431 


A SIMPLIFIED THEORY OF THE EFFECT OF SURFACES 
ON THE MOTION OF AN EXPLOSION BUBBLE 

A. R. Bryant 

October 1944 


A sitnpta physical SKplanatlon of tha affect of surfaces on ths dlsplaceiiant of an explosion 
bubble Is suggasted. TMs axplaitatlon leads to the same quantitative expression as the treatments 
given by Conyers Herring and by Taytor. It Is shown that In order to calculate the effect of any 
given surface It Is only necessary to determine the space gradient of the velocity potential due 
to the 'Image* sources 'Induced* by a unit source at the explosion centre. A table of this 
factor for a numbnr jf fymsT. surfaces !• added for cemvenlence. 

Introduction. 

The purpose of this note Is to suggest a simple physical explanation of the influence of 
surfaces on an explosion biXjble. it will be shown that this explanation leads, with comparatively 
simple mthemetlcs, to the same quantitative expressions as the treatment given by Conyers Herring (l) 
and Taylor (2). 


For purposes of explanation It will be convenient to consider a simple case first, viz. the 
Influence of an Infinite rigid plane on an explosion bubble. The generalization to any arbitrary 
surface follows at once. For the moment the effect of gravity will be Ignored, 


A. 


Figure t. 



I 


0 

X 


In Figure t, 0 is the centre of an explosion bubble of redlus a, assumed small compared to 
its distance d from the rigid plane AB. The motion of the water outside the bubble due to its 
pylsation Is t|» same os tiet produced by a point 'soyrea' at 0 nf strength 

e • (1) 


The effect of^thc plane surface aB is then ths Mime as that produced by a source, e at the mirror 
Image point o'. 


The iiTidge source e at will produce a velocity In the water everywhere directed radially 
away from or to 0^ according as the sign of e Is positive or negative, and this velocity Is 
superimposed on the radial flow due to the explosion bii>ble at 0. Hence the water In the 
neighbourhood of 0 will have a net velocity towards given by 


“l 


3 X I at 0 


( 2 ) 


where Is the 
co-ordinate being taken 


velocity potential due to the Image source at o' the origin of the x 
at 0 . 


432 


In addition the Imaje source produces a presence p' at any point In the water, For points 

not too close to 0^ the pressure p' Is simply • The pressure aradlent ^ In the 

4 d t ox 

ne I jhDournooo of o oue »o the iisjst awy.‘»o st 0 is thus. 


3eil = d 1 f 1 

Bx j ' Bt [ S X J at 


Now It Is a matter of common ex^rlence that a buhblc or holliM In a liquid In which there 
Is a pressure gradient due to gravity will rtove upwards In the direction of the pressure gradient, 

I.e. from rrqions of high pressure to regions of low» In the sane way the explosion bubble will 

tend to drift In the direction of the pressure gradient set up by the Image source at 0^: this 

will be away from the surface as If e Is positive. 


G. I. Taylor has derived by a simple physical argument an equation for the velocity of tho 
bubble U due to tho action of a pressure gradient go.. The argument runs as follows. The 
“floating power* of a spherical hollow of radius a due to the pressure gradient syp Is jTT a^ tp, 
and this Is therefore equal to the vertical momcmtim comnunlcated per second to tho water flowing 
round the spherical hollow. Tho Inertia of the mass of water effectively moving with tho bubble 
Is e?p (a well known hydrodynamical result), and Its momentum Is h(We j7r a’p U where U Is 
tho velocity of tho cor.tro of the bubblo. Thus ^ (| w ” 7 ^ '^Tom which follows 

equation (a). The same equation is obtained by Conyers Herring by his perturbation methild* 
using Taylor's equation for u 


It is clear that any pressure gradient, no matter how produced, will couae a similar drift velocity. 


Tihus ,a pressure gradient p ^ \ 1 di 

dt (. d X J at 0 
a velocity Uj towards 0^, whore 

u » .2 r 1 f 

^ ^ 3t[5x]aio 


due to tho image source will cause the bubble to acquire 


The total drift velocity of the bubble U towards the rigid surface Is thus the sum of 
U^ and Uj. 

In the particular case discussed here It can be seen that 4^ ” and the velocity u of 
the bubble towards the rigid plane distance d tcom the bubble 

U « “ sfa. - — ^ ^ i (a=!j) dt 

4d* fd'a’ dt 


. - 5 ^ + -_2— nVdt 

4 d‘ 2d a^ 


on Integrating by parts, F.quatlon («) is that given by Conyers Horrlng and by Taylor. 
Extension t o the Gener a l Case. 

The argument used above for the simple case of a rigid plane is quite general, and pan 
easily be extended to the general case. The result Is simply stated here. 

Let there bo any set of froe or rigid surfaces symmeirical about tho axis x of co-ordinates 
passing through the centre of the explosion bubble o. Assume that a distribution of Inoge sources 
Can be found which satisfies the boundary conditions at the given surfaces when a unit source Is 
placed at 0. Lot the velocity potential due to these Image sources be Then the drift velocity 
U o' the buoble In the direction of the x axis towards tho origin of co-ordinates Is 





strictly shaking a dipole source should t>e added to o to allow for the linear notion of the 
bubble (3), and further muUlpoles to allow for Its change of shape. The effect of "Induced’ Image 
sources due to these will be second order snail quantities if the effect of tho primary Inoge 
distribution Is a first order small quantity. Similarly, since the drift Is a first order smell 
quantity the error due to assuming the distance from bubble to surfaces to retrain constant will be 
a second order sirall quantity. 

The velocity Is osclUatory and Is directed away from rigid surfaces In the first half of 
the oscdlatlon, and towards rigid surfaces In the second half. Its Integrated effect over a whole 
cycle being usually nugiigible. The velocity Uj due to the pressure gradient set up by the image 
sources Is directed away from rigid surfaces as long as the pressure Is positive, l.c. as long as 
the bubble Is greater than the hydrostatic pressure at the same depth, when the bubble pressure 
falls belcw this hydrostatic pressure the pressure gradients are reversed and the velocity is 
then towards a rigid surface. Since the force exerted on a hollow vessel Is proportional to its 
votuse the bubble acquires moit of Its mamantum when It is large and as the pressure is below the 
oorml hydrostatic pressure when the bubble Is large the net of foot at the end of one ascii tuition 
Is a drift towards rigid surfaces. These statements may assist an understanding of the phenomena 
found by Temperley In calculating the effect of a deformable target plate on the motion of th" 
bubble, and In particular the ccrrellatton between the sign of the motion and the sign of the 
pressure In the bubble. 


broximafe Formula for the Disfilacement of the. Bubble.. 




The Integral for the momentum of the bubble towards a surface, and that for the vertical 
noraentum dbe to gravity both become nearly constant tewards the end of the first oscillation. It 
is In this region tlmt most of the displacement of the bubble occurs. Hence It Is a reasonable 
approxlmtlon to assume that the total dlsplacenents at the end of the first period due to the 
Siirfaoa and to gravity are In the same ratio aS the momenta. Denoting these two displacements 
by S and H respectively we hpva 

- * “1^1 (R.H.S. Non-dimensional) <8) 

H 3xjat0 ij a- dt 


Where tto intograls are to be taken over the whole period, S Is towards the origin of the 
origin of tho co-ordinate x. 


In obtaining (3) all lengths and times on the right h-nd sloe have been converted to Taylor's 
non-dimensional units, i.o. all lengths are divided by the standard length 1 “ (w/go) where H 

is tho onorgy of the miotion, p the density of water. »11 times are divided by the standard tlme,/T. 
— £ has the dimensions L~*. In addition the sarall oscillatory term 3a*S In the expression (7) ® 

W X 

has hesf! dropped, slnco it .T.ay be- exactly integrated and shown to be vary small nt the end of a 


period. In Table 1 the value of these Integrals are calculated for a number of cases, and compared 
with two suggested approximate formulae. 



1 ?? 





(1) Conyors iiorriny’s formjiao in non -dimensional form. 

(2) Fornulao given In the report "The Behaviour oT an Mndemioter Explosion Bubble*' 

The values of the Integrals givan In Table 1 were obtained from the numerical Integration 
of Taylor's equations for the motion of an explosion bubble In the absence of all surfaces. It 

has been assumed that the perturbing effect of the prosonca of the^surfaee on the radlu|/time 
Curve of the bubble Is sirall during the time when the bubble Is large and the two Integrals are 
growing. 

It win be seen that Conyers Herrlno's formula for a*S*dt is considerably In error, and 
an alternative formula viz,, 0.37 x 10“^ ®m**a empirically, Is put forward In the Table 

which gives reasonable agreemint. Inserting this approximation, and the one given In the last 
line of the Table, In eguatlon (s) now yields 

- “ *1-^1 (r.H.S. don-dimensional) (9) 

H, 3 X J at 0 "9 

To take the approximation one stage further the value of thj rise due to gravity obtained, 
may be Inserted. The equation given was 

Rise due to gravity h * — (dori-dlmcnslonal) (to) 

'0 ■ ° 

Hence the displaccncnt s of the bubble towards the surface In non-dintenslonal units 

becomes 

s * 1 1.89 a_^z (Non-dimensional) (u) 

o X J at 0 " 

-a 

But Is very nearly Independent of c, I.e. Independent of charge weight and also of 

Zp (dopth), Neplacing It by an average value giving thn best fit over tho whale range, (U) 
becomes finally, 

Non-diroenslonal displacement , 

towards origin of x co-ordinotes = - 0.37 2,^ 1 (Hon-diraenslonal) 

In first period d x j at 0 (l2) 

This 







- 5 - 


435 




This simplification asrees with equation (U) within 7< over tha range “ 7.0, 

C • 0.063 (1 02. at 6 feet depth) to • 2. C - O.ll (475 ID. at 93 feet). 



Let S be the abfolutc displacsjnent of the bubble at the end of the first period In any 
particular set up of charge and surfaces. Let s' be the bubble displacemont In a geometrically 
similar set up where all dimensions arc Increased In the ratio the charge weight being Increased 
in the ratlo\^. Then It follows at once from equation (12> that 

s' » A.* S (13) 

Equation (13) Is Independent of the depth at which the explosion occurs. If the 
displacement had scaled according to the w^/3 law s' would have been equil to\S, so that (13) 
Indicates that the displacement of a lirgo explosion bubbla Is less than the value obtained hy 
scaling up from a small scale experiment. 

CoKclusion , 

The effect of surfaces on an explosion bubble has been shown to be due to two principal 
causes, viz., the velocity Imparted to the water near the bubble by the Image sources, and the 
pressure gradient set up In the water by the Image sources. It Is found that to calculate the 
effect It is necessary only to calculate the space gradient, along the axis of symmetry, of the 
velocity potential due to the Image sources produced by a unit source at tha explosion centre. 

A simple formula, equations (ll) or (12), then enables the displccenent of the bubble In the first 
oscillation to be calculate. These displacements arc almost Independent of depth. Actable of 
values of the coefficient ^ In equations (u) and (12) Is appended for convenience. 

0 X 

Xe.fe renees , 

(\) Theory of the Pulsations of the Gas Bubble formed by an Underwater exploslpn. 

Report No. C4-sr20-0l0, 

(2) The Motion end Shape of the Hollow Produced by an Explosion In a Liquid. 

G.l. Taylor and R.M. Davies. 


-,:l 
' t 
,i 

: i 

V 


(3) cf. "The motion and shape of the hollow produced by an explosion in a liquid", 
G.l. Taylor and R.M. Davies, where the bubble Is represented by a point source 
and a dipole and the effect of the Images due to each is considered, 

(4) This result is also given by Savic. 



(5) Calculated from Savie's equation for the potential due to the Images by 
differentiating with respect to d his expression for W, 



43i 


Nature of Surface 


Rigid Infinite Plane 


Rigid Sphere (4) 


Rigid Sphere with centre 
In infinite Free surface 


Rigid Infinite Cy1 inder wi th 
axis in Infinite Frve 
Surface (s) 


Oinenslons 


Radius R 


. Rad I us R 


Rigid Infinite Cylinder {3} | Radius R 


Distance of Bubble Centre 

at bubble 
° " centre 

d 

from bubble to piene 


d 

bubble to sphere centre 

»3 

d (d*-R*i* 

6 

bubble to sphere centre 

1 2r 3 fd"fR‘*) 

4? (d'*-R‘*)*d 


bubble to cylinder axis 


Radius R I bubble to cylinder axis 





























THE RATE OF RISE OF LARGE VOLUMES OF GAS IN WATER 


H. N. V. Temperley and LL G. Chambers 
Admiralty Undex Works, Rosyth, Scotland 


British Contribution 


January 1945 




m 


THE RATE OF RISE OF URQE VOLUMES OF GAS IN WATER 


H. N. V. T8Bp*fl*y Q» Ch«Bbgf« 

January 1948 

• **•••* 


Svmmaty . 

won. OMCrlbert by Taylor and tavl.s (i) Is oKter^sd to voU.«d 0^ (!« “L 

9 . 11 ons. m. necessary large voUnes of gas obtained by overturn ng a buc.at 
by * dune, eharaes of burning cordite. The resulting bubbles were pwtognsph»0 "> the glase-tronted 
tanks of tb. M.A.E. Establlshnwnt at Olen Fruln. The relation U - */, '' «% 
radius, found experimentally and thanretleally by Taylor and Bavles. was * ’ 

cap radi! of up to iB cm., and It was concluded that .such bubbles are near the limit of stability. 

The work appears to be In general sgraeitent with existing knowledge of the behaviour ©f the 
gaaeoua products from vory deep explosions. It Is concluded that the bubbles from such oxplwlons 
break up Into comparatively small fragments, once the oscillation has ceaaad. 

An attempt Is made to Improve the theoretical solution of the hydro-dynamic equations found 
by Taylor and oavles, but It appears that the convergence of the method Is rsther slow. 


Tntroduetion , 

The work described in this report was carried out In contlnuatton of that by Taylor at^ 

Devlas (il. The largest volume of gas used by them was of the order of loo cc. and It w<» ™sl.^ 
to extend the work to nweh larger volumes of gas to gain a mors complete idea of wMt Wppens In the 
fln^f liero^sn underwater explosion. Thus, u* volume of gaseous products release ona gram 
of Amatol (neglecting water) Is given by Taylor and Davies as 650 ec. Thus we conclude that an 
ounce of explosive will liberate roughly four gallons of gaseous products at atirospherle pressu ^ 

,n order t» deduce what happens during the final stages of an underwater ^Ihe 

Whether such large volumes of gas cah exist as coherent bubbles, br, If not, what Is the voluno o 
largest bubble that can exist and rise through the water without breaking up. 


Hethods oi carrying out t he Tr.i<tl. 

The relative fragility of the glass windows in the Glen Fruln tanks precluded the use of any 
fom of high explosives and therefore two alternative methods were employed. 


(a) 


W 


he method Oescrtbod by Taylor and Davies (l) was tried on a larger scale. a bucxet 
capacity x gallons) was lewcrsd upside down to the bottom of the tank, a sinker being 
ttached to the handle. A small out-of-balance weight (4 lbs.) was attached to the end 
• B Short lever pivotod at the bottom of the bucket. While the bucket was being lowered 
his lever was supported by a string, and the bucket remained Inverted with the air trapped 
nderneath it. When the string was released the lever fell to o horliontal position and 
he out-of-ba.,-.nce weight tipped the bucket upwards 'emptying out- the air. 


A ounce burning cordite charges were Ignited electrically by means of a -puffer*. Twp 
types of conxalner were used, a -pill-box* type of bakelite container which broke easily 
when the charge was fired, and a stout brass cartridge-case the dnly exit from which was 
a hole of i" diameter. 


The 




- 2 - 


Ths resulting bubbles were pbotngraphed by cine-ciuners (i6 frames/second nonilnal spued) 
distances being determined by comparison with the edges of the panes of glass (ii feet), and the mesh 
of the protective wire-netting inside the tank (3 inches). The tiiiB-scate was established by 
photographing dials driven by a phonic motor controlled by a tuning fork. 


3" Descriiiion of Bubbles , 


The out-of-ba 1 «nco couple on the bucket had been adjusted in preliminary experiments near tbs 
surface of the water, so that all tne air was ‘poured out‘ simultaneously. If the couple Is too 
small, the air escapes in the form of si.'^ll bubbles, while If it Is too large the bucket goes beyond 
the vertical before all the air can escape, when the weight was properly adjusted, the air ‘poured out* 
caused a considerable upheaval and ‘whitening* of the water, similar to the ‘plume* from a deep 
explosion. when the bucket was 10 feet or more below the water surface, It was found that the air 
cane up as about 4-10 large bubbles and a large number of smaller ones. The disturbsnee produced 
at the surface was very slight, it was concluded that the original large bubble had broken up. 
when the experiments proper were begun iii the glass-fronted tank It was fourd that the large bubble 
broke up alaiast Immediately into a small number of large bubbles and a multitude of very snail ones. 

All the large bubbles were of tne characteristic *mu$hrocm‘ shape descrlhed by Taylor and Davies (1) 
and rose at speeds of the order of 2 feet per second, sonie were observed to break up as they floated 
upwards and it was also thought that some wore colliding and coalescing, but the latter Impression 
provided, on examination of the films, to be a mistake. Two'bubbles sometimes settled down into an 
apparently stable configuration, one below tho other, but with axes of synmetry offset, and rose 
together, go larger complexes were seen. 


The bubbles produced by the burning cordite were very similar to the air-bubbles, and the 
general sequenca of events seemed to be very much the same. it did. not appear to make any difference 
to the general nature uf the phenomena whether the cordite was fired in the esslly breakable pill-box 
or the strong cartridge-case, indeed, one of the largest bubbles obtained In the trial was obtained 
in the latter cl rcums'tances, which one would think were the most unfavourable. Soms close-up 
photographs ware taken in the smaller tank at Gian Frutn in order to get the earlier stages of bubble 
furnetlon in more detail, but the earlier stages were obscured by what appeared io be a cloud of small 
bubbles. A few photographs were taken from above of a bubble 'breaking surface* but exhibited no 
feature of Interest. The surface phenunena ware confined to slight ripples (Figure i(b^). 

Two features noticed by Taylor and Davies (1) were apparent In the records. First, the bubbles 
are often ‘lop-sided,* i.e. not perfectly symnetrlcsl about an axis, the ‘lopsidedness*, when It exists, 
often persisting throughout the life of the bubble; secondly, in many of the records, there Is an 
Indication of a wake lying approximately tn the sphere of which the bubole is a cap, similar to the 
effect shown by Taylor and Davies (1) In Figure 1 of their report. In our case the existence of a 
wake la augg«st«d by clouds of small bubbles following each large one (Figure 1(a)). 

4, Analysis of the Records . 


The trajsclorles of 31 bubbles of various sizes were plotted, within the errors of the 
experiment the velocities were all constant In time. In spite of the fact that the depth of water was 
33 feet so that the bubbles must have expanded to twice their volume during 'the rise. A few of the 

• «ajwv*vt witvarwhi 1 itw t-uo • s wis« v’ « vovvtsavsvit i»w ana 1 1 ww iiigciaiiivi ouvwb o imau 79 twl%r* 

but there was no definite upMird or downward curvature. The Images of the bubbles were somewhat 
blurred, but in some cases it was possible to esiiiiBi e th e radius of curvaiura of the upper cap of 
the bubble, in order to cheek up the relation u • found by Taylor and Davies (1) for their 

snail bubbles. The blurring of our Images niay be due to the condensation mist on the glass panes of 
the tank, which was difficult to remove completely. 


The radius of curvature was measured by projecting a magnified imago of the bubble on to 
squaried paper and estimating tho radius from the length of the chord and tlv» distance from the chord 
to the top of the bubble, owing to the blurring it was not passible to say how nearly the bubbles 
are to parts of spheres, but It appeared that the departure was not great, and that the astimetes of 
thu radii should be accurate to 2Ct. ins radius was measured at savaral dlff$r3rit points on €aCh 


trajectory 


- 3 - 


439 


trajectory and a mean taken. The results are plotted In Figure 2, together with Taylor and Davies' 
points. In one Tilm the time-scale was uncertain, and the points obtained from It are shown 
separately on an arbitrary velocity scale. They have beari Included because one of these points was 
from a double bubble of the kind described above. The results Indicate that Its upward velocity Is 
definitely less then that of a single one of the same mean radius (Figure J). 

Leaving this doubtful film out of account It will be seen that the renalning it points agree 
with Taylot and Davies' relation within the experimental error. Taylor and Davies' values for the 
radii of the caps of the bubbles extended from i cm. to a cm., while ours extend from li cm. to i5 cm. 

The question of the stability of bubbles of various sizes was examined by measuring the velocities of 
g bubbles that ultimately split up, but no correlation between velocity and stability could be traced, 
the bubbles appearing to split at all velocities in the range studied. This seems to indicate that 
we are near the limit of stability, which Is perhaps fixed by the relative Importance of surface 
tension snd hydrodynamical forces. By tending to keep the surface email, surface tension would act 
as a stabilising Influence, but would become loss Important for larger bubbles. Taylor and Davies 

(l) mention that conditions had to be adjusted carefully to obtain their bubbles. This agrees with 

what we have found with our rather larger ones. 

If this Interpretstion of the results Is correct, it would seem that the final upward velocity 
of the products of an explosion, after the oscillations have ceased, snd the bubble has broken up into 
sitiali ones. Is of the order of 2 - j feet per second, compared with ti.2 feet per second Inferred by 

Taylor and Davies for a goo 1b. charge on the assumption that the explosion products remain’as one 

bubble. 

It Is perhaps worth mentioning that the rate of rise of bubbles of exhaust gas from a torpedo 
Is known to be of the order of i feet per second (again of the sama order of magnitude). 

Theoretical Const derations 4 

although it Is clear from Taylorand Davies* and our photographs that these bubbles have a 
wake, the niotton of the remainder of the water is probagly Irrotational and It Is of interest to 
examine whether there are any solutions of the hydrodynamical equations which enable the velocity 
and pressure conditions to be satisfied along a cap of a sphere of limited angle. A start on this 
problem Is made by Taylor and Davies (l), who show that the ordinary solution for a sphere In a 
uniform stream satisfies the conditions for continuity of pressure as well as velocity as far as terms 
Involving (where 6 Is the polar angle referred to an axis through the cap of the bubble) provided 
that veldclty and radius are connected by the relation u * */-v^ which has been conflrmad 
experimental ly. if we take a mors general velocity potential of the tons , 

♦ u r cos e. 

r r* 

ropresenting a combined source and dipole in a uniform stream. It is possible to satisfy the continuity 
conditions as far as terms Involving 5“, provided that we assume that the profile of the cap Is of the 
fens R ■ a ♦ bj ♦ b^*. (it Is unnecessary to Introduce odd powers of ff). Alternatively, if we 
asstime a, zero, so that the bubble behaves as a simple source rether than as a dipole source, we can 
still satisfy the equations as far as terms of tha order The various solutions are tabuiate.! 
in Table i. 



A„/Ua* 

Aj/Ua^ 

bj/a 

b^/a 

uA^ga 


(1) 

0 

.50 

0 

0 

.67 

.67 

(a) 

1.00 

A 

V 

* .126 

0 

1.00 

.87 

(y) 

- i.2j 

* i.lz 

- eOT3 

- .019 

*00 

• 54 


Taylor and Davies 


source In stream 


The , 



440 


-li- 
the lest column is obtained by putting in the radius of eurvat-jre of tha cap at its top point instead 
of a. By the usnnl formula we have 


a 



a 


It is evident that the relation between the velocity and the radius of the cap Is sensitive to slight 
departures from the spherical shape of the cap. (it Is known that the virtual mass of an ellipsoid 
of given volume is sensitive to the exact shape of the ellipsoid). It would therefore seem that 
one would have to go to high powers of 6 , and introduce a corresponolngly large number of spherical 
harmonies in the velocity potential In order to determine the exact theoretical shape. it Is 
probable that one ould obtain a multiplicity of solutions, and one would then have to determine 
which are stable by assuming small perturbations. Asa matter of fact, substitution of the apparently 
better solution ( 3 ) ror (tj would worsen me agreement with experiment. 

we have introduced a certain degree of arbitrariness by the fact that the origin of 
coordinates is unspecified. The same profile might be specified in quite different ways (as a 
function of 6 ) for different choices of origin. it will, In fact, be noticed that the discrepancies 
between the various values of U^'gp are much smaller than those between the various values of UA'^ 
so that It may be that the three solutions we have found are ail really first approximations to the 
same actual profile. These points seem well worth further Investigation, but it has not been 
thought advisable to hold up the issue of the report as the investigation would be lengthy. The 
profiles corresponding to the various solutions are plotted in Figure u, it might be thought that 
it would be an easy nvntter to decide experimentally between them, but even curves 2 and 3 can be 
Drought nearly into coincidence over tha relevant range of angles (about 60®) by shifting the origin of 
co-ordinates, and choosing the scale so that the radii of curvature of the caps ara the same. 

6 . Conclusions, 

Taylor and aviec formula relating upivard velocity and radius has been conflnned for radii 
up to iS cm,, which is probably near the limit of stability. The available evidence suggests that, 
after an underwater explosion bubble has ceased oscil lat ing, it splits up into many small bubbles of 
about this order of size. 

Reference, 


(t) The rate of rise of large volumes of gas in water, Taylor and Davies. 










! V : ' 

Asvac.ias.f raj R'. R _ im. ■ . , 

^S.-- M" , UHK>«OV,jN_^ UWfTaj. 

)S> THf. W'Jtc. fiv S>r '4T »■ (V Tti Th e J’JrMtQl*? ■ 

t»o>HT Mftfn'ir.o OMij! ;=?>oi-<i'i;j TO ft c«up...ir. tJupe^usi. 





TKH attraction of an underwater explosion 
bubble to a rigid disc 


A. R. Bryant 

Road Research Laboratory, London 


British Contribution 


February 1845 


THE ATTRACTION OF AN UNDERWATER EXPLOSION 
BUBBLE TO A RIGID DISC 


A . R. Bryan t 
February. 194S . 


Summar y, 

The attraction of an explosion PuPble towards a rigid disc has been calculated for two 
important cases, via. (l) the disc fixed, (r) the disc moving along the llvie johnlng disc and explosion 
centre. The case in which only part of the disc Is noving has been treated in an Appendix, 

For the fixed disc the attraction falls off more rapidly with distance than the attraction of 
an Infinite rigid plane. At one disc radius the attraction is one half that of the Infinite rigid 
plane at the same distance. At one disc diameter the attraction is one seventh that of the Infinite 
rigid plane. 

The attraction of the moving disc d.epends on both its velocity and acceleration. That part 
of the attraction which is due to Its motion falls oft more rapidly with increasing distance than the 
part due to Its rigidity. 

It is suggested that the attraction of a bubble to a rigid disc is a reasonable approximation 
to the practical case of a target like a Sox Model. It is pointed out that the motion of the Box Model 
as a whole due to the explosion pressures may have an appreciable influence on the displacement of the 
bubble, 

Introduetion ., 

In considering the damage to finite targets caused by underwater explosions It is desirable to 
calculate, at least approximately, the displacement of the bubble towards or away from the target. 

So far the only finite rigid surface whose attraction has been calculated is the sphere. For targets 
like the O.h.C. box model, or the ■drum" model used in the U.S.A., where a target plate is surrounded 
by a rigid “skirt* or baffle of finite extent. It is suggested ttat the attraction of an explosion 
bubble to a rigid disc would be a bettor .ipproxinot ion to the experiment .cl conditions than the 
attraction to a sphere. The problem of a rigid disc is treated in the following note. 

The following assumptions have been mede:- 

(1) The bubble rc.mains spherical thre-ughout its .rot ion, 

(2) The naximum radius of the bubble is small compared to Its distance from the disc. 

(3) The velocity of displacement of the bubble Is small and Its contribution to the 
attractive forces is neglected. 

The Attraction of an lixtlosion Bubble to a :Hxed Ri^id SH.se . 

The method of calculating the attraction of a fixed rigid disc is as follows. A point source 
of unit strength is placed at tho explosion centre. A potential (Pg is found which, when added to the 
potential 0 ^ due to the point source, gives the correct boundary condition over the surface of the 
disc. This potential may De regarded as due to image sources "induced" by the original point source. 
The required “attraction coofficienf is then the value at tho explosion centre of the space gradient 

' ^ ^ positive in a direction away from the disc along the axis of syirmetry. It was shown 

in an earlier paper* that the velocity of the bubble towards the surface at any time is the product 

of two 

•A Simplified theory of the Effect of Surfaces on the Motion of an Explosion Bubble." 


446 


- 2 - 


of two factors, one the “attraction coefficient’.^^, which is a geometrical factor, and the other 

an integral expression involving the radial motion of the budhie and independent of the goometry of 
tho surfaces. 

Th e Velocity Pc-tential Equations , 

The most suitable co-ordinates in which to solve the problem are oblate spheroidal 
cc-ordlnates r, s, v/j, as shown In Figure l. tB ts the disc, radius c, and a, B, are the foci of 
the confocal oblato spheroids, r = constant and the hyperboloids of one sheet s • constant. OP is 
the axis of syirnietry, and the explosion centre is at x on this axis, dlstanco d frosi the disc. 't 
is convenient to choose o< r^'”. so that s varies between - i and'-i 1, This choice makes the $ 
co-ordliate continuous in the region of the field. It is to be noted that s changes sign on passing 
through the disc. 

Tho relationship between tho spheroidal co-ordinates r, s, and cylindrical co-ordinates x, fi, 
origin at o and the x co-ordinate positvo in the direction of OX, is 


c r s 

(1) 

c [(IV r2)(l- s*)]* 

(») 


Since the problem Is entirely synmetrlcal wltn regard to</> It will not appear in the equations. 

The disc *B Is thus the surface r » o. Denoting the potential due to the point charge at 
X (r « rjj, s = 1, i/* “ Dy 4>^ we have* 

‘ Jo '"o’ ^ ^ 

Where and <j^ are the Legendre functions of the first and second kind, and i » /-l. 

To this must be added a potential Il’ich satisfies Laplace's equation, vanishes at Infinity, 
and has no singularities In the region of the field. Thus 

* 4 

The potential 0 « has to satisfy the condition that at the disc, l,e, the surface r • o. 

Its gradient normal to the surface is zero, since the disc Is both rigid and fixed. This gives an 
equation from which the coefficients may be determlnad. Thus 

4 ^*n ”n^ * r ^ ’n ’’n^ ’’n ® 


whence 


An -■ (2n + 1) 0^ (ir^l 


n odd 


(it ts 


Static and Dynamic Electricity. Smythe, MeSraw Hill Wt, p.l65. It me/, be verified that 
the potentials given in (Ja) and (Jb) sa.tisfy Laplace's equation In those co-ordinates, and 
reduce to Heine's expansion of ffp-rpy, or respectively, along the axis of syimetry 

s ■ 1 (see Whittaker and Watson, sth Edition p.3Z2). 






(it Is to be noted that the function 0„ bos a branch line a'oris the real axis between -1 
and +1, and that the value of 0^^ (ir) at r • o is really tho limit of (l5), as 8 tends to ;ero 
through positive values). 


The desired potential due to the •Induced* images is therefore 

<^2 ’ 7f^ J " 3) * l<"-o) 9 ^ * 


The "attraction coefficient" of 
at the explosion centra X, 


the disc Is *^2 = T ^^2 at s = 1 and r ■ r • I, t.c. 
r c 'ir—— 0 c' 

ax dr 


I Jat X 77 c* 


2 (im ♦ 3) g,_ ♦ l*'''o' oL t^'''o* 


Equation (7) may be put In finite formas follows, Heine's development of |/t-z Is: 
(Whittaker and Watson, p.32l) 

^ • f (2n + 1) P. (z) 0 (t) (a) 

‘-r pao " n 

The series In (8) Is valid for all points z lying Inside an ellipse In thn complex plane 
passing through t and with foci at ±l. Cfianging the sign of z throughout and subtracting this new 
expression from (8) gives 

f. («m + 3) Pj. ♦ Q. * ■ 

nmo ^ (t^ - z^l 

Dividing both sides by -2 (t - z)^, and Integrating both sides with respect to z from -1 
to +1, yields 


J (4m e 3) Oa„f «-4 /‘ 


J (t*- z*)(t - z)2 


I , A - A 1 

-p los^rr-ij- w (i- t^)^' 


Since the reversal of the order of integrating and sunning may be justified for all values of 
t not on thp branch line of the g functions, i.e. the portion of the real axis between -1 and +1. 

Pinally, replacing t In (») by (ir^), and taking only the principal value of the complex 
logarithm In (9) as necessitated by the bourdary condition at the surface of the disc, equation (?) 
far ths ‘attraction coefficieni" of the fixeo rigid disc becomes 


I / - 

s I > 


3 X y at X 77 c*i 2r ^ 


l!r„ (1 ♦ r^2) 


It nay be verified that as d tends to zero, the attraction coefficient in (to) tends to -1/ud^ 
the attreption coefficient for an infinite rigid plane, kecapltulat ing the result obtained in 
koto A0M/210/ARB, the velocity U of the bubble towards the disc I» therefore 




Where a is the radius cf the bubble at tfme t, and 3<^/3 x is given by (to). 


Comparison 


The extension of the foregoing to the cese of a rigid disc moving with velocity v along the axis 
of ayimatry, I.e. towards or atvay from the explosion centre, will now be node. The co-ordinate system 
Is fixed In space and the surface r » o Is made coincident with the disc at the Instant considered. 

The velocity potential due to a disc of radius a moving along Its axis with velocity v awrv from 
the explosion centre at r^ “ d/c Is 

<#13 ‘ ~ (Ir) s (12) 

It this Is added la <(> • tp^ * 'I*® complete velocity potential for a moving rigid disc and a unit point 

source Is obtained. Since this added potential is Independent of the strength of the source the 
analysis given In R.R.l. Note aDn/210/aRB must be modified slightly. Pressure gradients and velocities 

along the axis of symmetry will be additive so that the drift velocity or displacement of the bubble 
due to the motion of the disc may ba calculated separately and added to that due to a fixed rigid disc - 
equation (it). 

Poliowing the physical arguments of the above mentioned paper the effect of this added velocity 
potential is two fold. First, th« motion of the disc imparls a drift velocity 3(^/5 x towards the 
disc to the water in the neighbourhood of the explosion centre. Second, a pressure gradient 



is set up In the water in the neighbourhood of the explosion centre and this gives the bubble a drift 
velocity towards the disc 


2 





dt 


Adding these two drift velocities* separating out the geometrical factor, and inserting the value of 
^ from (12) gives for the drift velocity of the huOOle towards the disc due to the disc's motion . 




V * dt 

- TT - 0 77 1 ^ 


V ♦ -j J a — tu 

a 0 


(15) 


wnere * u/v * 

since Ih . jLl±k along the line s ~ 1, i.e. along the axis of symmetry. 
3 X p® 3 r 


For convenience of use the geometrical factor In the first bracket in (l3) has been plotted 
in Figure 3. It will be seen that the Influence of the motion of the disc falls off more rapidly with 
increasing distance than the attraction due to the disc's rigidity. 

If the acceleration dv/dl is a constant Independent of time It may be removed from the integral 
in (i3). This integral Is now the same as that occurring in tho equations of motion of the bubble 
under the Influence of gravity and in the absence of surfaces; in fact the velocity and displacement 
of the bubble due to the acceleration of the disc arc a constant fraction k of the velocity and 
displacement of the bubble under gravity, where 


k 




a dv 
9 3^ 


- 6 - 


449 


a is the factor In the first hracket in (13) which has been plotted In Figure 3. 

Application to Pox Modal E xperiments . 

In Box Model and similar experiments where the target is slung in such a way that It is free 
to move as a whole under the influence of the explosion pressures acting on it, the effect of such 
movement on the dis liacerrwnt of the bubble rMy be estimated by tirjans of equation (13). fhe following 
approximate numerical example suggests that this effect may be of real Importanc.'. 

The area of the target plate and ’skirt' of the R.R.L. box model Is 5 square feet, i.e. for 
the purposes of using equation (l3) the equivalent disc would have a radius of 1.25 feet. Consider 
the case of a l oz. charge, 3 feet deep, fired at a distance of 1.5 feet from the target plate. It 
will be supposed that the effect of the explosion pressure.s on the box model as a whole is to give It 
an Initial velocity away from the explosion centre- which Is rapidly reduced to zero by the drag forces 
In the water. 

It Is not practicable with present knowledge to calculate the initial velocity and the deceleration, 
but In order to estimate the importance of the effect arbitrary values will be assumed. Let it be 
assumed that the box model Is brought to rest by the drag forces with a uniform deceleration of bg, and 
that It comes to rest In 50 milliseconds. This corresponds to an initial velocity of 5,4 feet/second 
and a total displacement of the box of l.» inches. These two latter figures do not seem unreasonable 
for a box model slung at the same horizontal level as the charge. 


The displacement of the bubble due to the motion of the box model alone nay be calculated from 
(13). The overall displacement is away from the box and is found to be aPprox irately one half the 
rise of the bubble under the influence of gravity at the end of the first oscillation, I.e. It Is 
displaced about 6 inches. 

Strictly speaking equation (X3) only holds for cases where the maximum bubble radius Is small 
compared to the distance from the disc, end this condition is considerably exceeded in the numerical 
example abeve. The example does, however, suggest the d.-.-sirability, either of measuring the overall 
movement of the box model so that this effect ray be estimated, or of fixing the box radei so rigidly 
that the effect becomes negligible. 


(i) The attraction of a fixed rigid disc for an explosion bubble has beer calculated and is founo 
to decrease rapidly with distance. If the explosion centre is one radius distance from the 
disc the attraction is one half that of an infinite rigid wail at the same distance. At a 
distance of ono disc diameter the attraction is only one seventh of that duo to an Infinite 
wall at the same distance. 

(2; The attraction of a rigid disc moving towards or .away from the explosion centre has been 

calculated. The effects due to the motion of the disc fall off with increasing distance more 
rapic^ly than th^ attraction from a stationary r}oi<f Bot.h thg and thf» 

acceleration of the disc give terms in the equation for the velocity of the explosion bubble 
towards the disc, 

(3) It is pointed out that in experiments with box modqls or similar targets with flat plates 
aurroundoi ny rigid flanges the foregoing analysis Is relevant. In particular the motion 
of the box model as a whole due to the explosion ray have an appreciable influence on the 
displacement of the explosion bubble. 


Appendix 


450 


- 6 - 
APPENDIX. 

T he A {tract ion of a fHind Oise of uhich Part is Hovin/f , 

The attraction of a rigid disr of which the centre portion moves with a given velocity while the 
outer annulus reowin';, fixed IsS of seme practical Inttr.st as boing rather like the case of a fixed rigid 
box model with a ri: l(3 "yklrt" in which th'j t-irg-.t platos moves as a result of the pressure in the water. 
This case will now be oniw,.M 3 fornully, thought it has not been found possible to reduce the solution to 
a simple finite form suitabl'* for computation. TN; case wh».rc the rigid ■skirt" extends to infinity has 
been treated by TempcrlGy. 

Let the velocity point of the disc be v. Only the case of radial symnetry Is considered 

and over the surface of the disc r » o, the other spheroidal co-ordinaio s Is « function only of the 
radial distance p of the point from the axis. The r. latign |$ 

o = c [» - * 

Henco if V is a given function of p it may be written a function of s and, with the usual restrictions 
on the form of the function, nay be expanded In a series as follows. 

fO 

sv » s1(s) = X a (• (s) (in) 

n=o " ■' 

where a^ = ^ t (15) 


The putential ^ Tils tee boundary condUion that the normal velocity over the surface r = o 
is everywhere equai to v is 


<h 


-ic 


2 


(16) 


This potential is to oe added to <p » + ttj to qive the complete potential for the disc and the 

unit point source. The "attraction coefficient" in equation (13), i.a. the term In the first bracket, 
is replaced in this case by 


= i Lt') 

3x/atX c Bryatrv d/c = r^, 


oo 

2 


n=o 



S = 1 


(17) 


The special case of a piston of radius R moving with velocity v in a fixed finite circular baffle 
of radius c is given by 


f(s) 


\ < ! 5 I < 1 

0 < I S I < \ 


f-) 


where X 

In this case the coefficients a to be inserted in equation (l7) are 


a„ ■= -V] 


• (l - X-')v 

'|2S4-r'’n * ^ * Pn*=TmrT^ ''n " 


T W= ' l) '’n ■ ^ 


') 


for n “ 3, 5, 7, 


1 


(18) 


etc. 




lATIO OF THIE ATTRACTION COEFFICIENT OF A FIXED RIGID DISC TO THAT OF AN 
INFINITE RIGilO PLANE AT THE SAME DISTANCE 






DISC MOVING ALONG ITS AXIS 



THE EFFECT OF AN ADJACENT DEFORMING TARGET 
UPON THE BUBBI,E DUE TO A SUBMARINE EXPLOSION 


LL G. Chambers 

Admiralty Undex Works, Rosyth, Scotland 


British Contribution 


November 1946 






THE EFFECT OF AN ADJACENT DEFORMING TARGET 
UPON THE DUBBI.E DUE TO A SUBMARINE EXPLOSION 

LI. G, Chanfoera 

November 1S46 


■Stjaiaarv . 

Reference Is irade to previous aork on the subject. An alternative theory is put forward 
to account for discrepancies which have arisen in Box Model work and calculations are descrloed 
based on certain Box Model shots and discussed. it is concluded that the effect of the rigidity 
of the target on the bubble Is small and may. In general, be neglected, 

notation and Symbols us$d. 


The suffix 0 indicates values at t • 0. The quantities are non-dimensional, being defined 
in terms of the units introduced by G I. Taylor in the report "Vertical notion of a spherical 
bubble and the pressure surrounding It" 

a ■ radius of bubble, assumed to remain spherical, 

b ■■ equivalent radius of plate, 

c n an explosive parameter, 

d “ depth of centre of bubble below target, 
h • central deflection of target, 

K * a constant of integration, chosen to satisfy the initial conditions, 
t o time. 

u * velocity of bubble 

V ■ upward velocity of centre of plate. 

i « depth of bubble below virtual surface. 


It Is well Known that the mechanism of damage due to an underwater explosion is extremely 
complex, even in the comparatively simple case of a single plate rigidly clamped at its edges, and 
that the phenomena involved are many - for example, the elastic properties of the steel, the 
occurrence of cavitation either at or away from the steel-water interface, and lastly the effect 
of the motion of the gas bubble, it is with this last phenomenon that this report is mainly 
concerned. 

It has been shown by Conyers Herring and by others, that the bubble Is liable to migrate, 
the nature of this migratio' depending on the nature of aujacent surfaces. On ths whole, the 
Dubblo is attracted by a rigid surface and repelled by a free surface, although there Is initially 
a repulsion away from a rigid surface. Thus, for a given charge and distance, there nay be a 
particular strength of plate such that the bubble migration is negilgible during the very 
Important first oscillation of the bubble. 

If the plate was weaker than this, the net effect would be a repulsion, so that the first 
mininum of the buPble would occur farther from the plate than the original explosion. if the 
plate is stronger, the net effect would be ae attraction, which inureases, rapidly as the distance 
from the plate decreases, so that the first minimum of the bubble would occur very near to, or 
even in contact with, the plate, and might thus give a large contribution to damage, while the 
contribution in the repulsive Case would be negligible. 

It was found that in certain Box Model snots inconsistencies arose and this was thought 
to be due to some Intrinsic instability in the dependence of the bubble movement on the movement 
of the plate. Thus in considering the attraction of the bubble towards the target, it Is 
desirable to know whfthnr the notion of the target plate in the battle effects the bubble or 
whether we can assume with sufficient accuracy the target plate to be rigid. 


A, 


458 


- 2 - 


A difficult)' Is encountered as water is conyiressleie. This question was attacked b).' 
first considering the plate to start at rest and secondly by considering the plate to be given an 
impulsive velocity equivalent to the experimentally measured impulse per unit area due to it lb, 
of T.n.t, In this case the motion of the Box Kcdel plate was subject to the following 
asstmptions!- 

!a) The ’skirt* of the Box Model was assumed to be an infinite wall, 

(b) The tension of the plate, during the- process of stretching, was assumed to be 

constant at the yield point, the plate being reprcst-nteu as a piston backed by a 
spring closing hole In an Infinite wall. This reprasentat ion was deemed to 
hold after the stretching when elastic recovery makes the plate nnve out again. 

It was concluded by Temperley that a defornable structure may act either as a free surface 
or as a rigid surface, and also that the nature of the effect may be reversed during the interval 
of the cxpsriinent. The valid objection has been made against this treatment that effectively 
the whole infinite baffle is moving. An altornative derivation of the equations of motion, due 
to Temperley is presented hero, together with a discussion of the results of Integrating the 
equations in two cases. 

TViec ry. 


The target is treated as a fixed rigid plate and the fnotion of the target plate is allowed 

i • ran -w 4 < 4 •> A <e Aia^/*a uihAAA ^ AAAf K • « f> A t K« f i Ka # 1 auf 4 

a hemisphere with centre at the source is equal to the volume actually swept out by the target. 

The conditions of continuity of pressure and velocity -at the gas-water interface give, 
on equating terms independent of cos 6?, and terms in cos 0, respectively. 



15 , 

a“ X - z V a8 t 1 

i2 1 .,2 . a^ a + 2 a a* 
“ ■ H “ + 33“ 

(i) 


Si e uiisf 

0 d^ 2d dt ud' 

and 

2a » 3 aU 

♦ ab' ♦ — sisL ♦ 3 

ltd' * P 

(2) 

3 0* 

■ rp 

(a V * a V) 


where Taylor's units are used throughout. 


To aid in the computation we form a pseudo-energy equation. This is derived by 
multiplying (l) by it ir a *a, and (2) by~7ra^y, suostracting and integrating the resulting 
equation with respect to time. We obtain 


(a) 




dt ' k 



We nay note In passing that (a) presents several interesting features. It Is, in fact, 
similar to equation (2) of the report 'Vertical motion of a spherical bubble and the pressure 
surrounding If. The radial kinetic energy is increfised by a factor (l ♦ -^) due to the rigid 
surface as pointed out by Conyers Herring, The cross product a i does not occur In equation (2) 
of above mentioned report which is a true energy integral, k i$ not equal to unity, being 
analogous to the total energy of tho system, vmich Includes an unknown amount of energy In tho 
moving target plate system. 


Putting 




_ 3 .. 


45 ’? 


i 



Putting (2) in the form 

2a3 . _ 2_ (a^j) + (a^a) - 2s!|f 2- (aV) 

^ (It 4d* dt 20“' dt 

ws notu that putting the left hand side of (/5) equal to zero gives the case of zero gravity. 

Also we have 

2. (d - z) • V. 
dt 

This assufnes that d is measured from the centre of the bubole to the centre ot gravity 
target plate. it might be more consistent vsith the other assumptions to measure d from the 
initial position of the plate, in which case ^ (d - z) = 0. The values of d obtained frem the 
two assumptions only differ by /* v dt which is never more than a few inches, and since the 
other two equations only involve d itself, and not Its derivatives, the difference between the 
two assumptions is small. 


Experimental Data . 

fortunately there became available records of the Sox Nodel shots detailed below. 

The diagramnat ic set>up is as shown in figure ), in the actual calculations the set-up was 
simplifie(3 io iruj of Figure '* »«s assumed that the mutual effects of the target and 
bubble would be the same whether the target was above the charge or to its side. the conoieions 
of the experiments were;- 


Test 48 ; Depth of charge 
Stand-off 
Charge 


■■ 5 feet. 

* 2 feet. 

= 1 OZ-, of T.li.T. 


Tost 50 ; Depth of Charge 
Stand-off 
Charge 


=■ 5 feet. 

• 2i feet. 

• 1 oz. of T.n.T. 


In both cases the equivalent radius of the plate was S.25 inches, for this charge the 
Taylor unit of length is S feet, and the Taylor unit of time is 0.394 seconds. 


Oescription of Tables , 

Trie inti.'grat ion of the equations derivizd in paragraph 3 were carried out under the 
tellowing conditions:- 


0 . 01 « 

7.6 


( 8*1 ■ • 


.063 

.133 

1.4006 


There were four cases. 
Table 1: d„ 


.40 a) V = 0 

jSj V from Test 48 


Tabls 2: d^^ » .50 a) v » 0 


/?) V from Test 60 


The valuus of v used In cases v3 and ^ were derived from smoothed graphs of the deflection 
time curves. 



458 


- u - 


For the purpose o'f cunputation, a variable x was Introduced, defined by:- 


X 



dt 


!n fact, in the Tables the quantities involved are defined in terms of equal intervals of 
X, which obviously increases continuously with values of t, although the corresponding values of 
t are given. The computations were carried out to the point where the equations ceased to have 
any physical meaning owing to tr.e radius of the bubble becoming greater than the distance from 
the model. (In the case of 'test 50, this happened just before the first mlnimuiri of the bubble 
and the equations would have broken down In any case). As regards the accuracy of the so'utionsi 
the following should ha noted:- 

(a) r. warning must be given about the interpretation of the computca figures, 
in some cases mare figures than are justified have been retained in order 
to keep a fixed number of rteciiiBls. The computer is fully aware of this 
and precautions have been taken accordi.agly. 

(b) It is realised that U decimals in a and z are meaningless but It is 
essential to keep this number of figures in order to comprehend the full 
behaviour of the solution and to understand the structure of the differential 
equations. No great pains have Peen taken to maintain this accuracy, though 
owing to the inherent stability of the equation tor a, the last figure should 

be reasonably youd, evert to the c.sd. 

The tables give - for the rigid case a, t, z, and - for the case where there assumed to 
be a source a, t, z, h, v. 


Pi scussion of Jfasults , 

It will be seen by reference to Tables t and 2 and Figures 1 and 2 that the actual effect 
of the plate motion on tho Oubole appears to be fairly small. in fact trm Figure 1 It appears 
tiiat the effect of the mobility of the piste has not appreciably affected the depth of the centre 
of the bubble. This may be due to the fact that over the greater part of the period under 

consideration, i.o. from t = .02 to t » .17, the motion of the plate is comparatively snail, the 
bubble being fairly large over this time. As Is well known, the bubble acquires momentum, chiefly 
when large, and at this time the plate Is practically stationary. The motion of the plate has 
little effect on the radius. The maxima occur as far as can be seen practically at the same time 
and the maximum radius Is diminished by about 2* by the plate being assuned to be non-rigid. 

In both cases the bubble begins to flatten itself against the target before the minimum 
Is approached although at this time the approximations cease to be valid in any case. 


Considering Table 2 and its graphical representation Figure 2, it may be seen that, 
although the plate has sprung back, and has even come back beyond its original positlcn, the effect 
on the bubble io‘ sciTicwiiat t hc sams SO in thc cSse where the plats remained disneri inward. 

The differences between the rigid case and the moving plate are very slight, as regards 
the movement of the bubble, and less than the probably accuracy of the theory. This might be 
expected as the subsequent movement towards the plate is governed mainly by tho momentum constant. 
This agrees with photographic observations of the bubble moving towards tho box model when a 
rigid plate theory gave excellent fSsuUs for both Inch and inch plate,. 

It should be noted that in the actual experiments the Box Model and charge are at the 
same depth, whore gravity is approximately cancelled out by free surff,ce and bottom. * 


Conclusions .. 


- 5 - 


4S9 




Co nclusions and suggest ed_ furjUl£Z-hSlh.’ 


It Is concluaed or. the v.hoU that the buftble is not -very sensitive o the motion of the 
taraet plate, there being a Might tendency for the bobbie to follow the ^tion of 
Plato, ana for the buoole not to expand quite as much as .t would if the 
It is intended to determine by photography or otherwise the behaviour of the oubole m two 
experiments, the conditions of wHith are Identical except that the two targe plate, ore of 
widely varying strength, the motion of the plates Peing rtoordod and correUloo with the 

behaviour of the bubble. 

Acknouil edeements. 

The theory on which this report is based is due to Mr. H.N.V. Tsmpcriey. 













TABLE 2 . 


(Teat 50 ; d = 0 . 50 ) 


RIGID PLATE 

a 

tio”^ 

z 

.0160 

0 

7.6000 

.0353 

36 

7.6000 

.0698 

156 

7.6002 

.1077 

422 

7.6006 

.1460 

875 

7.6014 

.1832 

1545 

7.6026 

.2183 

2 iMV 7 

7.6041 

.2505 

3584 

7.6057 

.2789 

4949 

7.6070 

.3026 

6520 

7.6075 

.3210 

8265 

7.6066 

.3333 

10141 

7.6037 

.3395 

12096 

7 . 598 if 

.3395 

14078 

7.5902 

.3333 

16033 

7.5790 

.3210 

17909 

7.5644 

.3032 

19656 

7.5463 

.2800 

21234 

7.5247 

.2530 

22613 

7.4992 

.2228 

23776 

7.4691 

.1906 

24717 

7.4330 

.1579 

25445 

7.3882 

.1420 

25735 

7.3612 

.1269 

25981 

7.3301 

.1133 

26169 

7.2944 

.1019 

26365 

7.2534 

.0931 

26517 

7.2075 





NON RIGID PLATE 


z 

h 

7.6000 

.0000 

7.6000 

.0008 

7.6001 

.0030 

7.6004 

.0102 

7.6013 

.0168 

7.6030 

.0125 

7.6048 

.0110 

7.6065 

.0100 

7.6078 

.0118 

7.6084 

.0096 

7.6078 

.0044 

7.6052 

-.0011 

7.5999 

-.0054 

7.5915 

-.0020 

7.5799 

-,0025 

7.5651 

-.0019 

7.5470 

-.0002 

7.5252 

.0020 

7.4987 

.0223 

7.4684 

.0192 

7.4319 

.0196 

7.3861 

.0211 

7.3584 

.0211 

7.3265 

.0210 

7.2897 

.0209 

7.2477 

.0207 

7.2014 

.0205 












TABLE 2 . 

Test 50, d = 0.50) 


EIGID ELATE 


tio”^ 


.0160 j 0 

.0353 36 

.0698 156 7.6002 

.1077 422 7.6006 

.1460 875 7.6014 

.1832 1545 7.6026 

.2183 a}47 7.604} 

.2505 3584 7.6057 

.2789 4949 

.3026 6520 

.3210 8265 

.3533 10141 

.3395 12096 

.3395 14078 

.3333 16033 

.3210 17909 

.3032 19656 

,2000 2123/., 

.2530 22613 

.2228 23776 

.1906 24717 

.1579 25U5 

.1420 25735 

.1269 25981 

,1133 26189 

.1019 26365 

.0931 26517 


422 7.6006 

875 7.6014 


& 

tio"^ 

.0160 

0 

-0351 

36 

,0698 

156 

.1078 

422 

.1459 

876 

.1823 

1542 

.2169 

2436 

.2485 

3561 

.2764 

4909 

.2997 

6459 

.3177 

8178 

.3300 

10025 

.3360 

11952 

.3354 

13901 

.3284 

15817 

.3152 

17646 

.2967 

19342 

.2734 

20867 

.2461 

22193 

.2159 

23305 

.1840 

24201 

.1518 

24889 

.1364 

25162 

.1220 

25394 

.1093 

25590 

.0991 

25758 

.0918 

i 

25905 


NON HISID HiATE 


z 


0 7.6000 .0000 

36 7.6000 .0008 


























SETUP OF BOX MODEL FOR KRL. SHOTS 


F»G 




FIG 



I 



THE PRESSURES PRODUCED BY EXPLOSIONS UNDERWATER OF 
SMALL CHARGES NEAR A RIGID BOTTOM 


G. Charlesworth 

Road Research Laboratory, London 



; British Contribution 

* i 

f ] 



THE PRESSURES PRODUCED BY EXPLOSIONS UNDERWATER OF 
SMALL CHARGES NEAR A RIGID BOTTOM 


0. Chartfeswof th 
December 1943 


Su mmery . 


Pressura-iirae curves have been obtainedt by means of piezo-electric gauges, at points 2 . a 
and 0 feet above 1 oz, charges of P.A.6. fired underwater at distances from 0 to j feel from a rigid 
bottom. The results have been compared with those observed previously with a gravel bottom. 

The shock wave tall was not reduced so much tor the charge near the rigid bottom as when it 
was fired near the gravel bottom. as before, it was found that lower intensity pulses of unknown 
origin occurred some U to 7 milliseconds after both the shock waves and bubble waves, their occurrence, 
but not their form, being independent of the presence of the bottom. 

The forms of the bubble waves with a rigid bottom were similar to those obtained with a gravel 
bottom, but with the charge near the rigid bottom were more intense and of shorter dii>atioh. At a 
given distance from the charge the pressures and impulses in both the ist and 2 nd bubble waves, when 
plotted against the distance of the chirge from the rigid bottom, had maxima for some value of this 
distance between o and 2 feet, with the gravel bottom this effect was only observed for the ist 
bubble wave impulses. whereas for the gravel bottom no second bubble-wave was detected when the 
charge was i foot i Inches or less from the bottom the corresponding distance from the rigid bottom 
was 6 Inches. 

The 1 st bubble period Increased as the charge approached the rigid bottom; with the gravel 
bottom the period decreased. 

Introducti on, 

The object of the tests was to termine the underwater pressures produced by exploding charges 
near to a rigid bottom arvd to compare the results with those previously obtained near a gravel bottom. 

Experimental , 


Site;- The tests were node in a concrete tank In a depth oT p feet 6 Inches of water. 

The bottom of the tank was flat and the “rigid bottom* used in The tests consisted of a steel plate 
3 feet square and iJ Inches thick placed flat on the bottom of the tank. 

Charges ;- as before, i oz. charges of p.a.G, were used, detonated by Ko. 8 A.S.A. detonators. 
The charges ware cylindrical in shape with height equal to diaireter. They were placed .vllh their 
axes vertical and detonated from the top. 

Pressure m easurement pressures were measured by means of piezo-electric gauges recording 
photographically u^irig cathode-ray cseillographs. 

Arr angement of tests ;- The gauges were suspended vertically above the charge at distances 

of 2, 3, H and 5 feet from it. Tests were made with the charge at distances from 0 to 3 feet 

from the bottom, 


Results 


466 


I 






i 


i- 


Sesults , 

The forms of the pressure-time curves recorded at a point 5 feet above the charge, for various 
distances of the charge from the bottom are shown In Figures i and 2; the shock-weve peaks have not 
been recorded and portions of the records between the pulses have been omitted. The records exhibit 
the same general features as those obtained in earlier work where charges were fired near a gravbi 
bottom though there are certain differenves as mentioned below. 

Shock waves: - The recorded parts of the shock waves were similar in form to those obtained 
previously except that with the charge vary near to the bottom the reduction in the tail was not so 
pronounced. a few measurements were made of the magnitude of the peak shock wave pressures produced 
with the charge on the bottom. The results Indicated that there was an Increase In peak pressure 
above the 'open water' value. Further investigations are required to determine more precisely the 
action of the bottom In this case (e.g. to determine whether the charge plus bottom acts as s single 
charge cf greater weight or as two separate charges). 

Waves of unknown o'cigln i- Pulses similar In form and magnitude to those observed in the 
earlier tests were recorded some S to 7 milliseconds after both the shock wave and bubble waves. 

This Is further evidence against these waves being produced by reflections from a denser substrata 
since the twu sets cf tests were made at different sites. 

Bubble waves :- In most cases tivc bubble waves were observed. in the previous tests, only 
one bubble wave was observed for charge distances of t foot 6 inches or lees from the bottom, whereas 
here two pulses ware recordad at 1 foot from the bottom. At distances of 1 foot 6 Inches or loss, 
there was a difference in wave form in the two cases, e.g. with the charge fired on the rigid bottom, 
the bubble wave was more Intense and of shorter duration than «dren fired on the gravel bottom. 

The naximin pressures and impulses In the 1st and 2nd bubble waves at points 2, ), s and s 
feet above the charge are shown in Figures 3. u, S and i for various distances of the charge from the 
bottom. 


i P 

i I’i 


i 

I * 


The pressures and impulses in the 1st bubble waves with the charge 3 feet from thu bottom are 
about the same as observed before. The present results, however, show an increase in both pressure 
and Impulse with the charge about 6 inches from the bottom. This effect was not previously observed 
for the pressures, but was observed to some extent for the impulses. 

The variations of pressure and impulse in the 2nd bubble waves with distance of the charge 
from the rigid bottom were appreciably different from those with the gravel bottom. Thus Figure 5 
shows that the pressure has a maximum value for the charge about 2 feet from the bottom and Figure i 
shows that the impulse has a maximum for the charge about 1 foot from the bottom whereas previously 
both pressure and Impulse decreased with distance of the charge from the gravel bottom. 

NO measurements were made of the rise or fall of the bubble, it Is proposed to make these 
m^suraments using '.he underwater camera, which wilt also give information on the shape of the bubble. 

Bubble periods ;- The periods of the i:t and 2nd bubble oscillations are shown In Figure 7 
as functions of the distance of the charge from the bottom. Also shown in Figure 7 is the theoretical 
curve for the 1st oscillation of 1 oz. of T.N.T. neglecting the effects of the free and rigid surfaces. 
It Is seen that the 1st period increases as the charge approaches the bottom, until the charge is 
6 inches from the bottom where the period is a maximum. This Increase is expected theoretically if 
the bottom behaves as a rigid surface, with the gravel bottom no such Increase In period was observed. 
The bubble periods observed hers are greater than those obtained in the gravel pit. This mwlbc 
due to the proximity of the sides of the tank tu the explosion 

C onclusions . 

The principle effects observed with the rigid and gravel bottoms are suiiwrised below for 
charge distances of 3 feet or less from the bottom. 


Rigid 





- 3 - 


467 


•/ 

•\ 


r, 

h 


I 


t 

1 


i I 


I u 




Rigid bottom 

Gravel bottom 

Shock wavs 

SOfM reduction in tail with charge 
near the bottom. 

Ao for rigid bottom but effect more 
marked. 

Wave of unknown 
origin 

Oceur,<i at 4 - 7 milliseconds after 
shock wave and bubble waves 
regardless of proximity to bottom. 

As for rigid bottom. 

1 st bubble wave 

pressure a maxioium for the charge 
about 6 Inches from the bottom. 

No maximum. pressure decreases as 
charge is brought nearer to the bottom. 


impulse s maximum for the charge 
about 6 inches from the bottom. 

indications of maximum impulse for the 
Charge about 6 Inches from the bottom. 

2 nd bubble wave 

pressure a maximum for the charge 
about 2 feet from the bottom. 

impulse a maximuifi for the charge 
about 1 foot from the bottom. 

No maximum for either pressure or 
impulse, both of which decrease as the 
charge is brought nearer to the bottom. 


NO bubble wave for the charge 
« Inches or less from the bottom. 

No bubble wave for the charge i foot 
t Inches or less from the bottom. 

Period 

1 st bubble period increases with 
decreasing distance from the bottom 
until the charge Is 6 Inches from 
the bottom where the period Is a 
maximum. 

1 st period decreases as distance from 
the bottom decreased. 


/<■ 









DISTANCE OF THE CHARGE FROM THE BOTTOM - FEET 


Fiq3. MAXIMUM PRESSURES IN THE I BOBBLE WAVES AT POINTS 2.3,4. 
AND 5ft. ABOVE THE CHARGE FOR VARIOUS DISTANCES OF 
THE CHARGE FROM THE BOTTOM 








distance of the charge from the bottom - Feel 


Fia4 IMPULSES IN THE f BUBBLE WAVES aT POINTS 2,3, 4. AND 5fl 
above the charge for VARIOUS DISTANCES OF THE CHARGE 
FROM THE BOTTOM 







aximum pressure 







fMPULSE - Lbsfs tqia 




OtOi 


003 


f , 

/ 

I 

I 

I 




4tt. dbovr the charge 


ao5 


ji-if 



3rt 4bove ihp ch^rqr 

4. 


■ I 2 3 

distance of the charge from the bottom -Feet 


Fja.ft IMPULSES IN THE ^^SUBBLE WAVES AT POINTS 2,3,4, AND 5ft 
ABOVE THE charge FOR VARIOUS DISTANCES OF THE 
CHARGE FROM THE BOTTOM 


PERI 


lO 


I otcilUtion 


Th«or»tic4l period of I oscilUtion for loz. T.M.T. 
OfuKUnq effects of boilom and free surface 


2 oscitletion 



4 2 

distance of the charge from the bottom -F eet 


FI 9.7 OBSERVED PERIODS OF THE I AND 2 OSCILLATIONS OF THE 
BUBBLE PRODUCED BT FIRING loz. CHARGES OF PAG. AT 

Various distances from the bottom 







SECONDARY PRESSURE PULSES DUE TO GAS GLOBE OSCILLATION 
IN UNDERWATER EXPLOSIONS 
I. EXPERIMENTAL DATA 


A. B. Arons, J. P. Slifko, and A. Carter 
Underwater Explosives Research Laboratory 
Woods Hole Oceanographic Institution 


American Contribution 


January 13, 1948 


Reprinted from THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA 
Vol. 20, No. 3, pp. 271-276, May 1948 


475 


the JOURVAI- OE the acoustical SOCSETY of AMERICA VOLUME 20, NUMBER 3 MAY, 1948 

Secondary Pressure Pulses Due to Gas Globe Oscillation in Underwater Explosions. 

I. Experimental Data*t 

A. a. Arons,** J. P. Slifko,*** and A. Carter 

Underwater Explosives Research Laboratory, Woods Hole Oceanographic Institution, Woods Hole, Massachusetts 

(Received January 13, 1948) 

Pressure waves emitted by the osciilaling gas globe in underwater explosions of T.N.T. liave 
been recorded at depths great enough to render small the perturbation effects due to migration 
of the bubbles under the influence of gravity. The first eight periods of oscillation have been 
measured and the pressure-time curves analyzed in order to obtain peak pressure, positive 
impulse, and energy flux resulting from the first two pulses. The scaling of pressure pulse 
parameters with charge size is examined. 


I. INTRODUCTION 

1.1. The detonation of an explosive charge 
results in the conversion of the initial solid 
material into a globe of gaseous products at 
exceedingly high temperature and pressure. This 
is followed by expansion of the gas globe and 
propagation of a strong shock wave through the 
fluid medium surrounding the charge. 

In underwater explosions, the succeeding phe- 
nomena are associated with the oscillation of 
the gas globe, which expands to a maximum 
radius and then collapses under the influence of 
the hydrostatic pressure. A pressure pulse (called 
the first bubble pulse) is emitted while the 
bubble is near minimum size, and second, third, 
etc. bubble pulses are emitted as the bubble 
proceeds with successive oscillations. A typical 
pressure-time curve for the entire, phenomenon 
is reproduced in Fig. 1. 

1.2. The gas globe produced in an underwater 
explosion tends to migrate vertically upwards 
under the influence of gravity, the migration 
being most pronounced in the interval when the 
bubble is near its minimum radius. In the process 
of migration, some of the potential eneigy is 
converted into kinetic energy of vertically mov- 
ing water. T'his energy is not restored to the 
bubble as it collapses, and hence the amplitude 
of the emitted pressure wave is less than it 


* This work was performed under contract NOrd 9500 
with the Navy Department, Bureau of Ordnance. 

t Contribution from the Woods Hole Oceanographic 
Institution No. 430. 

** Present address; Department of Physics, Stevens 
Institute of Technology, Hoboken, New Jersey. 

**• Present address: Naval Ordnance Laboratory, Wash- 
ington, D. C. 


would have been had no migration occurred. 
This effect does not scale with charge size in 
the same way as other parameters because of the 
constancy of the acceleration of gravity, g. 

From the foregoing, it is evident that the 
pressure pulse will be at a maximum under 
conditions which make the bubble migration 
negligible or cause the bubble to be in a “rest 
position.” It is known that for relatively small 
charges a rest position occurs at a certain critical 
depth below the water surface owing to the 
balancing of gravitational effects by tiie repulsion 
from the free surface.* However, accurate pres- 
sure-time measurements under these circum- 
stances aie not possible because proximity to 
the free surface causes serious interference from 
the surface reflection of the pressure wave. 

Since migration decreases with increasing depth 
of detonation, it is possible, by performing experi- 
ments at sufficiently great depths, to make the 
migration effects relatively small, or in some 
cases completely negligible, and thus determine 
the parameters of the bubble pulse unperturbed 
by the effects of nearby surfaces and bubble 
migrations. 

1.3. The object of the present investigation 
was to obtain such unperturbed measurements 
in order to test the scaling of bubble pulse param- 
eters with charge size and to obtain better 
impulse and energy flux data than have hitherto 
been available for the stationary bubble. 

Cast T.N.T. was used in three charge sizes: 
0.505 lb., 2.507 lb., and 12.01 lb. Measurements 


• B. Friednian, “Theory of underwater explosion bub- 
bles,” Report IMM-NYU 166, Institute for Mathematics 
and Mechanics, New York University, September 1947. 


271 



T t T t t T t T t 

Shode Calibration Fint Second Third Fourth Fifth Sixth Seventh 

Wave Step Pul^ Pulie Pclst Pulse Pulse Pulse Pulie 


t Uc. Timing Wave * 

Fio, 1. Pressure-time record showing shock wave and bubble pulses. Charge; O.SOS lb. T.N.T.; 

Gauge dist.: 2.25 ft. Depth: 500 ft. 


v/ere made at a depth of 500 ft. in water having 
an over-all depth in excess of 650 ft. (For com- 
parison several 0.505- and 2.507-lb. charges were 
fired at 250 ft.) Under these conditions, surface 
and bottom effects were negligible. The half- 
pound charge is deemed to be the smallest T.N.T, 
charge which can be reliably detonated under 
these circumstances. The twelve-pound charge is 
the largest that could be tolerated at a distance 
of 500 ft. below the vessel performing the experi- 
ments; thus, the range of charge size was the 
widest attainable with the available experi- 
mental equipment and with T.N.T. as the 
eixplosive. 

All pressure-time measurements were made at 
a distance frosn the charge such that the value 
of W^/R was constant and equal to 0.352, W 
being the charge weight in pounds and R the 
charge to gauge distance in feet. A typical pres- 
sure-time record is reproduced in Fig. 1. Periods 
of pulsation haVe been measured to the seventh 
and, in some cases, to the eighth bubble pulse. 
Peak, pressures, impulse, and energy flux have 
been obtained for the first and second pulses. 

PBSIODS OF OSCHLATIOK 

2.1. Theoretical analysis shows* that to a first 
approximation the period of oscillation of a gas 
glob? is given by 

T=KWyZo*>*, ( 1 ) 

v.'here A— constant of proportionality, TF= charge 
weight, and Zo=absolute hydrostatic depth. 

If the energy associated with the oscillation 
were constant (i.e., if there were no dissipation 
or acoustic radiation), the period would also 
remain constant. However, appreciable energy is 
lost, principally in the neighborhood of the 
bubble minimum owing to the emission of the 
pressure wave and other dissipative effects and, 


consequently, the period of successive oscilla- 
tions decreases. A summary of the experimental 
results is given in Table 1 in terms of the pro- 
portionality constant, K, defined by Eq. (1). 

2.2. It will be noted in Table I that the values 
of K\. for the 2.5- and 12-lb. charges are sig- 
nificantly lower than those for the 0.5-lb. charges. 
The reproducibility of the period measurements 
and the fact that charge sizes were alternated in 
a definite sequence during the shooting program 
make it very unlikely that the difference is due 
to systematic experimental error in the measure- 
ment of T or Z. The discrepancy can be in- 
terpreted only as a slight but significant depar- 
ture from the ideal cube root law of variation of 
period with charge weight. (Although correction 
has been made for the effect of the tetryl booster 
present in each charge, one cannot be certain of 
the accuracy of the correction, :md it is therefore 
possible that the observed discrepancy may be 
due to the relatively larger proportion of bijoster 
in the half pound charges.) 

Examination of 2C» as a function of n shows 
lliat the period drops off sharply during the first 
three oscillations and then decreases slowly but 
steadily in such a way that the results do not 
yield a value of K which could be regarded as 
an accurate limiting value, applicable to the 
ultimate small amplitude oscillations of the gas 
globe. 

111. PEAK PRSSSUHE: FIRST AND 
SECOND PULSES 

3.1. Peak pressures of the first and second 
bubble pulses are summarized in Table II. 

As will be indicated in Section V, the cylindri- 
caUcharges were oriented with their axes in a 
horizontal plane. Pairs of gauges were placed 
above, bclov/, and to the side of the charge in 
positions denoted by Gl, G4, and G2,3, respec- 



I 





477 


SECONDARY PRESSURE PULSES IN UNDERWATER EXPLOSIONS. I. 

Table I. Summary of period constants for T.N.T. Tn=‘KnW‘^‘/Za^l‘, Tb** period of «th o'cillation (sec.), W^charge 
weight (lb.), A “depth of charge balow surface (ft.), 2o=absoliite hydrostatic depth “Z-h 33 (ft.). 


/ Period consiaiils K% 


(lb.) 

(ft.) 

A'l 

K, 

Ai 

Ki 

A| 

K, 

K, 

Kt 

0.50S 

250 

4.36 ±.01* 

3,14±.03 

2..56*.02 

2 38±.03 

2.16i.04 

2.08±.04~ 

~1.9^^T~ 

— 

0.505 

500 

4.36:h 01 

3.06±.01 

2.51±.01 

2.32±.01 

2,15±.02 

1.97±.02 

1.90±.02 

l.Pds.l 

2.507 

250 

4.27” 

3.11 

2.53 

2.35 

2.25 

2.02 

1.94 

— 

2.507 

500 

4.26db.01 

3.06±.01 

2.48±.01 

2.31d:.0J 

2,17±.01 

2.02±.01 

— 

— 

12.01 

500 

4.29±.01 

3.19±.01 

2.56±.01 

2.39±.01 

2.16±.01 

2.04±.01 

1.94±,02 

lv73i,0-5 


* The ntated error the standard deviation of Ute mean. 

^ The error Is not indicated since oiay two shots were made under these conditions. Results ot the two shots agreed very closely. 


lively, all the gauges being positioned off the 
cylindrical surface of the charge. It was assumed 
that the difference between the average pressures 
at Gi and G4 was due only to displacement of 
the bubble toward Gl, and die pressure was 
assumed to vary inversely as the gauge to bubble 
distance. This treatment neglects any effects 
resulting from possible asymmetry of the pressure 
field, but it appears to be justified by the fact 
that when pressures at Gl and G4 are corrected 
to the same distance as the G2,3 position, all 
the values agree within experimental scatter. 

The values given in Table II are the averages 
of pressures measured at all three positions and 
corrected (by means of the apparent migration 
result) to the given distance from the center of 
the bubble, Wi/0.352 ft. 

Because of uncertainty as to the exact location 
of the true base-line of zero excess pressure on 
the individual photographic records, the records 
were read with respect to an arbitrary base- 
line and were then adjusted to the theoretical 
baseline value calculated by means of the theory 
summarized by Friedman.' This is based on a 
calculation of the negative pressure (below sur- 
rounding hydrostatic level) existing at the time 
of the first bubble maximum, and for the condi- 
tions represented in Table II the values are —80 
and — 120 Ib./in.’ at depths of 250 and 500 ft., 
respectively. 

3.2. Ideal scaling would require that all the 
peak pressure values at a given ratio of W^/R 
be equal regardless of charge size. Examination 
of Table II shows that this is not the case. 
The principal cause of the observed variation is 
undoubtedly the migration of the bubble. Theo- 
retical calculation of the influence of migration 
on peak pressure' predicts very much less effect 
than is observed, the expected decrease in peak 


pressure for the worst case (2.5 lb. at 250 ft.) 
being less than 1 percent, whereas Table II 
indicates IS percent, if AP for the half-pound 
charges at 500 ft. is used as a basis of comparison. 
Further discussion of this result will be found 
below in Section IV. 

3.3. The results summarized in Table II indi- 
cated that the pressure field was cylindrically 
symmetrical with respect to the axis of the 
charge. A number of shots were fired with 
the charge so placed that gauges in poaition G2 
faced the cap end while gauges at G3 faced the 
butt end of the charge. Gauges at Gl and G4 
were still above and below the charge, respec- 
tively. In Table III the pressures at G2 and G3 
are compared with the pressure at G2,3 (off the 
cylindrical surface) given in Table II. It will be 
noted that a marked degree of asymmetry exists, 
particularly in th.e case of the srnal! charges, 
with the pressure from the cap end being sig- 
nificantly higher than the pressure from the 
butt end. 

IV. POSITIVE IMPULSE AND ENERGY PULSE 

4.1. For convenience in describing and work- 
ing with various portions of the continuous 

Table II, Apparent bubble migration and corrected 
peak pressures. APb— excess peak pressure of nth pulse 
(Ib./in.*), R ‘“distance from bubble center of point where 
AP, is measured; R — TVV0.3S2 (ft.), IFi-charge weight 
(lb, T.N.T.), X”depth of charge below suriace (ft.). 
Ah. ••vertical displacement of center of bubble from initial 
charge position at time of nth bubble minimum (ft.). 


w 

Ob.) 

Z 

(ft.) 

APi 

(lb./ln.*) 

^^t•) 

AF, 

0b./in.i) 

(ft.) 

0.505 

250 

1120±16* 

0.32 

22S±S 

0.51 

2.507 

250 

1020±10 

0.80 

213±5 

1.12 

0.505 

500 

1210±10 

0.11 

250±5 

0.16 

2.S0V 

500 

1160^10 

0.32 

245±S 

0.47 

12.01 

500 

1140±10 

0.86 

260dh5 

0.79 


* stated error Is standard deviation of the mean. 



478 


ARONS, SLIFKO, AND CARTER 


Table_ III. Bubble pulse peak pressures lor various 
charge orientations. Notation for position of measurement: 
02,3 — gauge faciog cylindrical surface of charge, 02 — 
gauge facing cap end of charge, 03 — gauge facing butt 
end of charge. Gauge distenco from center of bubble; 
ii = Rr|/o _352 (ft.), ^-.acharge weight (lb. T.N.T.), AP„ 
=exceas peak pressure of «th pulse (Ib./in.*). 


w 

z 

APi(lh./in.>) 

APidb./in.U 

(lb.) 

(ft.) 

•G2 

G2.3 

G3 

G2 

G2.3 

Gi 

0.50S 

soo 

K16S 

1210 

109S 

285 

250 

230 

2.507 

SCO 

1200 

1160 

1150 

280 

245 

240 

12.01 

soo 

1200 

1140 

1120 

295 

260 

250 


pressure-time curve illustrated in Fig 1, an 
arbitrary subdivision has been adopted. The 
shock wave is defined as that portion of the 
curve lying between the shock front (where 
the time, f = 0) and the point of first bubble 
maximum (i-T’i/2). Similarly, the first bubble 
pulse is the portion between first and second 
bubble maxima, i.e., Tx/2<t<T%/2, etc. 

4.2. The impulse delivered by the pressure 
wave during a fixed interval of time is defined : 

1 = I (2) 

During a bubble pulse, the pressure is initially 
negative, becomes positive, and then again nega- 
tive. It can be shown from acoustic theory that 
the integral in Eq. (2) taken over the entire 
pulse as defined above must be very nearly 
equal to zero. (Actually, there is a small negative 
residual.) Of principal interest, however, is the 
magnitude of the positive impulse, the integral 
being taken only over the region of positive 
pressure. A summary of average positive im- 
pulse values is given iii Table IV'. 

4.3. Impulses measured at Gl vertically above 
the charge are systematically high and those at 
G4 below the charge are systematically low, as 
would be expected on the basis of the migration 
results given in Table II. As with the peak 
pressure (Table III), the impulse is higher at 
points off the cap end and lower off the butt 
end than off the cylindrical surface of the 
charge. 

Using the G2,3 position (off the cylindrical 
surface) as being representative of the unper- 
turbed impulse, it is seen that at each depth the 
impulse scales according to the ideal similarity 
law within the experimental precision which is 


Tablk IV. Positive impuhe; First and second bubble 
pulses (no correction applied for bubble migration). Dis- 
tance of point of me.isurement from center of initial charge 
position: i?= lFI/0.352 ft., IF =>charge weight (lb. T.N.T.), 
.Z=depth of charge below surface (ft.). 


w 

Ob.) 

2 

(ft.) 

Reduced positive impulse, i/W* Ub. ieo./ln.* Ib.^) 

01* 

First pulse 
02 02.3 03 

04 

01 

Second pulse 
02 02,3 03 

04 

0.503 

260 

1.31) 



1.34 



1.32 


— 0.81 — 

0.45 

2.507 

250 

lAZ 

— 

1.33 

. .. 

1.22 

0.71 

— 0.87 — 

0,47 

0.505 

500 

1.10 

1.22 

l.lfl 

1.12 

1.15 

0,45 

0.48 0.44 0.40 

0.39 

2.507 


(.11 

U7 

1.12 

1.04 

107 

nsi 

0.48 0.4S 0.44 

0.40 

i2.Ql 

500 

1.15 

1.15 

1.17 

1.05 

i.n 

0.51 

0.49 0.47 0.4$ 

0.46 


* See Seation 3.3 end Teble HI for key to position notation. 


of the order of 4 percent. This agreement indi- 
cates that the pressure differences noted in 
Table II are confined to a very narrow region in 
the immediate vicinity of the peak of the pulse 
and do not affect the remainder of the pressure- 
time curve appreciably. 

Thus, the effect of migration on peak pressure 
is more than would have been expected on the 
basis of the theoretical predictions, but the other 
parameters of the pressure-time curve remain 
virtually unaffected. Since the impulse obeys 
the ideal scaling law, it is presumed that the 
peak pressure also would if the effect of migration 
could be reduced still further, but this is not 
conclusively demonstrated and further measure- 
ments would be necessary to test the presumption. 

4.4. Theoretical analysis' shows that to a first 
approximation the positive impulse in a bubble 
pulse would be expected to vary as the inverse 
sixth root of the absolute hydrostatic pressure 
at the depth at which the bubble is located. 
A test of this prediction is given in Table V. 
Since the quantity IZ o ''"/ is essentially 
constant with depth, it is seen that the prediction 
is confirmed within the available experimental 
accuracy. 

4.5. Integrations of the pressure-time curves 
were also peiformed for the purpose of deter- 
mining the irreversible energy flux which is 
given by acoustic theory as : 

1 /»'» 

F- I (Af»)W, 

Vta 

where Ap = excess pressure (lb. /in."), f = time 
(sec.), poCo= acoustic impedance of water (slug 
ft./in." sec.), and F=energy flux (in. Ib./in."), 








SECONDARY PRESSURE PULSES IN 

The Integration was made over the entire bubble 
pulse as defined in Section 4.1, including both 
the positive and negative phases, and the aver- 
aged results are summarized in Table VI. 

4.6. Energy flux measured at Gi vertically 
above the charge is systematically high and that 
at Gi below the charge systematically low, as 
would be expected on the basis of the migration 
indicated in Table II. As with the peak pressure 
(Table III), the energy flux is higher at points 
ofif the cap end (G2) and lower off the butt end 
(G3) than off the cylindrical surface (G2,3). 

First pulse energy flux measured in the hori- 
zontal plane off the cylindrical surface (G2,3) is 
appreciably decreased by migration losses only 
in the moat severe case — that of the 2.5-lb. 
charge at 2.50 ft. The remaining scatter in this 
column of Table VI is probably due to experi- 
mental error. 

4.7. From the successive period measurements 
given in Table I^ it is possible to compute the 
total energy lost by the bubble during the in- 
tervals defining the first and second pulses : 

-(“)'] (4) 

where = total energy associated with the nth 
oscillation; = period on nth oscillation. 

Taking Bi as 490 cal./g. of charge and using 
the values given in Table I, it is found that the 
total energy lost by the bubble during the first 
pulse is 300 cal./g. and during the second pulse 
85 cal./g. Assuming spherical symmetry about 
the center of the charge, the G2,3 results of 
Table VI were converted to obtain the total 
acoustic energy flowing through a spherical sur- 
face having a radius equal to the value of H as 
defined in that table. The resulting energies for 

Table V. Variation of ptBitive impulse of first bubble 
pulse with depth of detonation. ^»depth below surface 
(ft.), Za=^-h33 fft.), //IV*«r^uced impulse at ii= W‘/ 
0.352, iV- charge weight (lb. T.N.T.). 


z. 

l/wut 


(ft.) 

(lb. 8cc./in.« 

40 

1.66* 

3,1 

283 

1.34 

3.4 

533 

US 

3.3 


* Obtained at rest position for 300*g T.N.T .-charge near surface and 
corrected for surface reflection. This value would be expected to be 
•ystematicaily low. 


UNDERWATER EXPLOSIONS. I. 275 

the first two pulses are 120 and IS cal./g., 
respectively. 

It is evident that the measured acoustic radia- 
tion accounts for only a fraction of tlie total 
energy loss sustained by the bubble in each 
case. At this time there exists no concrete evi- 
dence pointing to the cause of the unaccounted 
energy loss, although various speculations have 
been advanced regarding the role of turbu- 
lence, chemical reactions among detonation prod- 
ucts, etc. 

V. EXPERIMENTAL TECHNIQUES 

5.1. The charges used were cylinders of cast 
T.N.T. boostered with pressed tetryl pellets. 
The weight of tetryl was converted to an equiva- 
lent weight of T.N.T. by multiplying by 1.03 as 
suggested by the results of various period meas- 
urements made at this laboratory. One gram was 
added to the charge weight to account for the 
explosive in the engineers special blasting caps 
which were used to detonate the charges. 

5.2. Tourmaline piezoelectric gauges’- • were 
used as the pressure sensitive element. The 
gauges were | inch in diameter and had an 
average sensitivity of 24 micro-microcoulombs 
per Ib./in.’. 

5.3. Eight piezoelectric gauges were mounted 
on A-in. steel cable in the plane of a 15-ft. 
diameter steel ring. The ring was suspended in 
a vertical plane and the charge was mounted at 
its center. Pairs of gauges were placed at equal 

Table VI. Acoustic energy flux; First and second bubble 
pulses (no correction applied for bubble migration). Dis- 
tance of point of measurement from center of initial charge 
position: R — 1FV0.352, JV^diarM weight (lb. T.N.Tl), 
depth of charge below surface (ft.). 


Reduced energy flux; P/1V^ (la. lb./in.* Ib.I) 


w 

(lb.) 

z 

(ft.) 

Cl* 

First puUe 
Gl G2.3 G3 

C4 

Gl 

Second puUe 
Gl G2.3 G3 

G4 

O.SCw? 

250 

161 



141 



122 

23 



IS — 

11 

2.507 

250 

154 


125 

— 

101 

28 


16 — 

11 

0.505 

500 

140 

164 

141 

1.55 

136 

17 

20 

17 14 

14 

2.507 

$00 

138 

14.1 

134 

i.n 

121 

10 

20 

16 16 

13 

12.01 

500 

150 

150 

142 

128 

120 

24 

21 

22 17 

19 


* bee Section 3.3 and Table lU tor key to position notation. 


• Clifford Frondel, “Construction of tourmaline gauges 
for piezoelectric measurement of explosion pressure wjtves,’' 
OSk-D Report No. 6256; NDRC Report No. A-378. 

• A. B. Arons and C. W. Tait, “Design and use of 
tourmaline gauges for piezoelectric measurement of air 
blast.” OSRD Report No. 6250; NDRC Report No. A-372. 



480 


ARONS, SLIFKO, AND CARTER 


m 

distances vertically above, vertically below, and 
horizontally on each side of the charge. 

The center of the charge was in the plane of 
the ring, and two charge orientations were used : 
(a) axis of cylinder perpendicular to the plane of 
the ring, (b) axis of cylinder horizontally oriented 
in the plane of the plane of the ring. 

Dimensions were so chosen that the wire 
structure supporting the gauges did not come in 
contact with the bubble and was never damaged 
or appreciably loosened by the fprees of the 
explosion. This insured against uncertainty in 
gauge position owing to displacement of gauges 
by the shock wave. 

5.4. The over-all depth of water was measured 
by means of a fathometer, and all experiments 
were performed in such depths that the charge 
was always at least ISO ft. away from the 
bottom. The depth of the charge was deter- 
mined by the measured length of suspension 
cable supporting the ring, the method fcieing 
very reliable since experiments were performed 
in a region of negligible tidal currents, ar«d the 
suspension always hung vertically downward. 
This is confirmed by the reproducibility of the 
period measurements given in Table I, 

5.5. Continuous 600 ft.-lengths of Simplex 
(F.O. 5879) signal free coaxial cable were used 


between the gauges and oscilloscopes, thus avoid- 
ing occurrence of spurious cable signal for an 
interval sufficiently long to allow faithful record- 
ing of the first two bubble pulses. Cable effects 
were further minimized by the relatively high 
sensitivity of the gauges. The polythene core of 
the cable rendered negligible any low frequency 
distortion due to dielectric dispersion. The cables 
were compensated in accordance with principles 
discussed elsewhere.* 

5.6. Recording equipment consisted of eight 
oscilloscope channels* on the schooner “Re- 
liance." In order to avoid low frequency distor- 
tion, it was necessary to increase the input 
impedance of the oscilloscopes by using cathode 
follower preamplifiers. The final over-all time 
constant of the recording circuits was about 500 
millisec. 

A llO-v d.c. motor with variable armature 
voltage was used to drive the rotating drum 
cameras. Writing speeds were varied from 125 
to 400 millisec. per revolution of the 10-inch 
circumference drums. 


* R. H. Cole, “The use of electrical cables with piezo- 
electric uauges,” OSRD Report No. 4S61; NDRC Report 
No. A-306. 

*R. H. Cole, D, Sbtcey, and R. M. Brown, “Electrical 
instraments for the study of underwater explosions and 
other transient phenomena,” OSRD Report No. 6238; 
NDRC Report No. A-360. 


SECONDARY PRESSURE PULSES DUE TO GAS GLOBE OSCILLATION 
IN UNDERWATER EXPLOSIONS 

n. SELECTION OF ADIABATIC PARAMETERS IN THE THEORY OF OSCILLATION 


A. B. Arons 

Stevens Institute of Technology 
Hoboken, N. J. 


American Contribution 


January 13, 1948 


Reprinted from THE JOURNAL OF THE ACOUSTICAL SOCIETY OF AMERICA, 
Vol. 20, No. 3, pp. 277-282, May 1948 


481 


THE JOURNAL OF TH7C ACOUSTICAL SOCIICTY OF AMERICA VOLUME 20. NUMBER 3 MAY, 1948 

Secondary Pressure Pulses Due to Gas Globe Oscillation in Underwater Explosions. 
II. Selection of Adiabatic Parameters in the Theory of Oscillation* 

A. B. Akons 

Department oj Physics, Stevens institute of Technology, Hoboken, New Jersey 
( Received J anuary U,. 1948) 

A summary is given of the theory of pulsation of a stationary gas globe in an infinite incom- 
pressible fluid. If three parameters appearing in the furmulatiun are selected by fitting the 
equations to three independent experintental results, it is shown that the theory fits the re- 
maining experimental results and can be used to predict bubble pulse properties over wide 
ranges of the independent variables. 


I. INTRODUCTION 

1.1. The theory of the pulsation of gas globes 
formed in underwater explosions has been treated 
by several investigators during the past few years 
and has been summarized by Friedman in a 
recent report of the New York University Insti- 
tute for Mathematics and Mechanics.' The theory 
in this formulation depends upon three parame- 
ters which cannot be accurately determined by 
means of o priori calculations, and recourse must 
be had to experimental information regarding 
certain properties of the bubble pulsation. 

Experimental results obtained in deep water at 
the Woods Hole Oceanographic Institution* afford 
the necessary information, and it is the purpose 
of this report to discuss the selection of parame- 
ters which make it possible to fit the theory to the 
experimental results over a wide range of the 
primary variables. 

n, rORMULATION OF THE THEORY 

2.1. The theory referred to above' has been set 
up to treat the general case in which the gas 
bubble migrates as a result of the combined effects 
of gravity and the presence of neighboring free 
and rigid surfaces. A brief summary of the formu- 
lation will be given here, modified for application 
to the special case of the stationary bubble (i.e., 
negligible migration). 

Consider a perfect sphere of radius ^4 expanding 


-* Contribution of the Woods Hole Oceanographic Insti- 
tution No. 431. 

* Bernard Friedman, “Theory of underwater explosion 
bubbles,” Institute of Mathematics and Mechanics, New 
York University, Report No. IMM-NYU 166. 

• A. B. Arons, J. P. Slifko, and A. Carter, “Secondary 
pressure pulses due to gas globe oscillation in underwater 
explosions. I. Experimental data,” this Journal. 


at radial velocity A' in an infinite incompressive 
liquid of density p. The total kinetic energy of the 
fluid external to radius A is given by 

r" p(RV 

T= I 4ir/?* dR = 2TTpA\Ay, (1) 

‘'a 2 

where the primes denote time derivatives. 

The potential energy of the system is 

I/=(4/3)w^*FoH-G(/l), '(2) 

where the first term represents energy stored 
against hydrostatic pressure (Pv being tiie abso- 
lute hydrostatic pressure at the depth of the 
bubble center), and the second term represents 
the internal energy of the gas in. the bubble. The 
zero of internal energy is defined as the infinite 
limit of adiabatic expansion, thus: 

G(24)=J^ pdV, (3) 

where the line integral is taken along an adiabatic 
and V represents the total volume of gas. 

It is assumed that the gas approximates ideal 
behavior: 

pV^ = ki, (4) 

where v is specific volume, ki is a constant, and y 
is the ratio of heat capacities. 

Combining Eqs. (3) and (4), 

/'"dV ifeiAfT 

G{,A) = k^M-< I — , fSl 

Jva V'' (7-l)^*<''“”(47r/3)T-' ' ' 

where M is the mass of gas. 

Denoting the total energy associated with the 
oscillation by E, 

2jrp.4>(^')“+ i/i/3)ieA^Po-\-G{,A). (6) 


277 


482 


278 


A. n. ARONS 


Following the convention adopted by Fried- 
man,* it is convenient to transform to dimen- 
sionless variables by using the following scale 
factors for length and time, respectively : 

L=(3£:/4tPo)», (7) 

C=Z.(3p/2Po)h (8) 

Combining Eqs. (5), (6), (7), and (8), 

= 1, (9) 

where a — AjL^ the dot denotes derivative with 
respect to non-dimensional time, and k is given 
by 

k= ; (10) 

« is the bubble energy per unit mass of gas (or 
original explosive charge). 

It will be noted that throughout the following 
analysis the total energy for a cycle of the 
oscillation will be considered constant. This is a 
very close approximation for most of the parame- 
ters since appreciable quantities of energy are 
lost by the bubble only during the very short 
intervals of time in the neighborhood of the 
bubble minima. The treatment of the peak pres- 
sure of the pulse, however, may be appreciably in 
error because of this approximation. 

2.2. Since the bubble is at maximum or mini- 
mum when d = 0, the corresponding maximum 
and minimum bubble radii are given by the roots 
of the equation, 

o3-t-fee-3(,-l)_l=0. (11) 


period of oscillation, 

a' 

da. (14) 

The actual period uf oscillation is given by T = Ct. 

Shiftman and Friedman have given a method 
of obtaining this integral* with a high degree of 
precision. P'igurc 1 shows a family of curves 
giving / as a function of k for various values of y. 

2.4. Bernoulli’s equation affords a relationship 
for the excess pressure at points in fluid : 

Ap/p=(dip/d7)-i(V^y. (IS) 

Here t is dimensional time and ip the velocity 
potential which, in the case of spherically sym- 
metrical incompressive flow, is given by 

<p = A’‘A'/R. (16) 

The second term in Eq. (15) is negligibly small 
compared with the first. Neglecting this term, the 
excess pressure is given by 

Ap = p(A^A'y/R = 2PoI.(oM) /3R. (1 7) 

An expression for (a'd) • can be obtained from 
the equations of motion; the most convenient 
method is the application of the Lagrange equa- 
tion to the energy relation of Eq. (6). The result 
is substituted into Eq. (17), yielding 

PoLta^d (y — Dfe") 

a+ _i (18) 

R I 3 a’T'-* J 

since 

2(oM)- =ad2-3tt-)-3(y-l)fco-»’+'. (19) 

When a is at a maximum or minimum, d is zero. 
The excess pressure at bubble maximum becomes 


The non-dimensional maximum radius is the 
root near unity of the above equation, i.e.. 




PoLr (y-l)^l 

1 

R L a.v®T-i 


( 20 ) 


ajv = [l— ’’jh (12) 

The actual maximum radius is given hyAu— Lau- 
The non-dimensional minimum radius is 

a„Sfe*'«i’-*L (13) 

2.3. Assuming the radius- time curve to be 
symmetrical about the time of bubble maximum, 
integration of Eq. (9) gives the non-dimensional 


Using Eq. (13), the excess pressure at bubble 
minimum is given by 


PoL (y-l)c^ ^n<T-n- 

R *2/«7-i)L (y-l).‘ 


( 21 ) 


In Eq. (20) it will be noted that (since y is of 
the order of 1.3, k of the order of 0.2, and an 
about 0.9) the first term predominates, the 
second term representing a relatively small cor- 


483 


THEORY OF GAS GLOBE OSCILLATION IN UNDERWATER EXPLOSIONS 279 


rection for the internal energy of the gas. Thus 
the minimutn excess pressure is negative relative 
to the original hydrostatic level, as would be 
expected, and is not critically dependent upon 
the equation of state of the gas. 

Because of the base line uncertainties in the 
measurements described in the preceding paper,* 
calculations based on Eq. (20) were used to select 
a baseline to which the pressure-time records 
were subsequently referred. 

2.5. It is seen from Eq. (17) that the non- 
dimensional bubble radius at at which Ap is zero 
is determined by the condition that {aH) ■ be zero. 
Applying this condition and eliminating d from 
Eqs. (9) and (19), one obtains 

4ao*=l-(4-3-y)A!ao-‘'*-«. (22) 

2.6. The positive impulse / delivered by the 
pressure wave is defined by 


into two parts. 


Apdr, 

aai 


rAPdr+r 


where is the non-dimensional minimum bubble 
radius, i.e., the radius when Ap is a maximum. 

Elimination of Ap between Eqs. (17) and (24) 
and conversion of dimensional time r to non- 
dimensional time gives 




d(«*d) 


27* 


j d(a^d), (25) 


where the integration is performed over the 
region lying between the point at which Ap rises 
to zero during the collapsing phase and the point 
at which it again falls to zero on the expanding 
phase, i.e., between points where the non-di- 
mensional bubble radii are denoted by soi and an, 
respectively. 

Because of the radiation of acoustic energy, the 
pressure-time curve is not symmetrical about the 
peak pressure ordinate. For convenience, there- 
fore, the integral of £q. (23) will be separated 


v/here Li and Ci are the scale factors of the first 
and Zii and Cs the scale factors of the second 
bubble oscillation. 

Since d is zero at the instant of bubble mini- 
mum, Eq. (25) becomes 

2Pt 

/=— [L,C,(a*d)«,-fL,C,(o*d)oj]. (26) 

iR 

(In Eq. (26), d is understood to represent only the 
magnitude of the quantity, whereas in Eq. (25) 
the symbol has an associated algebraic sign.) 

The appropriate expression for (o’d)o can be 
obtained from Eq. (19) by setting (a*d)' equal to 
zero ; the result is 

r ('y-T)fe|* 

(a*d)o=1.732flo'Jl 1. (26a) 

L oo*> j 


Fig. 1. Non-dimensional 
bubble period, t, versut pa- 
rameter k for various values 
of 7. 



aSIMATK SMAMITm.k 


484 


280 A, B. ARONS 


Tabli! I. Comparison of experimental impulse values with 
calculations based on Kq. (do). 


Zti 

i/W^ (lb. see. /In.* IbJ) 

(ft.) 

KxpcrimeiUal 

Krom Kq. (3ft) 

40 

1.7* 

2.0 

283 

1.33 

1.31 

533 

1.15 

M2 


•This value Is known to be 8ystcmat!ca)ly low because complete 
correction could not be made for surface reflection effects. 

'in. SELECTION OF PARAMETERS 

3.1. The theory developed in the preceding 
chapter contains three parameters related to the 
properties of the gas in the bubble. These are the 
adiabatic parameters 7 and ifei and the total 
energy t per unit mass. Although the values of 7 , 
ki, and e can be estimated roughly on the basis of 
a priori considerations relating to the properties 
and behavior of the detonation products, such 
estimates do not lead directly to results of suffi- 
cient accuracy to make the theory a means of 
calculating reliable values of the bubble pulse 
parameters over reasonably ■wide ranges of the 
independent variables. 

In order to obtain an empirical fit accom- 
plishing the latter purpose for T.N.T., recourse 
must be had to the experimental results reported 
in the preceding paper^ and to radius-time curves 
obtained from high speed motion pictures by Dr. 
J, C. Decius and his co-workers at Woods Hole. 

3.2. In general, it is seen from Eq. (14) that 
the non-dimensional period of oscillation depends 
on 7 and k. Since the latter parameter is a func- 
tion of the hydrostatic pressure, as given by 
Eq. ( 10 ), one would exjject the actual bubble 
period (T -■ Ct) to deviate from the ideal inverse 
5/6 power dependence upon the depth of deto- 
nation. Experimental results over a v/ide range of 
depth (m. 10 to SOO ft.) indicate that no «uch 
deviation is observable within the precision of 
experimental measurements. 

Reference to Fig. 1 shows that the theory 
would be quite consistent with the above result if 
the value of 7 is taken to be in the neighborhood 
of 1.25,* since along this curve a wide variation 
in the value of k has very little elTect on the non- 
dimensional bubble period f. This is equivalent to 

• This value is identical with the one obtained by Jones 
in Britain as a result of the theoretical study of the com- 
position of the detonation products and their equilibrium 
during the early stages of the bubble expansion. 


making t nearly independent of depth and allow- 
ing T to vary as as given by the scale 

factor C. 

3.3. Substitution of the appropriate values in 
Eqs. (7) and ( 8 ) gives the following expressions 
for the scale factors ; 

L^U33ti(W/Z,)i, (27) 

C = 0.373e*(W-Zo‘'‘), (28) 

where L is in ft., C in sec., « in cal./g, W is charge 
weight in lb., and is absolute hydrostatic depth 
in ft. If Z is the depth of the charge below the 
surface, Zo — Z-\-33, 

The experimental data for the first period of 
oscillation of the T.N.T., bubble is represented 
by 

T=i.36Wi/Zc'‘i*. (29) 

Since T = Ct, combination of Eqs. (28) and (29) 
gives 

e‘i=11.7. (30) 

It will be seen below that, for the depth range 
which has so far been accessible, k lies between 
0.1 and 0.3. From Fig. 1, / in this range has an 
average value of 1.483 along the 7 = 1.25 curve. 
Putting this value of t into Eq. (30) gives 
e=490 cal./g. 

3.4. There still remains the problem of select- 
ing the third parameter, ki. Fitting it to the peak 
pressure data is the most sensitive method. 
Expressing Po in terms of the hydrostatic depth 
Zo, setting 7=1.25, «=-490, and combining 
Eqs. (7), ( 10 ), and (21), 

3.55(10') Wi 

(1-4&'), (31) 

R 

where AP„ = excess peak pressure, lb./in.“, 
lF=charge weight, lb., t = bubble energy, cal./g, 
^ = /SZo''"V(‘>'“l)*'', Zo = absolute hydrostatic 
depth, ft., P = radial distance from bubble 
center, ft. 

The term (1—4^^) in Eq. (31) represents a 
correction factor which differs from unity by only 
three percent at depths as great as 1000 ft. It can 
therefore be disregarded for all practical pur- 
poses. The parameter /3 represents a numerical 
factor in the general expression for k, and jS must 
be selected in such a way as to provide a fit to the 
experimental peak pressure data. 


485 




( 


THEORY OF GAS GLOBE OSCILLATION IN 


Measurements of AFm for the firat bubble pulse 
in deep water under conditions of very small 
bubble migration give values® of about 1200 
lb. /in.® at W^*/-R = 0.352. Using this result in 
Eq. (31) gives^=31.8, and the general expression 
for the peak pressure becomes 

Wi 

AP„ = 34S0 — (!-#'), (32) 

R 

and k is given by 

fe = O.OSS2Zo’-'. (33) 

IV. APPLICATION TO OTHER MEASUREMENTS 

4.1. In the preceding section, experimental 
period and peak pressure results were used to 
determine the values of the three arbitrary 
parameters appearing in the theory. It is now 
necessary to ascertain whether the same parame- 
ters will also fit the rest of the available experi- 
mental data, consisting of (a) the maximum 
bubble radius, (b) the time and corresponding 
bubble radius at which A/> = 0, and (c) the posi- 
tive impulse in the first bubble pulse. 

4.2. High speed motion picture measurements 
of T.N.T. bubbles give the maximum bubble 
radius as 

A»=12MW/Z,)i, (34) 

where .d*f is expressed in ft. This represents the 
average of a large number of measurements over 
a wide range of depths (co. 100 to 600 ft.). The 
accuracy of the measurement is believed to be 
about ±2 percent, and within this scatter there 
seems to be no systematic variation with depth of 
the numerical constant 12.6 in Eq. (.34). The 
experimental non-dimensional maximum bubble 
radius, a^, is therefore constant and equal to 0.92 
if e is taken as 490 cal./g in calculating the scale 
factor. 

I'he theoretical values of au obtained from 
Eq. (12) vary from 0.94 to 0.90 for the range of 
depths cited above. This predicted variation is 
not confirmed experimentally, but since it is 
small and of the order of the magnitude of tiie 
experimental error, the agreement between ex- 
periment and the theorct ical fit can be considered 
quite satisfactory. 

4.3. The pressure-time curves® which yielded 
the value of AP„, used in the preceding section. 


UNDERWATER EXPLOSIONS 281 

also provided measurements of the time (meas- 
ured from the shock front) at which the excess 
pressure dropped to zero during the initial bubble 
expansion and the time at which it returned to 
zero from the negative phase as the bubble 
pre 'ceded to collapse. When compared with 
radius-tirne curves obtained from high speed 
motion pictures, these data show the non-di- 
mensiunai bubble radius ao to be 0.62 when 
Ap = 0. Within the precision of measurement, 
this value appears to be independent of charge 
size and depth of detonation. 

The corresponding theoretical value of ao is 
obtained by solving Eq. (22) using y= 1.2S and k 
as given by Eq, (33). It is found that Oo is very 
insensitive to variation in the depth and has 
values of 0.62 to 0.61 over the range Z5 — 83 to 
533 ft. This is in excellent agreement with the 
experimental observation of 0.62. 

4.4. The positive impulse delivered in the first 
bubble pulse can be computed from Eqs. (26) and 
(26a), providing an adequate assumption can be 
made concerning the magnitude ol! the scale 
factors Li and Ca- Both of these factors depend 
upon ta*, and «a is known to be less than ei, owing 
to the radiation of acoustic energy associated 
with the emission of the bubble pulse. If period 
measurements are adopted as the criterion, one 
would expect as a first approximation that 

(ea/e,)* = ra/Ti. (35) 

Experimental results give Tt/Ti = 0.72 as an 
average. 

T'hus Eq. ( 26 ) can be rewritten 

I/Wl = n WVRKl - 1.59;fe)», (36) 

y having been set equal to 1.25 and «o to its 
average value of 0.615 as determined above; the 
assumption is made that to a first approximation 
(o®d)in is equal to (a®d) 02. 

In Eq. (36), // IT* = reduced positive in, pulse 
(lb. sec. /in.® lb. t), Zo = absolute hydrostatic depth 
(ft.), 1T= charge weight (lb.), f? = radial distance 
from bubble center (ft.), and ^ = 0.0552Zo'®~*. 

A comparison of experimental values® of 7/ IT* 
with those calculated from Eq. (36) is given in 
Table 1 for various depths Za- It will be noted 
that the agreement is quite satisfactory. 

4.5. From Eq. (1 3) it is also possible to make t 
theoretical prediction concerning the mininuim 


485 


282 A. B. ARONS 


bubble radius. Combining Eq. (13) with the ex- 
pression for k, the non-dimensional minimum 
radius becomes 

a« = 0.02102a*, (37) 

and the dimensional minimum radius, 

/*«-0.286l^'», (38) 

where Zo= absolute hydrostatic depth (ft.), 
.4a, = minimum radius (ft.), and lE=charge 
weight (lb.). 

Unfortunately there is no direct experimental 
evidence available to check Eq. (38), since the 
bubble minimum cannot be measured on the high 
speed motion pictures owing to obscuration by 
carbon particles, etc. left iiehind in the water. 

V. SUMMARY 

5.1. It has been demonstrated in the preceding 
sections that if the parameters y, *, and k are set 
equal to 1.25, 490 cal./g, and O.OSS22o'^~', re- 
spectively, the theoretical equations provide a 
satisfactory fit to all the available experimental 
data. This fact increases the utility of the theory 
in that it should now be possible to make reason- 
ably accurate predictions of stationary bubble 
behavior under various circumstances, and the 
same parameters should apply to Friedman’s 
more general theory* which includes effects re- 
sulting from migration and the influence of 
neighboring surfaces. 


It should be emphasized, however, that any 
detailed physical interpretation of the parameters 
7 and k is tenuous at best. It is somewhat sur- 
prising that so satisfactory a fit to the peak pres- 
sure and impulse data can be obtained with the 
same parameters which fit the maximum bubble 
radius and period data. One would expect, for 
example, the true pressure-volume relationship to 
deviate appreciably from the ideal gas adiabatic 
which was used in the theoretical formulation, 
and the deviation, would be most pronounced 
during the high pressure, high temperatures phase 
in the neighborhood of the bubble minimum. 

In view of the multitude of approximations 
and assumptions involved, it would appear to be 
more rational to regard the numerical values of y 
and k as parameters affording a useful fit of 
simplified theory to experimental results rather 
than as actual physical properties of the deto- 
nation products. The distinction may appear to 
be subtle, but it is probably significant. 

VI. ACKNOWLEDGMENTS 

6.1. The author wishes to express his gratitude 
to Dr. J. C. Decius and Dr. B. Friedman for per- 
mission to reproduce the curves of Fig. 1 and to 
the Underwater Explosive Research Laboratory, 
Woods Hole Oceanographic Institution, Woods 
Hole, Massachusetts where all the experimental 
data quoted in this report were obtained. 


THE THEORETICAL SHAPE OP THE PRESSURE PULSE PRODUCED 
BY AN UNDERWATER EXPLOSION BUBBLE 


A. U. Bryant and LI. G. Chambers 
Naval Construction Research Establishment 
Rosyth, Scotland 


British Contribution 


1950 


THE THEORETICAL SHAPE OF THE PRESSURE PULSE PRODUCED 
BY AN UNDERWATER EXPLOSION BUBBLE. 


487 


A. R. Bryant and LI. G. Chambers 
Novel Conatruction Reaearch EstabLishment 


1S50 


Summary , 

This report presmus a methoe wnureby the actual shape of the pressure pulse prpducea By 
an unaerwater explosion PuPble may po oellnsateo with ilttie iabour, it is shown that a single 
univorsoi curva may, with aoproorlAte adjustments of the oressure and tins scales, be made to 
fit reasonably closely the theoretical curves obtained by lengthy numerical integrations Tor a 
wide range oT charge weights ana oeoths. A fonnuta Tor the Deah pressure In the pulse and 
curves Tor the mlnloum buDole radius and the haK-ccrloO of the Dulse are given, 

/ntroduction. 


The theory dovelooed by Taylor (roferenco l) describes the oscillation and the rise of 
the bubble sreduced By an ungerwater explosion in terms of two simultaneous non-linear diTferential 
equations whicn so tar have only been integrated for a very limited number ot cases. in 
reference I, a nutibcr tf oporoxlmate solutions were developed for certain maximum and minimum 
values of the variaolcs, it was found oossible for Instance, to derive an analytic expression 
for the eeaa oressure In Che oulse oroduced when the buoble passes through its mlhimuni size. 

The present caoer oxlonds this work by prosonting an approximate method for calculating with 
little effort the actual shape of tn® oressure culse. For convenience In actual use, the 
Important formutao and the use of the graphs Is summarlspo with an example, in the last section, 

7'fi eo rv . 


If a Is the non-olmenslonal radius of the bubble at time t, and z Is Its death relative 
to an origin 33 feet above S3.v-level, tnc olfferentlal equations oerived by Taylor are 



(i) 

(z) 


In these equations non-oimenslcnal lengths are used, Oerived from real lengths In feet 
by dividing by the length scale factor L » 10 M* where H Is the charge weight In lb. T.H.T. 

(or squlvulent weight of T.N.T. on an energy basis ir some other sxplosivo Is conslderea). 
Hon-olmensional times are convortoo to real times by mulliolying by the unit of t ire /(lig) . 

In equation (l) the quantity G/w is the ratio of the energy left In the gas to the total 
energy of the motion ano for T.N.T. It can oe written. 

G/vi a ca“* (J) 

Where c = 0. 075 (4) 

Two assumptions are new made which reduce equation (i) to an equation with only one 
Itenenaeni. varichls. It is assumed first that for tiraas fairly close to the instant wnen the 
Dubole re.aches Its minimum size the integral in (z) may oe regarded as sensibly constant at a 
value m Which has bem called tnc ’moment™ r.anstant" since, acurt from a numerical factor, it 
is equal to the m.irentum :i the w-iter in a vertical direction associated with the moving bubble. 


fhe . •*, 


488 


- 2 - 


The reason for this assunution is that near the time of the minlmjn) raoius the IntograiX} is very 
small comcarca to its value Jurino th^ large oart of the oscillation the oubole is large.. 
Accordingly equation (2) is reolacoo near tho time of the minimum radius by 


Methods of corriDuting m in tnb orusunce of various surfaces or in ooen water arc described In 
references- 2, 3, and 4. it shoulo bo realised that In what follows U coss not natter whether 
this momentum arises solely m the result of the buoyancy forces acting cn the bubble cr from 
the attraction .r reoulslon exerted by rigid or free surfaces. 


The second assumption mada Is that the term containing 7, in equation (l) Is negliglolr- 
comcirea to the otner tenris during the short oerlod near the mlnlnum radius when the pressure 
Dulse Is frainly oroouc^u. The rcasonuDloness of both those issunotlons Is best demonstrated 
by comparison :f tho results arising from this locrcxlmate treatment with the results calculated 
fcy numericHl integration of the full differential equations. This cahoaris^n will be made below. 


with these two assumotions equation (i) reduces to 



(6) 


The minimum radius follows at once from (6) Oy setting the quality under the square root 
equal to zero, giving 


(I - ca ( 7 ) 

4^11 

Equation (7) Is not easily scivea by successive aporoxinations. Moreover, since It contains the 
parameter c which oeoends on cnnrqe weight, ana the momentum constant m It Is not easy to oortray 
It graohlcally. However It Is shewn In Apcenalx l that by a suitable change of variables a 
universal minimum radius curve may be clotted. Such a curve has been plotted In Figure i. 

The pressure jiul sc . 

In the fu rcioulll equation tor the excess orsssuro p in the water the term tailing ott 
as the Inverse tirst oowur ot tns clstancc will orecomlnatc over other terns with higher Inverse 
cpwars excect quite close to the buDble. At a distance r from the centre ot the buPPle this 
term Is 


0 a i X I ) 
r ot 

U^Ing (b) in this oxpreasion, =and aftor seme roductUn, we get 


-_L_ [, . * CM-* ♦ ] 


(R) 


Thus D is given oy equation (8) in terms of the radius a with c ana m as parameters* 

As It stands (Q) Is cf little use since wo nave no analytic exoresslon for a In terns of tho 

time t, Howevur, it nay be Shown that the nsaxlimim oressure arises at the Instant of the 

minimum radius and by sub3t|tutln>j from (7) Into (8) wo get the following exoresslon for the 

maximum oressuro p . 

m 


rc 



(9l 


(it is convenient to have the crqssure eyoressed In IbJ^ln,^ and the Olstance In feet and this 
will result if the quantity on the right hand side of (7) IsnuiUlplIcd by 43.4 M^), 


ThH tiD.jvn forif-ulao wr^r? fl.?riv?o in rofersnee 2 and ire reoeaten here for convpnlence. 


In order to use cf (8) to obtain the shaoe of tha oressure pulse as well as its 
05?- It !r- nflctissary to obtain ths radius a a? a. function of time t, i.e.. It Is-necessary 

to integrate (6) In the neignbourhoua of the minimum radius. There seems no simple way» either 
of integrating (6) by exoansion in series* or cf clotting or taoulating the resulting solutions 


over 


_ 3 - 


489 


C'V»r n larje enouju nna3 -.f the t«o Odiaeseters tc enable inleroolatlon to be carrleo out. 

However a Ijovy guess showed that the variety cf orassura - time curves resulting from (6) ano 
( 8 ) could De rcouced with fair aooroxtmatlon to a single universal curve by suitable aojustments 
0 ^ oressure and time scales* The orocess by which this Is achieved Is describee below* 

A value of c ■> 0.10 corrosoondlng to a charge of 100 lb* was selected us a useful starting 

oclnt* Equnticn ( 6 ) was then Integrated numerically, ns described In Appendix 2* for the 

fall.wing values of m* vli., I0*'m ' 0, 25, 6 », too, 226 anJ UOO* Fcr each of these values of m 
the pressure function given by the exoreselon on the right of (e) was cariouteo from the values of 
a and olstted against time. Insoectlon of these pressure time curves showed that whereas for 

small values of m the curves were narrow with large values of the peak Pressure, for large values 

of m the curves wore broao with low values of peak pressure. nevertheless the curves all had a 
certain similarity* The experlmsnt was therefore tried of adjusting the oressure scales In each 
to give the saiae nunidrlcsl value for the peak cressure* wnlle the time scales were adjusted to make 
the family of eufves coincident at th* oslist share the pressure had fallen to half Its maxlniim 
value. 


Thus fcr each of tne values of tha iiomentum constant m the quantity o/p was plotted against 
the quantity t^t^ (tne origin cf tha time scale being taken at the Instant of minimum radius] where 
t^ Is the time taken by tne pressure to fall from Its maximum to half Its peak value* It may be 
celled conveniently the 'half-oerlod*. Two sdcn curves for tne extreme cases m > o, and m • o.mt 
nave been plotted In Figure 2. it was found that In the region where p/p^ is greater than about 
0*2 thsso curves fitted each other very closely* In fact the closeness of fit Is orobaoly better 

than It Is reasonable to exouet as b.)twsan orossure-tlmo cu.'vss obtained from exact sslutlon of 
equations (l) and (2), and from actual pxoerl.monts* Accordingly It seems quite adequate for all 
orectlcal ourooses to take any one of these oressure-t Ime Curves, expressed In this manner, as a 
universal pressure time curve - as regards shsoe - for the oulse oroducea py the collapse and 
re-exoansloq. :f the exslcsl.n buPble*. 


So far we-have only considered one specific value of the oarevnetcr c deoenolng on the 
charge weight. In Accenalx 1 It la shown that tnese solutions may oe converted Into solutions 
for any other gasirgd value of tne Charge weight by a simple linear transformation of the length 
and time scales and fpr a ejrresoonaing altered value of the mcmentuii constant m* Since, as 
states above, and aemsnstrateo In Figure 2 , ehango In m makes an almost negligible difference 
to tha ahnae of th« pressure time curve when sxp.-sssso i,-. temis of ins oeak oressure ano tne hair 
aerki It fellows that all soluticne for -.tner values cf c and m (within reasonable limits) will 
lie vury close t. tha two eufvea In Figure 2 , 


Also plotted on Figure 2 are points oPtalneo from the full Integration of Taylor's equations 
for four eases ranging from l gram 3 feet deep to «60 lb, at a Oeptn of 60 feet* It will be 
observed that the aooPoxImatlonS uacO-in this rwoort nave not resulted In errors of more than about 
five per cent cf tne peak pressure* Tne agreement will Pecomo progressively werse at times longer 
than two ana one half times the half period due t; the neglect of the tenn In a In equation (l) 
which becomes more Important the larger the raolus* Nevertheless the differences of the order of 
five per cent arising due to the asproxlmatlons made are n small nrlre te pay ts Dp able tc 
delineate the shace of the main part of the oressure time curve for a very wloe range of charge 
weights and deaths without recourse to the very laPorlous Integration of the full equations* 


In order that FIgurti 2 may yield a pressure-time curve in real units It Is necessary to 
be able to calculate the peak pressure ang the half oorica tj - creferaoly in real units* 

By using F.^ure t to got a, and Inserting In equation ( 9 ) tne peak pressure may be easily obt.'.lnod* 
The values of t^ obtained rrse the numerical Integrations mentioned above have been collected and 
elctted as a function of tne momentum constant ra. As exolalned In Aooendix t this curve may now 
be made aoollcaole to all values of the parameter c (and hence to all values of the charge weight) 
by relabelling tne aoselesa m/iooe* and oy rslaoslllng tha orolnatea t./(»e) .^* However, this 
latter quantity can be converted to real times by multiplying t, by thi time SMle faetor/(xo M*/g) 
Thus the ordinates of this graph may also be labelled »*67 These values of the halt period 

have been olottoo In Figure 3 (after a further change cf scale length to remove the numerical 
footer »*67) using open circles* Also plotted are 'our values oPtalneo from the full Integration 
of Taylor's equBUon».( it will be seen that except for largo values of m the agreement between 
the more exact values and tnpse obtained by the approx Imal Ions of this resort arc vary close 
Accordingly the curve drawn In Figure 3 may 00 regarded as a universal curve for optalning the’haif 
Dsrico cf the bubble pressure pulse. ^ 


The 


since the equation of radial mst|j!i (e) In symetrical with regard tj the time of mini 
radius the resulting oressure time curve Is also symstrlcal and only one half of It I 
given In Figure 2 


Large values of m only arise in general for large snallow charges when large vertical 
migration of the Pubble occurs, witn considerable departure of tne bubble from sch.rlclty 
end the pressure ouls-s oroduced are relatively feeble a.nd unlmpcrtant. 


i 

i 

1 


i 

! 

i 

I 

j 

i 

) 


3 

!i 

? 

6 

j; 

f 

1 


! 

I 


For convonierce in use thei relevant t^(|uations ana directions for the use oF the d^nphs In 
obtainin 9 the orcssurc oulso to do ovocctcd frc.A a given charge at a given OQotn are ccUectea 
here together with a nurrcrlcnl 

(i) The two oaramotors c nno m rr^ist first do calcylntfiO, For t.n.t* ano aoproxlmatel v 

’ it if. 

for oquivoient wslgnts of other explosives on an onergy biSlSf c = &»075 w*'*® wherr m Is thf Cfjar^j 
weight in 1 b, The rrtotndriti/n constant iiuy De calculated tram for<r.u1ae in references 2 j 3 ano u 

for the case cf a charge in ooan wntor or near various surfaces. In ooen wster an approximate 
excri'ssicn for m Is 


0. 167 . 

m = --IT (1 - c 2 ..*) 

Where 2 ^ Is the initial oeoth below a point 33 above sea-level, in non-dimensional units. 
To convert to these nor>-dimon 5 lcna 1 units divloo all lengths by I = lO and all times by the 
unit of time (L/g)^. 

(it) Tnc mlniiitum radius in nor>*dimensicna1 units ano its value A in feet ntay be 
read off the curve in Figure 1 for the calculated value of tne oaranteters c (or m) and 


(Ill) The oeak pressure p in non-dlmonslonal units is next calculated from 


(1 - 3 c 

wnere r U ttie noif-aifr«n3lcna\ alstance from the point to tne centra of the ouDBle. If the 
distance is required in fc.'t and the pressure in lbt/sq.in. the right hand sloe pf this exoresston 
Should be muUiolied by U3.4 


4 7T ^ 


(Iv) Tnc hi'iif period T, , defined is tnv time for the pressure to oroo from Its mixlmum 
to one half its ffaxlmun value, is read eff the Curve In Figure 3 for the particular value of the 
parameters c anu 


(v) Finally either of tne two curves In figure Z may be taken as giving the prossuro 
time curve (or strictly as half the pressure time curve since tns curve is theoretically syrrmstrlcal 
about the tinx^ of tho oodk cressure). By (Tiultiplying the numerical values of the orolnates of 
the curve by tnc valiis of the osbU pressure calculated in (iU) above and the values of the 
abscissae by thv: half period T^ caicoiatso In (iv) above this curve bec»3mes the thooreHcal 
pressure time Curve in ^.bsoluto units* 

AS an ('X^jnuie censidsr n charge of 200 lb* t.n.t* at 200 feet depth, Tne length scale 
factor L - 37*5 fest and c = 0,1046* The notvdimenslonal flcclh is thus 233/37,5 » 6^21 = 

From (I) aoove the value uf tr fer open water (i*e* in the absence of surfaces) us 0,00454* The 

factor lOO = 1,094 so that m/lOO * 0.00415, From Figure 1 the minlffium radius 

“ 0, 067 (icc)'^'^ so that 3j = 0,0713 ano the nininuim radius In feet is 2,67, Inserting this 
value of in the fomwle for the peak pressure tne quantity rp^ is evaluated to be 20,35. 

The value of 43*4 ^ Is 613 so that in feet and 1b,/sq,in. units the quantity rc^ is 12490. 

This means for example that at a distance of 30 feet the peak pressure in the culse is 
acoroxlmately 624 lb,/sq,ln. Fin^.lly from Figure 3 the value pf i 5 seen to be 0,6B 

for this value of m and c, which givts T^^ - 4,0 milllseccnOs. Tne Shape of either half ct the 
pressure Urts- curve is given by Figure 2 * 

References* 


( 1 ) sir Geoffrey Taylor. 'The vertical motion of a sohericai oubblo ano the pressure 
Surrounding It*. 

( 2 ) A.R, Sryant,. "The behaviour of an undsrwator explosion bubble, Acoroximattons 

based on the theory of ®rofs;553r G.l, T-iylor*, 

( 3 ) A.R, BPyant. *-> simolifled theory of the effect of surfaces on themeticn :t the 
oxidI: si :n bubble ", 

(4) a.R, Bryant, *The bf.havlaur :f an unoerwatar exolosUn bubble. Further 

.'!OCrrxjft)utl:ns*. 



) 




I 



491 


AO^ iHDIX X . 

TSTEGUflOS OF BOVATIOH fg| FOR ALL CHARGE mOBTS 

Let us suDposs that wa hav« IntsjrateS (t) for a sooctf to valua of the parameter c, 
say c • 0.10, CorreSDonalhj tc H » too lb, and for any specific value of m, say m ■ in, '„et the 
solution he aonotea by the suffix zero, t.o., Og, t^ satisfy. 



Then after substitution of these new variables In (I) we get 

2S = f-U- (i-ca'5-i^ <ieo c^irjOl * (III) 

at I 2 7T a'’ I isr 1 J 

It follows then that any solution a^, t^, of equation (a) for c « O.lO is transformaa 
by the relations (II) Into a solution a, t, of aquation (&) with any selected value of the oarameter 
e but for a value of the nofnentum constant m s (loo c^} in. It Is therefore only necessary to 
Integrate (6) for one value of c and far a range of values of n. 

For exoisole equation (7) far the mlnlmxn radius has been salved by tabulating end platting 
m far a range of values af a^ and far e « 0.10, By relabelling the aralnates aj/(lD c)*^^ and 
the abscissae m/lOO c^ the curve beeamos universally aspllcable for all values of c una hence 
all values of charge weight, since the mlnlnum radius Is mare often required In fest, a further 
Change may be mnae by writing =» A^/lO M* where Aj Is the mlnumim radius In feet. Mshing use 
of the relation (s) eetwoen c ana H we find that 



so that the ordinates In Figure t may be also labelled with this alternative expression. 

In thu same way, after ealeutatlng the value of the half period t^ for a range of values 
of m and for c = 0. lO It is only necessary to relabel the ordinates tj/(ld cj®^^ ano to relabel 
the abscissae m/tOO c* to have a curve giving against m for all values of the oararreter c arrf 
Charge weight. The notwllraorsl onal half period t^ may be converted to real units by using the 
time scale factor (10 M*/g)’ so that the ordinates of such a curve may also be labelled 
A.P7 T^^r In Plotting the Curve in Figure 3 the values of ti nave oeen multiplied by 

lOtO'M.67 so as to remove the numerical factor and give In milliseconds. 

It may bo remarked that In Figures 1 and 3 real lengths and times have been divided by 
the quantity This result would have Oeen expected If the usual system of non-dlr«nslona1 

units employed In dealing with explosive phenomena had been Introduced at the start Instead of 
the soecial noivdlmenslonal units Introduced by Taylor. in general the nob-dl mens tonal system 
usually employs: In dealing with shock-wave phenomena Is inappllcaole t: the mctlon and 
behavlpur of the buPOle owing to tho role of gravity. However, when dealing with phenomena 
near the time of the minimum radius, when the Internal gas pressure is the dominant factor the 
aooroximsting assumptions used above remove any exoltcit connection with gravity effects so 
that this system of nom-dimenslonal units can oe effectively eraolcysd. The effect of gravity, 
or In other words the norfdlmenslcnal parameter r,, Is nevertheless oresent since It largely 

UttiSriTiifiCS the voroiTiStcr m« 


m, 


A°»£Hg|X 2. 

METHOD OF HUKEXCAL MTEGRATIOK OF TUB EQUATWUS OF HOTIOff 

Insaectlon of equation (6) siiovss that venlle the dependent and Independent varlaoles can 
easily Oe sesarated> making solution 6y numerical quadrature oesslble, the right hand side has 
a zero at the start of the Integration (the origin of liras Is chosen at the Instant of mlnlniun 
radius). The Integrand In the quadrature process la therefore Infinite at the start, and the 
singularity Is not esslly removed. Accordingly the following process of numerical Integration 
was adooted. 

The firat five or six values of a at equal Intervals of t were comsuted t>y writing down 
the first few terms of a Taylor series fora near the minimum value Uj. Tno first and third 
derivatives of a with rssoect to t vanish at tne origin. The second and fourth derivatives were 
calculated by reoeated differentiation of the right hand side of (6), making use of the 
eondltlon (7) to remove the carameter m, The Tcllcwlng values wore obtalneoi 



» af“ ( 1 - * caf*) 


Jt ■ 0 

un ^ 1 


o 

II 

. JiL a. (1 - d ca,-^) 
8 * * 

’ Ml 


The time Interval for flach stec'was chosen so that dfter about six stoos the term In t** 
was still snail. The Taylor series was also differentiated and the resultant series evaluated 
at the last time Interval. This value was then comoared with the value of oafdt ealciilatsd 
directly from equation (6) with the apprcorlate value of a Inserted to verify that neglect of 
higher terras In the Taylor series had eauseo no aoercclable error. 

With these starting values for a, and the cor.'esoondlng values of da/dt calculated from 
(6) the process described on page 9*2 of ‘Interoolatlon and Allied Tables* tor Integration of 
differential equations of first order was carried out*. The fourth differences of da/dt were 
watchso put at no stage was their emtr ibui icn to the integration appreciable. 


'Interpolation and Allies Taples* oublleheo by H.M. Stationery office ;n behalf 
of H.W. Nautical Almmuc Office. 


























THE OSCILLATION OF THE GAS-BUBBLE FORMED BY A DETONATOR 
EXPLODING UNDERWATER 

DETAILED COMPARISON OF THEORY WITH EXPERIMENT 


H. N. V. Temperley 

Admiralty Undex Works, Rosjrth, Scotland 


British Contribution 


March 1944 


497 


THE OSCILLATION OF THE 6AS-BU3BIE FORMED BY A 
DETONATOR EXPkODINS UNDERWATER. DETAILED 
COMPARiSON CF THEORY WITH EXPERIMENT 

H. N, V. Temparlay 


March 1944 




.Summa ry. 

The discussion of th« dshaviour of Qa$*oui>&)«s from large charges has been extended to detonatorS) 
for which tia possess the much more detailed evidence given by actual photographs of the bubbles. 
Calculations huve therefore neen made for the repraicntatlve case of a gram charge of T.H.T. 3 feet 
below th« surface of the water, and the results have been compared both with experiment and also with 
the approximate formulae developed by Herring and Bryant. The agreement of the exact calculations with 
the approximate forttuloe is found to be such that, with one exception, the latter can be aoplied to 
detonators even though quite a high value of Taylor’s norv-dlmenslonal parameter Is involved. The 
following Quantities are compared with experiment;- 

(a) The rad i us-time Curve for the bubble ; Agro.‘mont Is .already known to be satisfactory, except 
near the minima where energy Is being lost in various ways. 

(b) The trajectory of the centre of the bubbl e; acth under gravity and In the presence of a 
single free or rigid surfaco. agreement is at any rate qualitative up to the first minimum. 

(e) T he shape of the buboie : Agreement is satisfactory, 

(d) The beha vicus of the bubble from a Charge fir ed In contact with a steel olat o! 

Agreemehl Is only rough, two serious discrepancies (which may bo different aspects of the 
fact that a steel plate can naroly oe regaroed as rigid for a contact charge) have been found, 

(e) Th e behav i cnis pf the bubble from a Charge fired between two steal olates ! Agreement 
is as good as one could reasonably expect. 

Taken as a whole, the position seems to be very similar to that found for large charges, 
phenomena which depend on the behavious of tho bubble near its minimum radius, (such as the minimum 
radius itself, or the secondary pressure-pul sos) cannot be predicted, while phenomena such as the period 
of the bubble, and the rise or fall of its centre up to drst mlnimuni can be predicted fairly accurately, 

j, Intrccluction, 


up to the present tir.ie, practically all underwater photogi .nphy of the gas-hubhle has treen of 
charges not larger than a detonator. How It Is known that, unless the pressure of the atmosphere above 
the surface of the water can oe reduced, such experiments give an indication of the behaviour to be 

expected if full-scale charges are fired at depths of the order of 500 feet, whereas we are most interested 

from a practical point of view in charges fired at depths of up to lOO feet. By working under a partial 
vacuum, Taylor and Oavies have obtained scale models of the bubbles from charges fired at depths of this 
order, but the remaining work with detonators seems to 1,-ick direct application to the problem of damage 
to ships by explosions underne.ith them, and to be more of the nature of fundf5mcntal research, up to 

the present iirou also, theory has concerned itself nv:stly with the behaviour of the byoble at these 

relatively shallow deoths, which can only be inferred experimentally from a vory long scalitig-up of 
Taylor and Oavic-s' results, or Indirectly tr.ro pressure-gauge mej-surements on the full scale. The rather 
anomalous position has thus arisen that the i« re recent developments in the them'y have not been applied 


to that 


498 


- 2 - 


to that region tor which experiments! knowledge is the most detailed, and the ealcglatlons reported upon 
here were put in hand in order to Ml! up this gaa. 


Details of the Calculations . 

The theory used in suDstantiolly that developed by Marring N.O.S.C. Report C« - Sr20 - OlO) and 
Taylor (•vertical motion of a spherical bubble and the pressure surround If), who both developed formulae 
to allow for the effect of gravity, and of the proximity of free and rigid surfaces. The 
dropping the assumption that the bubble remains spherical has been investigated by shiffmann (Applied 
Mathemtics Panel Memo. 37-S) and. in more detail, by Tempertey. The theory of the bubble formeo uy an 
explosion in r.entact with a rigid surface was also glvan by Tempertey. The theory of mo benav.our u. 
the bubble from a charge fired between two rigid surfaces is now pibllshod for the first time. 

The calculations were all made for a detonator equivalent to one gram of T.H.T., and a total pressure 
of 36 feet if water, I.e., the detonator 3 feet below the surface of the water at atmospherical pressure, 
in Taylcr-s non-dimensional units, which we shall use heneefwfward, the values of the relevant quantities 
are as fcllowas- 

Factur to convert non-dimensional lengths to feet (l) ■ 2.2 feet 

Factor to convert non-dimenslonal times to seconds (T) » -261 ssc-ftd 

Mon-dimenslonal total external pressure (t^) ’ K-J 

Mon-dimensional initial depth below free surface (d^) - 1.3 

Ca mfiarison of the Calculations uith approximate theory . 

These conditions were chosen because they were thought to be fairly representative, but they do not 
exactly correspond with any of the previous experiments. wrIght, Campbell and senior used a no. 6 detonator, 
which corresponds almost exactly to 1 gram of explosive, but the depth was only i.S feet and not 3 feet. 

j ...Afi. at 4 fo&t hilt ha ii«Aii a so, A detonator, which is stated by 

Ueuxenani carapuaii's o*poi ini*ii.» i..-..— — - -• .. ® ^ 

him to contain about 0.7 gram of explosive. The best that we tan do appears to be to conpare the results 

of this calculation with the approximate formulae developed by iierrlng and iryant, |f the approximations 

prove satisfactory for the two cases calculated (conditions as above, effect of free surface first neglected 

then taken Into account) we oan then apply them to other cases with confidence, without these calculations 

by Nautical Almanac office we should have to extrapolate the formulae from the range ■ 1.5 to 7.8 (over 


oy Nauvivai Almanac an^uiw novq ww* * o 

which they have been cheeked against the results of direct solution of tho hydrodynamic equations) to the 
region z • 16. an undesirably long extrapclatlon. The following table accordingly gives the results of 
this comparlsons- 


Table I. 




499 


Table I. Coaparlaon of n.a.O. calculations with formulae due tu Herri ng and Bryant . 


1 gram of t.n.T. at 
a depth of 3 feet. 

Free Surface absent. 

Free surface present. 


N.AeO. 

App rex i mat Ions 

N.AwO. 

Approximations 

Period :if ist 
Osclllat iun 

28. U 

2B.6 

27. U 

27.5 mlllisec. 

Radius at fi rst 
Haximunn 

.5QA 

eSOO 

.508 

. gno fabf 

Radius at first 
Mlnimun 

.055 

• 055 

.0U6 

.05' feet 

Maximum Vertical 
Velocity 

+ 900 

^ no 

4- 820 

+ 690 feet/ 
sec. 

pise of Bubble up 
to first Minimum 

♦el5« 

♦.255 

♦ .090 

♦.141 feet 

Vertical ‘momentum 
constant* m 

♦ 82 

-S- 85 

♦ 43.5 

+ 31 X 10*® 

•Pr* (max.) lbs./ 
sq. In. foot 

♦ 32« 

+ 325 

♦ 370 

♦ 335 

1 Impulse (lbs./ 
sq. in. foot sec.) 

.13» 

.12b 

.,139 

.124 


It will be seen tnat the comparison is quite satisfactory except that the figures for the rise 
of the bubble are not correct, the approximate formula for the rise of the bubble giving too high a result, 
subject to this correction, It seems that the use of the approximate formulae In this region is quite 
satisfactory, as. a basis of comparison with experiment. The approximate formula for the rise was 
arrived at empirically, and the fact that it is Incorrect in no way affects the validity of the remaining 
approximations, which can njw be used with confidence. 


(bmpar'ison of theory with experiment . 

The following quantities suggest themselves for this purpose:- 

'*•1 The radlus of the bubble .as. a function of tim e; Careful comparisons have been made by Herring, 
(using Photographs obtained by Edgerton), and by Lieutenant Campbell. Agreement is good except in the 
region near the minimum radius where there is reason to believe that the simple Incompressible theory 
breaks down, due to radiation of acoustic, energy and possibly to other causes also. The maximum radius 
and period between explosion and first minimum have also been compared with experiment by wrIght, Caitsibell 
and senior, who find exeelTent agreement. The minimum value of the radius is hard to obtain from the 
photographs, owing to the fact that motion is very fast in that neighbourhood, and also owing to the 
obscuring effect of debris in the water. The further history of the hobble, after the first minimum 
can also bo accounted for satisfactorily, provided that it is assumed that there Is a loss of onergy 
at each ml nimum. 



500 


_ 4 - 


«,2 The trajectory of tho centre of the bubbla ; a detailed eoffloarison la difficult unless a 
calculation t.s made in detail for each separate case. Lieutenant Campbell doss not si'fe any figures 
for the trelsctory of the bubble In the presence of the free surface only. wrIght, Campbell and senior 
give a detailed trajectory, but In their work the effect of the bottom of the tank must have been 
appreciable. It being only about U feet below the charge, (the tree surface being t* feet above It). 

They find that during the first expansion phase, tnere Is fas predicted by theory) a definite attraction 
of the bubble towards the free surface, repulsion setting In during the contracting stage. using the 
Road Reser,rch Laboratory formulae and considering the effect of the tree surface only, they find good 
agreement for both the total depression of the centre of the bubble up to the first' minimum but not for 
the velocity of the centre of the bubble at this point, we have seen that the formulae used would give 
a numerically too large value for the r|se or tall of the centre, and the close agreement actually found 
may be due to the mutual cancellation of this error with the fact that the effect of the bottom was 
neglected. The situation regarding the vertical velocity at minimum is not so clear, the agreement being 
very poor, wrlght, caitpbell and senior give in error a theoretical maximum velocity of Sb.S feet per 
second compared with an observed value of 6U feet per second, the correct throretlcol figure being 
645 fest/second. The explanation of this large discrepancy is probably similar to that of similar 
disagreements found by Lieutenant Campbell discussed below. 

A comparison can also be made with Lieutenant Campbell'S results on the bubble near a vertical wall. 
The horizontal motion due to this will then be approximately independent of the vertical motion due to 
gravity plus the effect of the free surface and bottom, we use the data In his Table 2 and Figures 20 
and 21. we need first of all an estimate of the energy given to tns water by a 40. 8 cap (apart from the 
energy appearing as a shock wave). This is best obtained from the maximum radius of the bubble, which is 
given by him as 5,05 Inches (average) which corresponds almost exactly to the figure to be expected for 
0.6 gram of T.X.T. in the cases where the wall is so near that the bubble touches it before reaching Us 
maximum radlus.the maximum volume is considerably less, than in the cases where It forms a complete sphere. 
The reason for this is not very clear, but It is possible .that energy is carried away as a compresslonal 
wave In the steel of the wall, to a greater extent than it Is in the water. Anyway, theory cannot at 
present be applied to these cases where the bubble actually touches the walls excapt to the limiting one 
where the chatfle Is fired in actual contact with the wall. This case will be discussed later on. we 
make the calculations for 0,6 gram of T.N.T. on the basis of the approximate formulae, dividing the figures 
for the ri,se of the bubble under gravity by a factor of t.6 in order to bring the approximate formulae 
Into line with the detailed calculations. The remaining approximations are used os they stand. 


Table II. Com p arison of Li e utenant Campbell's results with theory . 


0.6 gram Of T.H.T. 30 Inches below free surface, and at various distances 
from rigid wali. 


■■ M p 

Ulstanee from 
wall 

(Inches) 

Period of 

oscillation 

(Milliseconds) 

Displacement 
towards wall 
after one 
osc il lat ion 
(Inches) 

Maximum 

veloelt 

centre 

bubble 

(teet/s 

y of 
of 

ec.) 

Minimum radius 
of bubble 
( 1 nches ) 


Calco- 

iated 

0l>- 

served 



Calcu- 

lated 


Calcu- 

lated 

Ob- 

served 

in 

24.6 

25 

H 


420 


.57 

1.5 

16 

25.2 

26 

— 

BB 

TSO 

30 

.58 


12 

25.11 

26 

0 

D 

760 

70 

.74 

m 

9 

26. j 

27 

1,7 

■ 


190 

.97 

mi 

.... ‘ . 

27.9 

27 

2.6 

■ — 

3 

B 

140 

i.S3 

■ 
























- 5 - 


501 


it is clear from these results that the discrepancies are due to the fact that the bubble never 
closes up as much as Inccmpresslble theory Indicates, owing presumably tc the losses of energy that seem 
to occur near the minimum. If the observed values for minimum radius are used instead uf the calculated 
ones in tho calculation of the moKlmum velocity, much better agreement Is obtained. The values ■btained 
thus are shown In ora'.kcts, but they are only very r.ugh, as they are inversely proportional to the cube 
of the minimum radius, which Is not known accurately. 

5.3. T he shape of the bubble ; This can be dealt with very shortly. The calculations made by Nautical 
Almanac office applying the the.ry developed by Temporley shows that (contrary to what was at first expected) 
larga departures from the sphorical shape might be expected to occur even for the case of 3 dstonator 3 feet 
below the water. This would not occur (even if the Incompressible theory ware valid fight up to the 
minimum} until a few hundredths of e millisecond before the minimum, so that It w-juld be extremely difficult 
to detect experimentally. It may therefore be said that the fact that the photographs show the bubble 
staylijg very nearly spherical for several oscillations Is not In disagreement with thejry, 

4.4, T he behaviour of the bubble from a chatfle fired In contact with a steel slate ; Photographs-of tho 
bubble from a detonator fired under these conditions were also obtained by Lieutenant Campbell. The effect 
of gravity on a bubble In this case was investigated theoretically by Temperley, who found that, for a 
bubble above a horizontal steel plate the effect of gravity should be to make the bubble become pointed 
during the early stage of Its oscillation, and then to flatten itself against, the plate just before It 
reaches Its minimum. The photographs show that, although the bubble certainly does flatten itself against 
the plate, this process is fairly gradual and is spread over a considerable proportion of the oscillation. 

In contradiction with the theory. This discrepancy may be connected with the one already noted, that a 
cht^rge fired in contact with a steel plate does not produce so l.arge a bubble as a similar charge in mid- 
water. This matter seems worth further investigation, as It may have some bearing on the behaviour of a 
•contact’ cfiarge. 


A charge fired between two steel plates, (■the bubble that splits in half*! : Lieutenant Campbell 

Investigated the effect of firing a detonator exactly mid-way between two vertical steel plates. Although 
the bubble rushSd towards one or other of the plates unless the detonator was fairly accurately centred, 
it was possible to obtain two other types of behaviour. |f the plates wnro fairly far apart, the bubble 
behaved very much as if they were not there at all, if they were fairly close together, the bubble divided 
just before the minimum Into two equal halves, one of which moved rapidly towards each plate, and flattened 
Itself against it, f.r the No. 8 cap, the critical distance between the plates was f.und to Vie between 
and IB Inches, it was decided to investigate this phanomemn mathcnetlcal 1y, as It seemed tc afford a 
very good opportunity of making a fairly stringent check on the whole tfieory of the distortion of the bubble, 

we consider a bubble with its centre mid-way between two vertical rigid .surfaces, we neglect the 
effect of gravity and of the surface of the water. a motion of a few inches paralle.1 to the plates would 

not affect otion in a perpendicular direction very much. (At the time this theory was developed it was 
Imagined that the effect of gravity In distorting the bubble would not be very large, .nt nny rate until the 
first minimum is reached. as stated above, this is net quite true, but It Is sufficiently nearly true for 
the present Investigation to be valid). The method used is a modification of the original image theory 
due to Herring, we use spherical polar co-ordinates, nnd take, fs the & axis the line through the ce.ntre of 
the bubble perpendicular to the plates, we take the profile of the bubble to bo:- 

» ■ a + bj (cos 5) ♦ b^ pjj(cosC') (i) 

The odd harmonics win be absent owing to symmetry considerations. 


Allowing for possible departures from the spherical shape, we assume the velocity potential expanded 
as a series of axial harmonics. in order that the ooundary conditions at the rigid surfaces should be 
aattsfled, we introduce an Infinite series of inages exactly equivalent to the compound source that represents 
the bubble In an Infinite sea. If this proceeding were pentilssible, the velocity potential would take 
the ferm:- 


<r> 



m 

s 


n»2 


i 

(r^ ♦ n^d^ + 2nd r Cos 0 )^ 


1 

(r^ ♦ n^d^ - 2nd r Cos ^)^ 


+ similar series for Aj, Ajj etc. 


( 2 ) 


Where 


502 


- 8 - 


Where !d Is the distance between the surfstes. 

Such a velocity potential satisfiel Laplace's equation and the bnvndsry conditions at both surfaces, 
but 1$ not physically pennissible because It diverges. This difficulty was pi^intbd out by Herring, it 
can, however, be overcome as follows:- 

if we consider the region in which r < d, we may transform the image terms (involving spherical 
harmonics about the origins ± n d) to series of spherics) harmonics about the centre of thi> bubble, we 
then get, by well-known forraulae:- 


J!o * (cos g) f^o £ f -1^1. 


t similar series. 

A A 

we now strike out the terms which vary with the time only, such as -S- , — ( etc., as we can always 
subtract any term depending on the time only from a velocity potential without altering Its physical meaning, 
in particular, the new velocity potential will now be finite everywhere (owing to the disappearance of the 
terms making up the series £ ^ { and will still satisfy both Laplace's aquation, and the boundary condition 
at the rigid surfaces. This velocity potential must thoreforo be the appropriate one for our problem, 
we treat this velocity potential by writing down the conditions that the pressure and normal vetoclty should 
be continuous at the surface of the bubble Rosa b^Pj {cos6) * ...... 


The final form of the velocity potential Is 
^ A„ A,P, (cos6i) 2A„r^P (COS0) oo (i) (j) 

^ ■ JS- ♦ * % u ♦ —2 -f-- 2 — T 4 O-^ (2) 

r P ? 2 („3j ij, 

we notice that the perturbing term is of the order of''^ , so will be fairly sensitive to the 
distance apart of the plates. if we regard-^ and b, as small quantities, and neglect their powers and 
products, we get:- 

5 , ,1A- . 608 a 

, m • b ♦ 2a b, ♦ — f aP a (*) 

a ' dP 

From the equation for continuity of normal velocity. Substituting these values in the exprassion 
for the pressure, we obtain final lys- 

. ? * 9 

rg - g* • a a t a' (6) 


(we have negleetod the higher order perturbing terms). 


Bj + 34 bj - V bj + 


2.02 d (a* da 


Mhere Pg Is the pressure in the bubble, and eg; the total pressure In the water outside. Equation 
(9) Is of exactly the same form at the equation for a charge In an infinite sea, showing that, to a first 
approximation, neither the period nor the energy of the bubble will be affected by the pretence of the two 
rigid eurfaces. (This result is due to the .bssnee of terms lirvolvlng Legendre co.^fflclerits of odd order), 
we know that the effect of a single rigid surface Is both to attract the bubble and te Ineraase the period, 
whereas the introduction of a second rigid surface seems to cancel both effects. indeed, we see from 
llsyt*r<ant Campbell's results that for a bubble 12 inches from a single rigid surface the first oscillation 
takes 26 milliseconds, whereas for a bubble midway between two rigid surfaces 2* Inches apart, the first 
oscillation takes only 23 milliseconds in agreement both with theory and with the period of a bubble rlg.bt 
away from rigid surfaces. 


Equations ...... 




I 


Equations (5) *nd (6) have b«en integrated by the nautical Almanac office for a i gram charge of 
T.n.T. between rigid walls 18 inches and 12 inches apart. It Is found that b^ first rises to a positive 
maximum then diminishes and changes sign and becomes mmerically large. A positive value of becoming 
comparable with the radius of the bubble would Imply that It splits in half. The results can only be 
qualitatively correct, both because for such small values of d we cannot neglect the higher order 
perturbing terms, and also because ,i No. 8 cap seems to correspond more closely to a 0.6 gran charge than 
to a i gram charge. 'Some specimen results are as folfowss- 

Table III. Distor tion of Nubble due to riald walls alone compared with that du e to gravity a lone. 

I gram of T.N.T. 3 feet below surface. 


Tima (Mill i- 
seconda) 


4.9 I 9.T 1S.4 t 20.9 25.2 27.7 28.3 28. « 28.7 28.8 


Mean Radius 
of Bubble 
( i nchea) 


4.78 5.83 d.ll 5.54 4.27 2.6} 1.80 1.03 


bj (inches) 
due to* 
gravity alone 

bj (Inches) 
due to rigid 
walls i8* 
apart 


.OOS .013 


.005 .026 .074 .309 Breakdown 


+ + ♦ + - Break- 

.180 . 272 . 282 .190 .182 down 


bj (inches) 
due to rigid 
walls 12" 
apart 


.024 .048 .037 .304 


.647 .937 .948 .591 


Breakdown 


Ekiwrlmental facts. Period of Bubble In open water 23 milliseconds. 

Between plates IB’ apart period “ 22 - 23 mllllsoconds. Bubble does not divide. 


Between plates 12» apart bubble divides after 21 milliseconds. 


Maximum radius of bubble 5.1“ 


As already stated, the discrepancies in period and maximum radius are duo to assuming too large a 
value for the energy of the no. b caps. It wIT' be seen that for the plates 12* apart, b^ becones positive 
and easparabla with a, corrsspording to splitting of the bubble, ens half going to each plats, wtisreas for 
the plntos is* apart, b, eventually becomes naaative and comparable with a, corresponding to a flattening 
of the bubble along the plane midway between the plates. In the actual experiment gravity acts In a 
direction perpendicular to the line we have taken as the 0 axis, so that, by Itself, it would tend to flatten 
the bubble along a horieontal plane perpendicular to the plates, and would thus tend to assist the splitting 
effect due to the plates. By the use of tes'seral Instfiad of axial harmonics one could take account of the 
effect of gravity and the rigid walls simultaneously, but this refinemont saems hardly worth while, as we 
have already shown that the theory of this phenomena Is in qualitative agreement with experiment. 

Conclusions, 


The comparison of phenomena on the dstonator scale with experiment reveals very much the same sort 
of situation as axista for large charges. The simple theory, assunlng that the bubble remains spherical 


throughout 


1 



504 - 

throughout, enables one to predict with confidence such quantities as the maxiisum radius, period of 
oscillation, and the trajectory of the centre of the Dubble up to first minimum. Owing to a variety of 
causes, such as flattening of the bubble and dissipation of energy in various ways, phenomena which depends 
on the behaviour of the bubble near its minimum radius, such as the actual minimum volume and the pressure- 
time curve due to the oscillations of the bubble, cannot be s.atisf/ictorily predicted theoretical ly. 





June 1944 




505 


PHOTOGRAPHIC MEASUREMENTS OF THE SIZE, SHAPE AND 
MOVEMENT OF THE BUBBLE PRODUCED BY 1-OZ. CHARGES 
OF POLAR AMMON GELIGNITE DETONATED UNDERWATER 
AT A DEPTH OF 3 FEET 

K, a. Brycnt 

Jun« 1944 

• it******* 


Suwniarv . 

HusyrMwnts «f th« «lie, shspe and iravimant of tho bubble produced by i oz. of polar «mon 
geltgnltt datonated at a. depth of j feet have bean obtained photos raphi cal 1y> The maaaurenenta cover 
the firat oaclllatlon In detail and a nutrOer of typical photoprepha la Included. Tho results are 
compared with caleulatloni baaed on the theory of professor G. i. Taylor, In which the effect of the free 
surface It alco canal darod. 

The theoretical radlua-tlme curve agrees quite well with the observed wan radius up to the first 
oaclllatlon, although owing to large onorgy loasas the mlnlnum radlua observed Is somewhat larger than 
the predicted value. The downward velocity of the bubble la of the laiw order as that calculated 
except very near the mlnumurnwhen the very .high calculated values are not attalnid. The bubple conmences 
to flat tan much earlier than according to the theory of Temparley, and at the minimum la roughly 
hemispherical with a slightly concave upper surface and a spherical lower surface, like an Inverted 
mushroom. 

The failure of the bubble to attain the predicted maximum velocity is attributed, at least in 
part, to the flattening of the bubble. It la shown that tna vertical monientum In the water Is 
approximstely conserved If the effective mass of water moving with the bubble 1$ taken to be a half of 
that dlsplneed by a sphere wtiuso radius equals the greatest horizontal radius of the actual bubble. 

Introduction . 

The purpoat of the teats described here was to extend the single flash photographic technique 
prcvlcusly used with detonstors to l-o;> Chargee, it was desired to compare the behaviour of- the i°oz. 
bubble with thocretlccl predictions and to previds data which would assist In Interpreting experiments 
on damago produced by t-oz. charges. 

B*^»rintntal Method . 


Single photographs of the bubble taken on stationary film were obtained at a series of times 
throughout the history of the bubble. The bubble appeared silhouetted against a white painted metal 
5«fU-Ctor which was illumined by a very bright flash of short duration. The flash was produced by 
detonating a z feet length of Cordtex In a glass tube filled with argon. 

Shsn photographs near the minimum were required ths light flash detonator was fired by an 
electronic switch operated by the pressure pulse produced by the collapsing bubble, suitable time delays 
being obtained electrically when desired. For other photographs a rotary switch with adjustable pre- 
set contacts fired the main charge, tripped the camera shutter, and fired the light flash at any desired 
time intervals. in all cases a plezl-electrlc pressure gauge, In the water near the bubble gave a 
record cn a cathode-ray oscillograph from which the timing of the light flash tngUl be measured wit'h 
an accuracy of about 0.05 milliseconds. All experiments were carried out after dark and the shutter 


renained 


308 


* z - 


reifainetl open ?or about i second. 
Loeatjon of the Charge . 


The eharje wa5 suspendsd, datonator undsrnsath, at a depth ef j fiot in a total deoth of water 
of 9 feet 6 inches. in most cases a rigid steel rod rrameworlt was fixed in the .plane of the charge; 
this served as a reference framework in the photographs giving both a length scale and a fixed reference 
point from which the movement of the bubble could be determined as well as Its size and shape. 

in order to obtain a satisfactorily large Image of the bubble when near Its minimum, it was 
necessary for the camera to be about s feet from the charge. At this distance tho field of view was 
a circle of about JO Inches diameter; accordingly the reference framework had to be woll within the 
voluffis occupied by the bubble when large. The framework was therefore made of small enough section 
to produce a negligible disturbance of the bubble motion. a number of photographs were taken with no 
framewrh present, and the bubble shape and sl 20 wore exactly the same as when a framework was used. 

For photographs of the bubble near its maximum size the camera was 8 feet away, and the field 
of view about u feet diamutor so that a irore rigid framework with Its mentors further from the centre 
of the bubble could be used. 

itet hod of Me asurin^ the Phatographs . 

(a) V pluTie of the Bubble ;- it was assumed that, apart from sf«ll protuberances and necdle- 
like projections, the bubble was symmetrical about a vertical axis through Its centre. The horizontal 
diameter of the profile of the bubble was measured at a number of equally spaced levels along this axis 
of symmetry - any small asymetrical bumps on the profile being ignored as containing a negligible 
volume. The squares of these diameters wore then summed, the Intervals being small enough to justify 
this. 


It will be observed later that when tha bubble Is very close to its minimum Its upper surface 
appears to be actually concave. This ’mushrooming* of the bubble has been observed by Taylor with his 
'micro-scale* spark bubbles*. The profile of the bubble no longer coincides with the cross-section - 
the profile shows only the rim of the “saucer'-llke depression at the top of the bubble. However, the 
bubble Is not completely opaque and it Is thought that tho upper boundary of the ’splash* of light coming 
through the centre of the bubble, which has been seen on some previous photographs, marks the lower edge 
of this hollow deprejelon in the bubble. Hence an a proximate shape to the cross-section of the bubble 
may be sketched in and its volume enmputed. 

The apparent diameter of a sphere viewed at a finitm distance is always less than Its true 
diameter. This difference can be allowed for quite simply from geometrical considerations and in the 
present experiments only amounts to a for the largest bubble diameters. 

(0) The PBtltion of the “Centre ■ «.' Gravity of the Bubble i- The position of the "centre of 
gravity" of the volume occupied by the bubble was calculated by taking moments of the squares of the 
horizontal diameters about any convenient horizontal line. The chief error Involved in determining 
the total displacement of the bubble lies in uncertainty as to the absolute position, relative to other 
objects In the photograph, of the point where the charge was detonated. Errors In setting the depth 
of the charge were of the order of A inch and a random error of this order Is Inherent In every 
measurement of the displacement of the bubble. 


(t) 


The motion and shape cf the hollow produced by an explosion In a liquid. 
G I, Taylcr and R. m. Davies. 


" 3 - 


507 


(c) Thg Shaoo Co-efficlents of the Bubat e;- if R be the raflius vector from a point on the 
axis of a bubble to its surface, then a bubble of any given shape (aasuming axial symnetry) nay be 
rspresonted by s,n InfinUe set of “shape co-efficients" as fo'iows:- 

R • a ♦ bj^fj (cosfl ) ♦ bjpj (cos 0) ♦ etc. 

where b^bj-etc, are here called the "shape co-off Iclents" 

6 Is the angle between the radius vector and the axis of syimetry; d « o Is taken downward. 

Pn Is the Legendre polynomial of the nth order. 

To a first order In small quantities, o, will be zero if the origin of the co-ordinates Is 
taken at the cantre of gravity. since it vas desired to compare the observed shape co-ofriclents with 
those calculated theoretically In wnicn the origin Is usually Chosen so as to make h. always zero, the 
centre of gravity was chosen as the origin for measurements. 

The method of measuring the co-efficients a, b^, b^ etc, for a given bubble outline is considered 
In detail In an appendix. Briefly, tho method assumes that no co-efficients higher than b^ are necessary 
to express tho shape obsorved. Radii vsctores arc measured for seven equally spaced values of 0,.and the 
resulting simultaneous equations are solved tor the seven co-utf Iclents. it Is sufficient justification 
of tho method, here, to say that when the f-jll outline given by the seven co-eftlclents thus calculated 
Is drawn out It agrees with the observed outline with an error less than the uncertainty In delineating 
the.observed outline of the bubble, even for extreme cases so far found. 

(d) Accuracy of Measurement !- Length measurements can be made on the photographs within about 
2 or 3 », but absolute values of the position of the bubble are probably only accurate to within about 

i Inch, Values of the velocity of displacement of the oubble are therefore subject to rather large 
Inaccuracies, though this Is somewhat reduced by drawing a smooth curve thrcugl'. a .'.umber of observations. 

Exberimentat Results , 

The experimental results are plotted in figures i to 5. The timing of all observations near 
a minimum was maesured in milliseconds before or after the minimum rather than in milliseconds after 
detonation since. there was considerable variation in the value of the first pe’riod. The ptiiod varied 
between 74 and 80 milliseconds with an average of 77 milliseconds, and this value ha.s been used in 
platting the data in Figure l. 

The data for the second oscillation are scanty and nave been put in a table which follows. 


Table i. 


Average second 
period 

Second Minimutn 
Had i us 

Ci sploccment downward 
at s^^cond minimum 

•5» 

milliseconds 

3.8 Inches 

16.5 inches 


Description of the Photographs , 

In Figures 7. 6 and » a number of typical p.hotographs of the bubble are given in each of which 
the bubble appea.'s silhouetted against a white reflector. The explosive, being of a type which gives 
rise to no free carbon in the explosion products, resulted in a bobble which was fairly transparent so 
that in most photographs a splash of light appeared at tne centre of the bubble where the gns-yoter 


i nterfac 


508 


.. u _ 


Intsrfics was approximately normal to the light path between source and camera. in figure 7a, 
however, It was necessary to open the earner* shutter before firing the charge so that the Image of the 
dstonatsd charge Is found as a very bright patch at the centre of the bubble. 

The bubble remains vary nearly spherical, apert from small excrescences on Its surface, until about 
10 mllllssconds before the first rolnimufli (Figure 7d)= as the bubble collapses further It becomes 
flattened, but In such a way that the lower half of the bubble remains very roughly hemispherical 
(Figures 7e, f, Ba, b, c). in Figure ab the bubble Is nearly hemispherical with a flat tcp. while In 
ee the flat top has become concave so that the bubble resembles'en inverted mushroom. During the 
later stages of collapse the needle-like projections have boeome more ecesntuatsd relative to the 
bubble as a whole, and appear to be slightly more merked on the lower surface. The shape of the 

bubble at the minimum Is somowhat obscure, perhaps owing to the relatively large duration of the 
light flash - about idO microseconds. The f.aymmntrlcal lobe on the left of the bubble In Figures 8d 

and e Is probably due to e perturbation of the bubble when large since In these cases the dharge wis fired 
te the left of the centre of the framework. The framework In these was feletlvely stiff, and was made 
of channel iron: the bubble whon large enveloped the left hand member of the framework. in all 
other photographs at the minimum no asynmetry was found. 

That the bubble is moving downwards may be Inferred from the shape of the horizontal cross 
wires nearest to the bubble seen in Figures sd and e. These wires were terminated by stretched springs 
designed to keep them taut when the inwsrd rush of water pilled In the sides of the framework. In 
practice the transverse pressure exerted cn the wires by the muvlng water displaced them very 
considerably. The actual shape of the wires Is determined by the past history of the water velocity 
In Its neighbourhood, and it is clear from thair shape in Figures sd and e that the motion has been 
mainly radially Inward, but with a velocity vertically downwards superimposed on this flow in the 
Immediate neighbourhood of the bubble. 

In all the photographs near the minimum there appears to be e region just abeva the bubble 
where the water scatters the light rather than transmitting It In the normal manner. |n Figure Be, for 
example, the Intenaely bright line of light produced by the detonated cordtex appears to be Interrupted 
by this region; the wire In Figure Be becomes almost, it not quite. Invisible when passing through this 
region. it U suggested that this Is evidence of a 'wake* in the rear of the moving bubble. Perhaps 
this region Is filled with many small eddies and fine bubbles. 

After the mlnlmun-the bubble expands somewhat Irregularly - Figures 8f, »a, b and c, its 
lower surtace'shows small protuberances, but not the long needles which are such a prominent feature 
of the tower surface before the minimum. The upper surface of the bubble, however, shews a very 
marked array of approximately radial streamers or ’whiskers* which appear to be long trials of bubbles. 

The length of these "whiskers' Is of considerable Interest; in Figure 9a they occupy water on either 
side of the bubble which In Figure so was quite clear and free from such bubbles. It seems somewhat 
unlikely that these 'whiskers* have actually been pushed out so far In advance of the main surface of 
the bubble. These streamers (nay be composed of fine bubbles of gas, actually left behind in.iiie water 
by the collapsing bubble which were compressed and rendered pfacticeliy Invisible by the very high 
pressure existing In the water round the bubble when near Its minimum. 

The curious radial streaks of light In Figure* 7e are thought to have nothing to do with the 
bubble. It Is thought that they are produced by small white hot particles perhaps pieces of aluminium 
from the detonator — projected outward at detoiiatier,. Although the Sulerioid which operated the camera 
shutter was not energized until some iS milliseconds after detonation. It Is probable that In Inis case 
the Impact of the shockwave operated the rather delicately sot shutter trigger. This would account 
for the absence of the actual flash of the detonating charge, and tor the absence of any streaks near 
the centre of detonation. The Increased Intensity of ti.eso t'.reaks towards the end of their path would 
be due to the Increase of light-gathering power of the lens as the shutter opened. This same phenomenon 

has been observed on several occasions when the timing of the main light flash and thus of the actual 
bubble photograph differed very considerably. 


Otscusslon .... 


- 5 - 


5G9 


Discussion - Coin^a^t sow u>ith Theory . 

(a) Radius - Tiitie Curves ;- in Figures l and 2 the irean radius of the bubble - defined as 
the radius of a sphere having the same volume as the bubble - is plotted against time. For comparison 
with experimental data, values of the radius as calculated by the nautical Almahac Office have been 
plotted In Figures 1 and 2, These calculations refer to a non-dimensional depth •> 7.2, the bubble 
being assumed to remain spherical, and account has been taken of the presence of a free surface 0.6 
units above the charge. These correspond to a l-oz. charge with a bubble energy of uuo caiorles/gm. 
at a depth of 3 feet. ' The standard length for this case is 5 feet and the standard time o.jpa seconds 
yielding a value of 77 mllvlseconds for the period. This period is so close to the observed average 
period that the theoretical results have bean plotted without any adjustments of scale. 

It appears from Figure 1 that there is a discrepancy somewhoro in that the periods agree while 
the observed maxlrrom radius exceeds the theoretical value by about as. if the energy of tne motion be 
calculated from the pbsorved maximum radius, (assuming that the fraction of the energy left in the gas, 
amounting to about 15S, is the same as In the theoretical case) n figure of S50 calorics/gm. is obtained. 
The energy calculated from the period assuming the correctness of the theory, Is only uuo calories/gm, 

In assessing tne cause of this discropancy It does not seem likely that the maximum radius 
measurement is more than 2J In error, while the measurement of the average period might be as much 
as 31 In error. These two might conceivably contribute an error of 153 In the energy, in the worst 
case where the errors act In the saite direction. 

It is possible that the theory of the effect of the free surface is somewhat in error for a bubble 
as close to the surface as this, since thu theory Is only an approximation which neglects cubes and 
higher pon/ers of the reciprocal of tne distance of the free surface. According to the approximate 
theory used the presence of the free surface reduces the period from 86 to 77 milliseconds In this case; 
a large correction which if itself In error might account fer some of the discrepancy. It has also to 
be remembered that tho effect of the rigid bottom at a distance of 6 feet 6 Inches from the bubble has' 
been neglected In the theoretical calculations. This, however, would raise the theoretical period to 
about 81 milliseconds and Increase tho size of the discrepancy. 

It is further just possiblo that there is a considerable energy loss during the contraction 
stage, Though most of this would have to occur while the bubble is large In order to reduce the period 
to such an extent. it Is estimated from the ratio of the first to the second period that about 603 
of the bubble energy Is lost between the first and second oscillation, but It seems.more likely that 
most of this loss occurs near the minimum when the bubble is iroving iiiost rapidly in a vertical direction. 
The data plotted in Figure 2 bring out the quite sharp energy loss as lixJIcated by the much slower 
expansion after the minimum, than contraction before the minimum. 

(b) The Hiniwum-Radlus-. - Energy Considerations:- According to the theory of a spherical 
bubble tiha mlnliTitm radius should be 2,0j inches when the energy in the gas is Eli of the total, the 
remaining 19% being kinetic energy In the vertical motion of the water. The observed mean radius at 
the minimum, based on the volume. Is 2,7 Inches, This puts the energy In the gas at 643, assuming 
that the gas adiabatic Is substantially that of the theory, (f one assumes that the effective volume 
of vwiter moving with the bubble Is half the volume displaced (as is true for a spherical bubble and 
streamline motion) the kinetic energy ol vertical motion w.-rhs out at 1,93 of the total - taking the 
observed maximum velocity as about 18S ft. /sec. 

However, the shape of the bubble at the minimum is more nearly hemispnerical with a radius of 
3.9 Inches (see Figure 8d), |f one thus assumes that the effective volume of the water moving vertically 
Is one half that displaced by a sphere of radius 3.9 Inches the kinetic energy at the minimum works out 
at 5.63 of the total. (In fact the effective upiume of water moving may be even greater since a 
hemisphere moving rapdily through water Is likely to drag with It a volume of dead water lying 
immediately behind Its flat surface in addition to the water circulating round it). This brings the 
energy In these two phases to about 703 of the total. It is not unreasonable to suppose, therefore, 
that of the observed loss of energy between the two cycles of about 603 (based on the ratio of the 
first and second periods) half Is lost just before the minimum, the other half just after the minimun; 

When the vertical velocity Is high. 


In a ..... 


510 




in a recent paper 6. t. Taylor and oavlee d nave ano»n that a ameU bubble r.tilng^ ateadlly 
through water tenda to as<in« the shape of tha cap of a aphare wi th a turbulent wake tn Ita rear,, and 
with a drag coHsftlelent of betwadn 0,5 and i,j. using, thus,, a drag eo-efflelant of 1 the energy 
loss by turbulence, of tha ih};, explosion bubble has been calculated up to the first minimum. The 
observed value of the velocity la given In I'lgure s, and the. moiliiui horizontal radlua of4ha bubble 
in Table I below. It Is found that 621 of the energy lota occurs In the last i lalllisedond before 
the minimum, s6> of the loa.a occurring in the last millisecond. The total loss, however, only, 
amounts to about son calories, I.e, to about 3,71 of the total bubble energy, since the velocity 
enters as the cube In this calculstion. If the actual bubble velocity were doubla the assumed figure 
for the bat half or one millisecond the energy loss would bo about 2SS of the total bubble energy, 

II is also possible that the drag co-offlelent of the bubble Is somewhat higher than unity, perhaps 
owing to the needle-lIke projections from Its surface. 

(e) Veloelty-TIme Curve . Momentum considerations:- an attempt has boon mage to estimate 
the velocity of the bubble from the displacement-time curve. Figure 3 . As was remarked above, this 
velocity data is only very approwlmete. The estimated value has been plotted in Figure $, together 
with the theoretical curve. 

The failure of the bubble to reach anything Ilka the theoretical maxihium velocity is very 
probably due to turbulence, but also Itv part to the change of shape of the bubble. in Table 2 are 
tabulated fur a few times the observed values of the velocity u, the mean radius a, and the nexlmum 
horizontal radius a^ of the actual bubble. The vertical momentun factor 2 m « Ua^ has been tabulated 
In columns s and 6 calculated In two ways. tn tha fifth column the effective volume of water moving 
witii the bubble Is assumed to be half that 'actually displaced by the bubble; In the sixth column the 
voluns of water moving Is assumed to be half that displaced by a sphere uf radius equal to the greatest 
horizontal radius a^ of the bubble. 

It will be seen that In the first case the calculated momanium of the water decreases rapidly 
as the mtntnum is approached, whereas In the second method of calculation (I.e, assuming the volyiM) of 
water moving is half that displaced by a sphere of radius equal to the greatest horizontal radius of 
the bubble) the momentum appears to remain more nearly constant, as In fact it should. 


TABLE 2 


Vertical Velocity and Momentum Data. 


Time msecs. 

Vertical 

Mean 

Maximum 

Vertical 

Momentum 

before minimum 

velocity 

Rad 1 us 

Horizontal 

Factor ^ . 


U 

a 

Red i us 
.1 

Ua^ 

U(a*)^ 


(ft. /sec.) 

(ft.) 

(ft.) 



2.60 

33 

0.635 

0.675 

8.5 

10,2 

1.9K 

Ul 

0.555 

0.595 

7.0 

8.6 

i.3d 

56 

0.477 

0,574 

6.1 

10,5 

O.f/T 

87 

0.406 

0.466 

5.8 

8.8 

0.50 

lUO 

0.335 

0.440 


11>9 

0 

%B5 

0.225 

0.325 

2.0 

6.3 


(d) 

d The Rate of Rise of Large volumes of Gas In water. G. 1 . Taylor and R. m. Davies. 



511 


- •/ - 

(d) Shaw Chanfl« of the Butiblei ~ in Figure u the second harmo"ie shape eo-«ftleient 
fcs^i' ninttsd «* a fraction of the co-efftclent of zero order a up to just bafors thu minimum. The 
following table includes figures for the Higher order co-etficlonta also. It will be seen that b^ 
is always small; the snail shift of origin which would have to be made to make it vanish exactly 
should not affect the other co-efficients appreciably. 

TA8I.E 3. 


Harmonic shape Co-efflelenta during Contracting phase. 


Mill iseconds 
before first 
minimum 

a 

(tn.) 




'a 

9.S9 

1J.85 

♦ .025 

-.025 

♦ .014 

-.019 

5.6t 

U.2 

♦ .004 

-.057 

♦ .036 

-.032 

2.60 

7.5 

-.080 

-.136 

♦ .097 

+.002 

i,9>i 

6.58 

♦ D20 

-.182 

♦ .JIO 

0 

1.3S 

6.13 

».0»7 

-.220 

165 

♦ ,006 

0.8T 

U.69 

♦ .008 

-.3*5 

<•-147 

+ .068 

O.SO 

4.30 

♦ .065 

-.423 

+ .300 

-.068 

0.50 

u.to 

-.093 

-.465 

♦ .488 

-*l5l. 


For comparison with the observed measurements of the second harmonic shape co-efficlsnt h^. 
the value calculeted by the Hauticai aVnenee office using Temperley's equation has been plotted In 
Figure S. Temperlo^i's equation contains various time derivatives of the first co-efficient a, and 
the linear velocity u of the bubble. These values were taken from the earlier calculation in which 
the bubble was assumed to remain spherical; I.e. it was assumed that a and u are not perturbed by 
the growth of the higher shape co-efficients b^ etc. 

It will be seen that though the general form of the b . versus time curve is roughly the same 
experimentally as. is given by the N.n.O. calculation, b^^^ attains appreciable magnitude much earlier 
than In the theory. It seemed possible that this was- due to actual linear velocity u being greater 
than the theoretical one in the early stages of collapse of the bubble. Accordingly Temperley’s 
equation was reintegrated, using the observed value of the velocity u together with the theoretical 
values of the mean radius a and Its derivatives. This result has also been plotted in Figure s. 

It will be seen that this curve is now somewhat nearer the observed curve, but there Is still a 
considerable' difference. it Is th'mght that most of this discrepancy could be due to error in the 
veloeity-tlnle curve used. a final check of the correctness of Temperley’s equation must theroforo 
awi.ii a more accurate determination of the velocity-time Curve for the bubble. 

Cone lus t 3WS. 


(t) During most of the first oscillation the bubble remains very nearly spherical end 
agrees reasonably well with Taylor's theory when allowance Is made for the presence of the free surface. 

(2) During the last five milliseconds before the first ralnlmum the bubble flattens, its 
vertical diameter shortening, and the upper surface becomes flatter while the lower surface remains 
approximately hemispherical, close to the minimum the upper surface becomes concave and the bubble 
rasombleg an inverted mushroom. 


( 3 ) 



(3) The sign of ths shape change agrees with that predicted by Temperley's theory, but the 
change takes place considerably earlier than In the iheory. A reintegration of Temperley's equation 
using the observed velocity-time curve reduces this discrepancy sonwwhat, 

(а) The downward displacement agrees reasonably well with the theory, but is somewhat less 
at the first minimum than the predicted value. The observed maximum velocity of 185 ft. /sec. falls 
considerably short of the theoretical value of 890 ft. /sec. 

(5) This failure to reach the high velocity predicted may be partly due to turbulence, but 
may also be due to the flattening of the bubble. 

(б) The vertical irementum in the water appears to be approximately conserved up to the 
minimum If It is assumed that the effective mass of water moving with the bubble Is one half that 
displaced by a sphere of radius equal to the iraxlmom horizontal radius of the flattened bubble. 

(7) The vertical kinetic energy In the water at the minimum is estimated to be about 61 
of the total energy of the motion, In contrast to the theoretical value of 191. 

(8) The energy In the gas at the minimum is estlneted at 641 of the total Initial energy 
of the notion, 

(9) There is an Indication of a “wake* or region of disturbed water in the rear of the 
downward moving bubble when near its minimum size. 

(10) The bubble remains coherent for at least two complete oscillations. 


Appendix 


S13 





( 



- 1 . 



•I 






APPENDIX 

Harmo ni c Anal'ysis.of Bubble Shai>es . 

Ths silhouette photographs of the bubele give profiles which in general have the shape of a 
cross-section through the axis of syimietry, it was desired to deterrtilne the Co-eff Iclents In the 
series. 


» * “ * * 

in which s is the radius vector to the surface of the bubble from a given origin on the axis, 0 the 
angle Included between the radius vector and the axis. 


AS a first attempt it was decided to determine that curve in which only the first seven 

co-efficients were non-zero, and which fitted the observed section at seven equally spaced values of 

the angled, viz. o®, 30®, fiO®, 90®, 120®, 150® and 180®. The radii vectors R. R, at these 

o o 

angles were measured giving seven simultaneous equations, of which a typical one ts:- 
9, = a ♦ bjP^ (cos e„) + bjPj (cos f + b^Pj (cos 


0 . 


mr/6 


( 2 ) 


These aquations wore solved once and tor all tor the seven co-efficients in terms of the radii 

R„ to R., Tha solution is given here for reference. 

0 0 


a ” .034 r^j + ,243 + .w * .266 r^ 


.085 r^ + .660 r^ + .687 r^ 


.152 r^ + .776 r^ - ,267 r ^ - .661 r^ 


.237 + .513 fj - 1.364 r^ 


.442 r^ - .274 r^ - .945 r^ + .777 r^ 
.677 - 1,173 fj + .677 r^ 


.■372 r^j --.745 r^ +. .745 r^ - .372 r^ 


(» 


where r„ 


7 R. 


n . "l * '*5 
1 — — 






R - R. 
r. • 

® 2 


Rj - 


A fairly extreme example Is illustrated In Figure 6. The full line Is the oOserved cross- 
seetlor, of a bubble, the broben line is the curve given by the seven eu-eff icients as calculated by the 
above method. The agreement between the two curves is well within the limits imposed by the nctual 
photographs. The above method which is the simplest that could be devised, was accordingly used 
In analysing all the photographs obtalnod. 

Analytically, ttie Ideal method of calculating any given co-effIcient in the series (j) soy 
0^ Is to use the known relation 

b .io-l-i R (cos 0). d (cos 0) (4) 

" 2 6 • 0 

To USi^awse* 







l.< 


514 


- 2 - 


10 use tnis B has to Do measured at a nuriDer ot angles giving equally spaced values of cos (9., 
and the product RP^ (cos tf.) then integrated numerically. This method has the advantage that any 
r.' 0 -s(ficlent Is given without assuming that any of the others are zero. The main disadvantage Is 
that because of the fluctuotlons of (cos over the range the numOer of radii which tnust Do 

measured to give any reasonable accuracy in the nunoricai Integration is rather large. This point 
Is brought out in the following table in which the result of analysing the shape snown in Figure S 
by the two methods Is compared. Simpson’s rule was used for the numerical Integration, and g, n 
and 21 ordinates were measured. 


comparison of Methods of Shaae analysis 


Co-eff icient 

Method 

Equation (3) 
7 Rad i i 

Equation (a) 

9 Radii 

11 Radi 1 

21 RaOl 1 

a 

+ U.30 

♦ U.2tt 

+ U.2U 

* U.27 

‘■i 

* 0.2P 

♦ O.^tt 

♦ 0.35 

♦ 0.30 

•>2 

- 1.82 

- 1.70 

- 1.76 

- 1.73 


+ i.ns 

+ 1.00 

+ 1.13 

+ 1.17 


T 0.2S 

- 0.02 

** oao 

> 0.23 


- 

- i);28 

- 0.51 

- o.uo 

.. 

+ 0.51 

+ 0.85 

♦ 0.9 

♦ 0.56 


The table shows that the 7 ordinate method using equation (j) is quite adequate, at least 
up to 0^, and clearly involves only a third of the labour of the metfod using equation (u) and 2 i 
radii vectores. 



) 












I 

.1 



517 


9 


. VERTICAL DISPLACEMENT OF CENTRE OF GRAVITY OF 3'JBBLE PRODUCED BY ioz RAG 
DETONATED AT 3 ft. DEPTH 


9 0 



. RA.G. AT 3ft. DEPTH 


280 




velocity COWNWARDS of bubble from Ioz. PAG AT 3ft 


e«i 80 



Observed shepe — — Shepe using 7 hermonic 

coefficients only 


R»4 3040 7« F>(eosO)-l 82 P, (cos e) 
+ 1 29 P,(cos 0) “O 29 P* (QOi e) 

— 0 61 Ps (cos 0)+O'5l P, (cos 6) 


Pig. 6. COMPARISON OF OBSERVED BUBBLE SHAPE AND 
BEST FITTING CURVE USING 7 HARMONIC 
COEFFICIENTS ONLY 


- ti- - .^ItnvfiiC,. .. -it 






/ 


521 



MILLI&ECONOS AFTER ^b)4l SO MILLISECONDS AFTER 

DETONATION DETONATION 



f7e1 14 JO MILLISECONDS BEFORE ;7o)9 90 MILLISECONDS BEFORE 

^ FIRST MINIMUM FIRST MINIMUM 





I 


522 



(•«) 1-95 MILLISECONDS BEFORE (8 b) O 8S MILLISECONDS BEFORE 

FIRST MINIMUM FIRST MINIMUM 



Cb c) O JO MILLISECONDS BEFORE (8 d) AT FIRST MINIMUM 
FIRST MINIMUM 




/8t O a MILLISECONDS AFTER 
FIRST MINIMUM 

O BO milliseconds AFTER 


FIRST minimum 









/ 


523 



(9d'i Mb MlLLISecONOS AFUR •'. ! L. 

nWM MINIMUM M ’^-M 



(9c) 10' 60 MILLISECONDS AFTER (9<l) 0 15 MILLISECONDS AFTER 

FIRST MINIMUM SECOND MINIMUM 




A TECHNIQUE FOR MULTIFLASH PHOTOGRAPHY OF UNDERWATER 
EXPLOSION PHENOMENA 


A. R. Bryant and K. J. Bobin 
Road Research Laboratory, London 


British Contribution 


August 1 945 


525 


A TECHNIQUE TOR HUITI FLASH 
EKPLOSION 


PHOTOGRAPHY OF UNDERWATER 
PHENOME NA 


Ad R. Bryaitt and K. J. Bobi n 


August 1945 


Summary . 

i, nethod Is described tshlch enables a number of photographs to be taken at one mtttlsecond 
intervals of explosion bubbles produced by the detonation underwater of charges weighing about i oz. 

A total of forty to fifty pictures in two groups may be taken, the two groups being usually timed to 
cover the, period when the bubble is near its first and second minima respectively. 

Light from an Ardltron discharge tamp is focused by a large metal reflector placed underwater 
on to the lenses of a twin drum camera housed in a watertight casing situated at the same level as the 
charge, thus producing a bright field of light against which the bubble appears silhouetted. 

The electrical gear producir;y the discharges in the Ardltron Is housed In a laboratory near 
the experimental tank and connected by ao feet cf cable to the Ardltron which Is situated just above 
the. water. 

A description of the electrical circuits and the underwater camera Is given together with a 
set of photographs from a typical shot. 

iTtirodveiion. 

in two previous papers (i) a method has been described for obtaining single flash photographs 
of an underwater explosion bubble produced by small charges. In order to Oetermtne the behaviour of 
the bubble in the neighbourhood of targets such as the box-model, however, it was necessary to devise 
some method of taking a succession of pictures during the course of a single explosion. This note 
describes apparatus developed for this purpose. 

f 

Gene ral Arrangemen t , 

Figure .t shows dlagfaimiatically the experimental arrangements. A rptatlng ty/in-drum camera 
housed In a watertight casing, and with all controls electrically nnerated, is hntted on to a rigid 
steel framework which is Iwered by crane on to the sloping side of the experimental tank and clanged 
in position with the camera at the required depth. ^ Light from an Ardltron discharge Imp in a housing 
just above the water's surface, illuminates a large carved metal reflector and Is focused on to the 
camera lens as shown, so that the reflector appears in the photograph as a bright ground more than 
tilling the field of view. The charge and the target are suitably placed to appear silhouetted 
against this reflector. 

The Ardltron lamp is operated stroooscopicaViy by electrical equipment housed in a nearby 
laboratory, the lainp being connected to the equipment by high voltage concentric cable. Since dost 
of the bubble Shape changes and displacements occur when tiin bubble is small, and since thee are the 
times when the bubble causes damage, it was decided to restrict photographs to two periods near the 
first and second minima respectively. The apparatus has therefore been arranged to take a total 
of forty to fifty pictures In two groups, one on each conera drum, the pictures in each group being 
spread at 1 miflisecund intervals. 


HuUifiashing 


$26 


- 2 - 


Multi flashing Sijuipment . 

Th$ (lUcirical equlpnient for producing the light flashes in the arditron tamp U housed in the 
iaboratory for reasons of bulk and general convenience of servicing. This has necessitated the use 
of a transmission line St feet long to connect the equipment to the tamp. It has been necesssry to 
devise an etectricat circuit which woutd minimise the undesirable effects on the efficient operation of 
the Arditron due to the total inductance of the transmission tine. The circuit adopted Is shown in 
Figure i and its operation is as foliows:- 

The reservoir condenser, which sorvoa as the power supply, consists of 2S t-microfarad condensers 
connected In parallel and trickle charged to s,000 vclts by a neon-sign tranafonricr and ten HtTS 
weatinghuusa rectifiers In series. 

A seven-inch diameter Tufnot disc, rotating at 3,000 r.p.m., carries 20 short lengths of U B.A. 
brass studding screwed through the disc at equally spaced intervals round a circle concentric with the 
axis of rutatlon. The disc Is nnurited so that these lengths of studding pass In succession between 
two pairs cf fixed points, and the clearance being about 6/1,000 Inch. Thus every .miMisecond 
one of the twenty studs passes between the points and Sj. These fixed points are so arranged that 
there is half a roilllsecond Interval between studs passing between theifi, I.e. If a stud Is In line with 
at a certain instant then half a billisecond later a stud will pass Sj and vice versa- 

With the relay R closed, when one of the studs passes between the points s,. the gaps at s 
break down and the Intermediate condenser, capacity o.S mterofarad. Is charged up to the voltage of the 
reservoir. The e-ohm resistance limits the current In this charging stroke. 

Half a millisecond later, when there Is no conducting path at s^. a stud passes between the 
points Sji This gap breaks down and the Intermediate condenser discharges down the concentric cable 

thus charging up the final condenser, capacity 0.2 microfarad. After a time of the order of lO to 12 
microseconds, When the final condenser Is nearly fully charged, the voltage applied to the grid of the 
Arditjron reaches Its breakdc?wn voltage and triggers the tube which breaks down and a discharge passes 
between anode and cathode. 

The min requirement of the lamp is that it should produce a high intensity light flash of short 
duration, and It Is therefore necessary that the current pulse In the lamp should be of great intensity 
and short duration. The final condenser is therefore housed In the lamp housing as close to the arditron 
as possible, thus keeping the inductance and resistance of the discharge circuit as low as passible. 

It is estimated that the peak current in this discharge is from 3,000 to 6, COO amperes, and the pulse lasts 
for sbout a microseconO. This is followed by a much longer discharge due to the Intermediate condenser 

discharging down the cable and the lamp. The 30-ohm. resistance, however, limits the current in this 
part of the discharge to a maximum value of 200 apiperes, which decreases with a time constant of the 
order of is microseconds. Thus the pulse of current in the lamp is conjectured to be somewhat as in 
Figure 3* 


By suitable adjustment of thO lens aperture and the speed of the film it is possible to arrange 
that the intensity of light tailing on the moving rUm due to iha long 'tail* of the currenUt ims curve 
shall be less than the threshold value for the film, so that only the siort peak of the light pulse 
registers photographically. Howovur, In high lights, whore the level of the Illumination is locally 
very much higher than mat over most of the picture, It Is impossible to prevont some of this later 
portion of |he discharge registering photographically. This causes a short ’trial’ of light, extenoing 
in the direction of potion of the film, on each of the high lights. Thu pictures obtained by this 
circuit a.re nevertheieps much sharper than those obtained by discliarging a condenser of the same capacity 
as the final condenser down the BO feet uf cable with the lamp at its far end. 

The rotary spark gap isolates- the ianp from the intemedlnte and reservoir condensers during 
$lie recharging stroke. This two-stage process is repeated every millisecond until the relay B Is 
opened or until the voltage of the reservuir falls below a critical value. In nurwal operation two 
groups of flashes are produced, each consisting of go t.-. 25 flasnes. and thijse groups must be timed 
relative to tne firing of the explosion with an accuracy of three or four ffihl Isnconds, so that the 
relay h must' operate quickly and repeatedly. To this end a Post office high speed relay has been 


adapted 


- 3 - 


527 





adapted, a light Tuftioi arm being attaefied to the moving armature, it BA trass studs on the iroving 
arm aftO on s fisted Tufnol arm serve as contacts. A gap of just over .|th inch in the open position has 

been found sufficient when working at 0,000 volts to prevent the train of sparks passing across the 

points of the rotary gap. The total time of operation of this relay is about 10 milliseconds, although 
the contacts are only moving for about 3 milliseconds. 

The Camera, 

The underwater camera is shos'n diagraimatlcaily in Figure m Figure 5 Is a photosiaph of the 
csimera removed from its watertight housing, it Is, in effect, two separate drum cameras mounted side 
by side. This enables a syfflelont nunbor of pictures to be taken without having an abnornaily targe 
drum. It also permits the two groups of flashes to be timed solely with regard to the phenomena to be 

observed and without regard to the relative position of the two sets cf pictures on the film. 

The two 7 Inch diameter duralumin drums A carrying 35 m.m. film .ore mounted side by side on a 
eonmen spindle and driven at speeds up tc 3,000 r.p.ra. by a belt drive B from a IJ volt car dynamo C 
used as a motor. The ball races D in which runs the drum spindle are housed in a pair of brackets 
fomsd by angle iron members £ welded rigidly to the bed plate F. This design combines lightness with 
great rigidity. 

The two lenses 0 are Taylor Hobson “Speed Panchre" lenses with an aperture of f/j and a focal 
length of. SB m.ms. Between the lens and the film a single duralumin vane H, 6 / 1,000 inch thick, acts 
as a shutter and is activated by a solenoid and lever system i. The arrangement was the simplest which 

could be put in the available space. on the application of about is volts to the solenoid the shutter 

opens In about i 8 mlHlseccnds; on removal of the voltage the shutter closes in a similar tlna. under 
present conditions of use this gives ample time for one shutter to close add the other to open In the 
Intervals between the two groups of light flashes. Approximately J 5 pictures 1 Inch in diameter may 
be obtained on each drum. 

Each lens and shutter is mounted separately on a shaped brass base J which may be moved through 
an arc of about 6 ° round tha periphery of the drin. Thus one lens may be tilted upwards while the 

other nay be tilted downards. in Figure s the right liand lens is shown tilted downards, while the 
brass dust cover has been removed to show the solenoid which operates the shutter. Thi» arrangement of 
tilting lenses enables two different fields of view, one above the other, to be covered, a feature 
which nay be used to advantage when the bubble is known to be moving either upwards or do^ymards. 

The film is hold In position by single spring clips L, The camera unit Is mads light-tight 
by a light cover k. 

The camera proper is contained In a strong watartight housing which Is’nade In two pieces. 

The bed plate F of the camera bolts on to a heavy rigid plate m which in turn Is bolted on to the 

external framework holding the camera under water. The front portion of the watertight housing N 
is welded on to the plate H. This portion has a cempartment eontalninij the motor c end e plate glass 
window 0, 1 Inch thick, is bolted between rubber gaskets to the front of N. This window has been 
found to wUhslarid the explosive effects of a 1 oz. charge detonated at distances greater than z feet 
6 Inclfies from It, The rear portion of the housing consists of the watertight cover p which la also 
bolted on to the plate H to complete the seal. 

Lam^ Hou si tte . 

The Ardltron lamp Is mounted in a large car hsadiamp reTlector In such a way that the beam 
of light is concentrated roughly Into a cone of about 50 ° semiangle. The 0.2-mlcroTarad condenser Is 
mounted just behind the reflector and the whole Is mounted In a wooden box held just above the water 
surface. It has not been found necessary to use a glass plate in the water surface to prevent 
scattering of light by ripples., 

Underuater. Reflector . 


The Cocitruciion of the curved metal reflector irj Figure i. Is ahown in greater detail In 
Figure 6 . The basic framework consists of three rows of upright angle irons a. welded to thre^ cross- 


braced 


528 


- tt - 


i 





B 


i I 




braced lonsttudinal angle Iron monbers B. To tha uprights are bolted nine pieces of I Inch thick 
plywood Cut to a calculated curvature and bevelled. Narrow strips of 16 gauge duralumin plating are 
screwed on to the plywood sections, the strips having been cut and fitted by trial. 


Ideally the reflector should form part of an ellipsoid of revotution; with the laap and 
camera at th$ two foci. This requires that the position of both la,mp and camera should always be 
fixed in relation to the reflector, and that the lamp should operate underwater. Owing to the shape 
of the experimental tank these conditions are not easily satisfied over a range of operating depths, 

AS a compromise, the present reflector has been made with sections calculated for a paraboloid of 
revolut ion with axis horizontal the lamp being operated Just above the water surface, owing to the 
finite size of the car headlamp reflector in which the srdltran Is heusid, this optical system 
produces a satisfactory bright background for a range of operating depths d.iwn to s rest, as may b« 
seen from typical photographs In Figure s. it will be seen from these photographs that the illliimtnation 
tends to be patchy; this Is due to the ontlclasSlc curvature cf the strips of duralumin which fons 
the reflecting surface. The reflector will withstand repeated sxpisslons cf a 1 uCi charge 3 or b 
feet away. 


Timing Controls . 

The firing of the two groups of sparks, the opening and closing of the two shutters in the 
underwater camera, and the firing of the charge are performed in their correct sequence by a rotary switch 
carrjring 6 brass sectors mounted in adjustable rotating ebonite discs. The period of the bubble motion 
being determined by previous oscillographic records, or by calculation, the sectors of the rotary awiteh 
are each preset to operate at the correct time, allowance being made for the known time lags in the 
spark relay and the shutter mechanisms. 

The rotating spark gap Is driven by an Induction motor working off the sd-cyele maina. as the 
motor is practically on 'no loan* its speed is always very close to 3^000 r.p.nu (this has been verified 
by a tackemeterj. The speed of the druns In the underwater camera has to be adjusted to give proper 
spacing to the photographs, and for this purpose a remote reading electrical revolution counter has been 
devized whose operation is shown dUgrammatlcally In Figures 7a and 7b. 

in Figure. 7h, B is the spindle whose rotational speed is required. The brass disc A carries 
five small squally spaced soft Irjn Inserts C which pass In turn between the poles of o permsnent magnet 
D causing a series of voltage pulses to be induced in the solenoid e. This pulse consists of a positive 
and an Identical negative portion and lasts for about one twentieth of the interval between successive 
pulses. 


Figure 7b shows tha electrical circuit used to indicate the frequency, of arrival cf the voltage 
pulses produced by the solenoid on the camera. Valves vi and V2 form a straight forward “flip flop* 
circuit in which there is normally no current flowing in the anede circuit of V2. The arrival of a 
negative pulse of sufficient magnitude from the camera solenoid causes a square pulse of current to flow 
In the anode circuit of Vj, the duration and nagnitude of this pulse being gevarrad solely by the circuit 
constants and not by the shape or magnitude of the triggering pulse. At normal dfum speeds ttw needle 
of the diilllammeter in the anode circuit of V2 cannot follow the Individual pulses it receives but gives 
e deflection equal tc the time average of the current passing through It. The deflection is therefore 
directly proportional to the frequency of the triggering pulses. The circuit shown has a response of 
1 milllamp per l,000 r.p.m. of the camera drum, linear from 500 r.p.m. up tc at least 3,500 r.p.m. 

V3 is a thyrotron and tha assoclatsd circuit produces s very short negative pulse at the 
frequency of the a.C. mains, thus enabling the overall eurrenUf requency sensitivity of the 'flip-flop* 
circuit to bo standardized just before use. 

Typical Photographs , 

In Figures 8 and 9 are shown some typical phstogrschs obtained front the explosion nf a single 
1 oz. Charge of Polar anmon gelignite at a depth of 3 feot. The Interval between pictures Is 1 millisecond. 
The group shown In Figure 6 was recorded on one drum of fllin ai^ was timad to Incluus the first bubble 
mlniffiun; tha pictures in Figure 9 include the aecord minimum. The general appearance of the bucale 

is similar 




I 


'I 




■•V 

) 






I« similar to Uiat shown by ‘single-shot* photographs on stationary 'film given In a previous paper (i) 
It Is clear from the present photographs that the bubbte Is self-luminous tor a period of the order of 
9 min Isecond. near Us first minimumi the gas at the centre of the bubble being the first to become hot 
enough te glowi it now seems probable that the patch of 1!|ht seen In the centre of the bubble in the 
•single-shot* photographs of the above mentioned paper (Figures 7d, e, f, - 89 j b, c. d, e, i, - 
9a, b, c) Is not light from the lllumlret|j>n source passing through the bubble as suggested in ttmt 
paper, but the flash produced by the self-luminous bubble at Its minimum, since in obtaining these 
photographs the shuttsr was opened for a period including the first minimum, 

Rtftrance. 

(l) Photographic measurements of the size, shape and movement of the bubble produced by i oz. 
Charges of polar amnon gelignite detonated underwater at a depth of } feet. 





arrangement of photographic gear in experimental 




BO ft. UR 34 (Concentric 10,000 V 
coble with polythene dielectric) 



FiQ.2 MULTIFLASHINC CIRCUIT 



CURRENT- Amp«r«» 




532 





TIME — Microseconds 


F19J. CONJECTURE© Ci^RENT-TlME CURVE Rj AROTTRON LAMP 
WITH CIRCUIT OF FI^^Z. 








Couc,a dura!. 



Ft9.6 CONSTf3lUCT!ON OF UNDERWATER REFLECTOR 

















I*' 

il ? 




PHOTOGRAPHIC MEASUREMENTS OF THE ATTRACTION 
OF AN UNDERWATER EXPLOSION BUBBLE TO A BOX MODEL TARGET 



A. R. Bryant 

Road Research Laboratory, London 


British Contribution 



339 


PHOTOGRAPHIC MEASUREMENTS OF THE ATTRACTION 
OF AN UNDERWATER EXPLOSION 31183 LE TO A 
BOX MOPE I TARGET 

A. 8, Bfyant 
April 1946 


Summar y 

stroboscopic photographs at i mlHIsccohd Intorvals have been taken of the bubble produced 
when I oz, charges of polar Ai«iori-gal Ignite were tired at various distances from the target plate of 
the R.fi.t. Box Model. The photographs show that the bubble was strongly attracted to the target. 

For a charge distance of about tPk inches the bubble made contact with the target plate at the time 
of Its first miniinum radius. Graphs of the radius .ind displacement of the- bubble when near its 
mlninum are given as functions of the time for various cnerge distances 

The diap1ucdfr.cnt of tno bubble at the mi.niraura was In r.-asonablo agreement with values calculated 
on the assumption that tho flat target plate end rigid surrounding flange may be- treated ns a rigid disc 
of equal aroa, This agrtoment Is regarded as indicating that the theory provides c good estimate 
of the linoi'.f momentum associated with the moving bubble, except pcrh?.ps during a short time near tho 
minimum. The observed minimum radii worn considerably larger than the predicted values, and the 
maximum velocit Ins correspondingly lower than the predicted values, throughout the range of charge 
distances considered. 

I ntroducUcn . 

In early measurements of the deflection of the R.ft.i. Box Model target plate it ws apparent 
that the flxpicslon bubble was contributing appreciably to tiie domago (i). The following experiments 
were designed to determine photographically the movement of tne bubble towards the target in order to 
assist in an jnalysis of the damage and to confirm the theory of reforenew (g), with regard to the 
obbble's dispiacowont, 

Experimental '/iethod. 

The R.R.L. Box Model was rigidly held with Its target plate vertical at a depth of 6 feet in 
k.r.L, tank, and i oz. charges of polar Armnrwgel Ignite were fired at distances ranging from ?0 to 18 
Inches from tho centre nf thn target plate. The method of mounting the hox model may be aSsimad to 
hold the box absolutely rigid, Photographs at one millisecond intervals wen; takun of the bubble by 
a eamsra situated about id faet fro.ii and at the same depth as the charge. Tho arditron equipment 
dascrlbsd.ln reference { 3 ) was used to provide the stroboscopic light flash, and the bubble appeared 
as a silhouette against a bright background, a simple rod framowork attached to the rigid flange of 
box model served to hold the charge in position and to provide reference narks in tho photegraphs. 

A pian and elevation uf tne camera and target positions are shown In Figure 1 , Target plates of 
both inch and ^ inch thickness were used, 

in each sixrt a group of about go pictures was taken hear the tlrrio of the first bubble minimum, 
in a few dasc-s a further group were taken near tho expected time of the secono minimum, but the 
pictures were of no Interest as the bubble was then in all cases an Irregular mass of gas cl Inning to 
the surface of the target plate and with little observable tendency to expand or ecntraet. 


540 


- 2 - 


The depth of chsnje ana target plate centre was kept throughout the series at five feet as 
at this depth the effect of free surface and no!!! oottOB almost exactly cancel the Influaneo of 
gravity during the first oscillation. The migration of the bubble up to Its first minimum was thus 
almost entirely due to the attractive effect of the rigid box modoi. 

Veosurewetit of P hoto tfraphs . 

From the silhouettes of the bubbles, volumes and positions of the centres of gravity of the 
bubbles were estimated by the method described in reforence (4). «s the effect of gravity approximately 
cancelled the free surface and rigid bottom effects the bubbles ware assumed to be symmetrical about a 
horiiontal axis normal to and through the centre of the target plate. The actual appearance of the 
bubbles up to their first minima suggested that this was a reasonabU- assumption, but Inmedlately after 
the minimum the bubbles hecame considerably distorted and quite cle.’.rly n-.d no axis of symmetry. 

The mjnsurement of vo'u.mBc. .nnd positions of centre of gravity of the bubOIos is thus subject to 
considerable enor after the minimum size has been passed. 


'•Results , 

Typical photographs are shown In Figures 2, 3 and it in which the following features may be 
noted, eefero the minimum the bubble Is relatively smooth, apart from tnin needle like streamors 
at the surface, and is elongated In a direction normal to the target piaie, on renchlng the minimum 
size the bubble moves quickly in towards the target plate, becomes vary distorted in general shape 
and shows a very irregular uneven surface. A short time before the minimum the surface of the 
bubble farthest from the target plate develops 1 very narked •plume-like* structure, .root cloarly 
seen In Figure 4, which is left behind as the bubble moves In towards the plate. This plume appears 
to be related to the wake behlod the moving bubble. it Is not certain whether it consists of a 
trail of particles of dirt or unburnt carbon, or whother it is a stream of fine bubbles. The plume 
does not oscillnto appreciably In size and remains more or less stationary aftir the bubble has spread 
itself out over the target plate. This same ylurat. has beon observed in ether photographs of the 
bubble produced by 1 oz, charges In the absence of a target where appreciable movoment cf the bubble 
occurred, end was always situated In the rear of the moving bubble, 

in all shots where the charge was 23 inches or more from the target plate the bubble was 
self-luminous for a short time near Its minimum size, and this ‘flash* produced a bright streak bn the 
moving fllmi which may be seen In Figure 2, The production of this flesh was noted in reference (3}, 
and Is discussed b(e1ow. 

Tho quantitative measuremonts from the photographs are depicted In Figures 5, 6 anil 7, and 
numerical values set out In Tables, 1, II and III. Table 1 contains a summnry oT tho principal 
quantities determined for each shot. 

Radiusr-Time Curves , 

in Figure 5 the mean radius of the bubble, defined as the radius of a sphere having the same 
volume as the bubble, has been plotted against time for a range of valued of the initial charge 
distance. The zero of time lo chosen arbitrarily to exhibit the curves in relation to each other 
to the best advantage. For the larger charge distances the curves are similar to those for a charge 
in open water awa,v from any target. When the cliarge was Initially 22 inches or less from tho target 
plate the bubble and water acquired a great deet of kinetic energy due to the bubble's motion toward 
tho target; In consequence, the collapse of the bubble was somewhat Inhibited, the potential energy 
in the gas being less in these cases and the corresponding minimum radii greats.*. Moreover, the 
subsequent expansion of the bubble also seemed to be less rapid for these closer shots, which may In 
part have been the result of losing a considerable amount of energy to tho target piste. In addition 
to the probable loss of energy as the result pf turbulence. 

The observed values of mlnlmom radius are set out in Table 11, together with thnne eslrylaten 
from equation (7), reference (6) relating the minimum radius to the *momentum constant* ut. This 
iTiCiTientuiri constant m is a iTufr-diinenslonol constaftt proportional to the linear momeniuri acquired by the 




bubble ...... 


■h-wk' 


- 3 “ 


541 


hubble durifici the first oscillation, and has fteen calculated in a manner to be discussed presently. 

It will br observes that in every case the observed minimum radius is considerably greater than the 
calculated value. 

Because of the observed luminosity of the bubble gases it seemed of interest to attempt some 
calculation of the gas temperature at the instant of maximum compression. As data for the adiabatic 

of the explosion products of polar Ammon-gel Ignite were not available the adiabatic for the explosion 
products of T.li.T. given by jones (6) has been used to calculate the temperatures given in Table II. 

These temperntures are .-ather low in view of the very great intensity of the "flash* produced by the 

smallest bubbles - e.g. shot 124. It Is noticeable that the region of luminosity only occupied a 

fraction of the volume of the bubble, and In the shots which produced rather weak "flashes* this 

luminous region was very small compared to the total volume. it might be argued that contrary to the 
assumption in the usual bubble theories the pressure throughout the gas is not uniform, and this could 
lead to sctiK! Increase in prnssure in parts of the bubble. sine.;, however, the temperature (absolute) 
varies approximately as the one fifth pover of the pressure, it does not appear likely that the 

maximum temperature will be much above the figures given in the Table. 

Oi splaeement - Time Curve s. 

The displ scemenl of the centre of gravity of the bubble towards the target is plotted as a 

function of time before and after the occurrence of the minimum radius in Figure 6 for a range of 

values Of the charge distance. as may be seen from Figure 5 this time is not very clearly defined 
In the close shots. For the more distant snots the deflection time curve is roughly symmetrical 
about the time of minimum radius. in the nearer shots the bubble approached very close to the target 
plato, making contact v.lth it soon after the minimum, so that a marked asymmetry of the displacement- 
time curve Is to be expected. 

Cpiftjiarisnn of Oi.^ervecl IH s placement with Theo . 

In calculating the attractive effect of the box model it has been assumed that the flat 
target plate surrounded by its rigid fiat flange may oe approximated by a rigid fixed disc of 
equal area. Accordingly the "momentum constant" ra, defined as one hai f the cube of tho non- 
dlmenslonal minimum radius mult Ipl ied by the m-'.xl mum non-dimensional velocity, has been calculated 
using the fornwla for the attraction co-efficient for a rigid disc given in reference (2), The 
displaeem.nt it the minimum was thon calculated. This curve was obtained by plotting the results 
of all avait -ble ."ull inteyrotions of the equations of bubble motion. Tho displacement thus 
calcuMtid has been plotted in Figure 7 together with the observed values far »ii «hs phstoqrayhlc 
dhe-fs. The resiiiis for hotn ^ inch and j„ch plating do not appear to He on separate curves, 
end in plotting them in Figure r no distinction has been made. This is to be expected since the 
target plate is practicaily nationless, and therefore effectively rigid, during the period when 
the bubble is iarge, i.o, when it acquiros siijst of its momentum. 


The agreement between the observed and predicted displ.acement of the bubble shown in Figure 7 
requires to be interpreted with caution, in comroon with most experimental observations Of the 
bubble's behaviour near its minimum, there arc rather wioe discrepancies between theory and 
observation in regard to quantities such as minimum radius or iraxlmum ilnear velocity. Since, 
however, a considerable proportion of the resultant dispTicomont takes place some little time 
before the minimum, at least for the closer shots, the agreement in observed and calculated 
displacements might still be expected, provided that the discrepancies arose only very doss to the 
tiine of minimum radius. It is thus reasonable to regard the observed agreemert as indicating that 
the theory provides a good estimate of ths linear AMraentum of the bubble, except possibiy during 
i v.-Ty short time near the occurrence of the minimum radius. Moreover, it should be remembered 
that the theory given in ( 2 ) was derived on the assumption that the radius of the bubble when largo 
is small compared to its distance from the target. for « one ounce charge the maximum radius is 
about 18 inches so that tho theory is here being used w.;ll beyond the region of validity of this 
essumption. 


Hav i ng 


542 


- u - 

Having regard to the relatively long Interval between flashes, the difficulty of mnasurins iha 
volutne accurately, and the uncertainty concerning the correct value to be essumod for the "virtual 
mass' of the water movin-g with the bubble vary little accuracy is possible in a direct measurement of 
the momentum from the photographs. An attempt has, however, been made to estimate the linear fflofnentum 
at a tliTc near the minimum radius from the photographs and the values of the •momentum constant" are 
given in Table III together with the corresponding calculated values (the actual momentum In non- 
dimenalonal units is s E m. where m is the quantity denoted as the "momentum constant*). The observed 

momenta are at any rate of the same order of magnitude «a those calculated. 

This confirms the deduction nade from displacement results. 

ConglusioiiSt 


(1) The displacement of the bubble towards the target plate at the first minimum is l.n reasonable 
agreement with values calculated from the theory of the attraction of an explosion bubble 
towards a rigid disc. 

( 2 ) The observeo agreement is regarded as indicating that tl)6 theory provides a correct estimate 
of the linear momentum of the bubbln, except possibly during a very short time near the 
occurrence of the minimum radius. 

(3) There is considerable discrepancy In the predicted and observed v.-iluos of the minimum radius 
and the maximum velocity. This discrepancy in radius and velocity may, however, only occur 
for a srml 1 interval near the time of minimum radius. 


References , 

(1) Deflect ion-Ti me curves at the centre of box-model plates, resulting frem underwater 
explosions, 

( 2 ) The Attraction of an underwater Explosion Bubble to a rigid disc, 

(3) A Technique for Multitlash photography of Underwater Explosion Phenomena. 

(4) Photographic hieasurements of the size, shape and moverrent of the Bubble produced by 1 oz. 
Charges of Polar Ammon Gelignite detonated Underwater at a depth of 3 taet. 

(5) The Behaviour of an underwater Explosion Bubble. 

(d) The Bshavlour of an underwater Explosion Bubble further approximations. 







Minimum tiadius and Gas Te>nperature in HubbLe, 


Shot 

No. 

Diiatanoe of 
Charge from 
Target Plate 

Calculated 

Minimum 

Radius 

Ohsorved 

Minimum 

Radius 

Maximum 

Gas Temperature 

Nature of 
"Plash" 

124. 

30.5 in. 

1 .?4 in. 

2.9 in. 

365°C 

Strong 

121 

27.5 

1.97 

2.85 

380 

Strong 

120 

24.5 

2« 3^ 

2.9 

365 

Strong 

105 

23.5 

2.46 

3.1 

335 

Very faint 

114 

23.5 

2.46 

5.2 

320 

Medium 

126 

22.5 

2.65 

3.9 

2!40 

Very faint 

108 

20.5 

3.15 

4.8 

IBO 

No flash 

111 

20.5 

3.15 

5.0 

150 

Ne flash 

125 

19.5 

3.46 

4.75 

160 

No flash 

130 

18.5 

3.85 

4.75 

160 

No flash 














- 7 - 
TARLE III. 


Observed and Calculated liomentwn of Bubble at ’finimum. 


Shot 

No. 

Distance of 
Charge from 
Target plate 

Observed 

Maximum 

Velocity 

124 

30.5 in. 

260 ft/sec. 

121 

27.5 

310 

120 

24.5 

310 

105 

23.5 

250 

114 

23.5 

310 

126 

22.5 

200 

o 

CD 

20.5 

210 

111 

20.5 

200 

112 

19.5 

180 

125 

1 

19.5 

200 

130 

i 

18.5 

180 


Calculated 

Maximum 

Velocity 


1.13x10 ■' 0.25 X 10"-^ 780 ft/aeo. 









540 


liiumifiQtfi^ rsfUcior 



/ 


Rigid veriicat girder 



model 


Bubble 


-G3- 

Comero 


Ploft Vi<e» of Box Model ood Comero 



VerlicQl 

girder 


Elevation of Box Model Viewed from Comero 


Fig- 1 PLAN AND ELEvATiON Of BOX MODEL AND 
CAMERA ADJUSTMENT 

















tiott minirnum —TIME - Milfucconds — After minimu 

DISPLACEMENT OF C.C. Of BUBBLE TOWARDS TARGET 







3AVld i3DbVi WOM3 OD 319909 JO 30NV1SIQ 




Fig.7 POSITION OF ^BBLC AT FIRST MINIViUM 




MEASUREMENT OF BUBBLE PULSE PHENOMENA, lU 
RADIUS AND PERIOD STUDIES 


E. Swift, Jr. and J. C. Decius 
Utiuerwater Explosives Research Laboratory 
Woods Hole Oceanographic Institution 


American Contribution 


11 September 1947 


553 


( 


GONTiSNTS 




ABSTRACT 
LIST OF SU1BOLS 

I. INTRODUCTION 

II. EXPERIMENTAL METHODS 

1. Babble Per lode 

2. Depth 

3 . Ghiffga Weight 

4* Redlw Ueasurementa 

III. RESULTS 

Flrat Bubble Period iii Fras Water 

6. Second end Third Babble Ferloda in Free Watex^ 

7. Flrat Bubble Periods Close to a Free Surface 

8. Uaxhttuu Bubble Radii 

A. Free Water 

B. Radii Close to & Free Surfaae 

C. Radii for Second and Third Periods in 
Free Water 

IV. COfJlPARISON ® EXPEHIlffiaiTAL RESULTS WITH THBORI 

9. Resume of the Theory: The Role of the Adiabatic 
Parameters 

10. Determination of the Bubble Energy 
APPENDIX I 
LIST OF REFERENCES 


h 

V, 


/ 




1 


1 

1 

2 

2 

2 


3 

3 

11 

11 

15 

15 

18 


20 

20 


20 

26 

32 





I 


1 

i i 554 

i' i. 

I. u 


LIST OF FIGURES 




Figure Page 

1 RadiUB-Tlme Gtirve for 300 gm of TNT Depth of 300 ft 16 

2 Non-Dlmensloual Period vs Parameter k for Various T 

(Hatio of Speolfie Heats) 25 

3 Perlod/MaxLntuffl Radius (Koh'-DlmsBslonal} vs Parsmetar 

k for Varloua *r 27 

4 Composite, Non-Dlaienalonal Bedtus-Tlme Curve for 

TNT - 300 ft Depth 33 

5 Composite, Non-Dimensional Radius-Time Curve for 

TNT - 550 ft Depth 34 

6 Composite, Non-Dimensional Radius-Time Curve for 

Tetryl - 300 ft Depth 35 

7 Composite, Non-Dimenalonal Rsdius-Time Curve for 

Tetoyl - 600 ft Depth 36 

8 Compc^ito, Non-Dimensional Radius-Time Curve for 

Torpe3c-2 - 600 ft Depth 37 

9 Composite , Non-Dimensional Radius-Time Curve for 

Blasting Gelatin - 500 ft Depth 38 














This rspoirt presents measuraiaenta of the periods of o^oillatlon and 
the T'uill at m asd Jiu m size of bubbles of product gases from \indermter 
explosions* The elgnif loanee of these measurements Is discussed as It 
pertains to the energy of the bubble* Mo attempt Is made to relate this 
energy to the energy emitted by the bubble In the form of a presaura wave 
or "bubble pulse" at the time of the bubble minimum* 

i Isrge number of plezoaleotrloally recorded period measurements 
are reported for otaavgea wslghlng from 1 oa to l/Z lb* Maximum redli. 
obtained from simultaneous photogra|Mo raoords are also shown, Ths 
ohargea wars fired at depths ranging fl^om Z to 700 ft. 

An analyeis of the data Is made, based on the theory as developed by 
Shiffman and Friedman . Their numerical integrations of the period have 
been extended to include a wider range of the parameters which describe 
the internal energy of the product gases, ^y making reasonable assumptions 
in the calculations, the energy of the bubble is computed to be about §0$ 
of the total energy released by ths detonation. It is shown that there is 
some uncertainty in these oalcnlations due to the limited rosge of the 
experiments and ths lack of defixiite knowledge as to ths state of the 
gases in the bubble* 


: ! 




5&6 


LIST OF SYMBOLS 



8 . 

Ainn» Onn 

o 

C 

D 

E 

g 

H 





radius of gae tubble 
non-diffioaslonal radius of gae bubble 
jntuciniuEj radius of gas bubble for n-th oaoillatlon 
mJailfflUm radius of gas bubble for n-th oscillation 

adiabatic constant 

time scaling factor 

total depth of water 

total energir of bubble 

acoelaratlon of gravity 

depth to tha chargo 

radius constant for n^-th oscillation 

period constant for n>th osolllaticn 

parameter in energy aquation 

length scaling factor 

arbitrary constants in equation of curtate cycloid 
hydroatatie pressure at depth of detonation 
pressure inside the gas bubble 
detonation energy 
ratio of to Ami 

fraction of detonation energy left In bubble 

non-dimensional period of n-th bubble oscillation 

period of n-th bubble oscillation 

weight of charge 

depth of detonation + 33 ft 

ratio of specific heats 
density of sea water 
see Ref. (5) 

curve fitting parameter for empirical radius”time curve 
paremoter in empirical radius-time curve 


) 


) 


1 



OF POLSS PHpOWSHA, 111 

mum jm period sruDiiii 


I. IRfBOD?CX£ON 


£i imf itudjr of tha maohanlvm and en«rgv'»tioi of an iffies 7 t:st 3 Z> « 3 Q) 1 os 1 o&, 
oaa la Inavltably lad to a raaliaatloa of eonaiderabls aaomts of 

anoFgjr iaharoni in tha bubble of o 3 q>loalon product gaaaa. Sluoa 'Uia total 
dasgags dons bj an undafnatar asrploaion la cauaad by both tha abook wsra and 
the aubaitquaat bubble pulaaa« these effects uniat be separated for a proper 
Tinderatanding of the neohattiaaai Involved* To detemtne the exact quantities 
of energy present at various etagoa of the eiq>losion and to find the aasount 
of uaeful energy la a taak of great Bsgnltnde* In thla report are presented 
data idtich eere gathered in an attanpt to evaluate the energy In the bilbble* 

Cue of the eaaleat neaaurementa to make la tha measurement of the 
perloda of oaoiUatlon of tha hubble, that li' ae ti^ intervals betwea 
oucco60iva mlniTaa in the bubble radius. These are referred to aa the first 
bubble periodi aecond bubble period* and so on. Tha length of the bubble 
period la related to the energy left after the passage of the shook nave by 
tha aque.tiena dlaouased below. In general* It may be said that the longer 
the period, the greater tha energy. However, it la neoessary to have 
additiniu^ inforsMtion *— spesiflcelly, the aquatics of state of the predust 
gaaes — before an. exact osdoulation of the total energy can be mi^e* 

A. aecond lieaaure of the energy may ba obtained from a study of the 
maximum and minimum radii of the bubble. Photographs of the bubble at 
various stages of its oseiUatlon show that the outline of tbs bubble is 
rather olsarly delineated up to the first maxlinum, but that it is oomewhat 
leas clear at aubsequent maxlina, and is completely obticured carbon 
atreemera In the water at the nlnlma. Since the bubble radii at tho minima 
have not been measui’ed, it Is not possible to obtain a good valUe of the 
energy from radius mdhaurements alono« By a comblnatioi:|i of radius and 
period measurements, however, an attack can be madiji on a more precise 
ooloulation of the energy In tbs bubble* 

Thin report presents a compilation of data obtained in a rather oxtensive 
program of bubble period and radius msasu?smente,l* 2 ) These appear to b® the 
most complete and accurate data of this sort available at the present time. 
Using these measurements* calculations of energy and other parameters 
appearing in the bubble equations have been carried out. 


II. EIPKKUlEtlTAL limODS 


1. lubMff Peri t^B 

Periods were measured^) by recording oseillographlcaLly the signal iVom 
a pisdoeleotrio gauge exposed to the shock wave and bubble pulaea. The 
lo^ssure changes were recorded on moving film elmultaaeoualy with a 1 ke 

^ming wave. 


558 


Th* psaotaJoa of Bs^asypaaeat of tl» parioda dapsadad oa'tha praciaiou 
vri.tb iddab tha filta paoord asald ibs p#ad« This i <t agtisuiteu to ha about 
0.5^0 p[?eoieloa ia not Tsr; lilcaiy to ta attainad by this 

aathod beeausa of tha finita 'sidtb of tha tfaoa on iho film and baoauaa of 
tha dlffleulty of looatiag tha aaaot position of tho bubble pulae >aax.liaum on 
the filn* 

2f Ssptb 


Depths ware fouad fron aaasurad lansths of eabla, qs ^ jin the deapar 
shots, fpoa paacl,in$s of a Sourdos depth i^iuga. ?rom tbs ^sflcus data 
ooUaotad, it is astlaatad that tha pfaolsioD of memaupauant la about 'i$ and 
may b« as poor as ^ in the eerst oaeas* Lass ralianoa oan bs plaeed on 
those shots share the cable aoglo was great, i,a., greeter than 20^ from 
tha Tarticali and share no depth gauge sas usad« 


Since tha radius of tha bubble depends upon tha cube root of tha depth, 
it is not vary ssnsltlva to errors in tha ln.'^>tar* Hovevar, the period dspasda 
upon the 5/6 poser of tha depth, so only slirihtly more praciaion is attal^ble 
in tha period than is iaheront la <he depth aoasurament. 


% 




Tha half>pound obargae used ware weighed to tha nearest gran, the 
1 os chargaa to 0«1 gm. Since charge weight appasra in tha aquations as a 
cube root, tha srror node hare is quite oagHgibl#. 


A oorrsetion had to be made for tha booster in each case* This correction 
was made by raduolng (or Inpreiuihg) the weight of booster used In the cal- 
culation to an equivalent weight of tha explosive being studied. The factor 
employed was from the ratiov of bubble pairlod constants, by 

successive approximations If neoehsary. Any error made in this correction 
will introduce a negligible error in the final weight, since the weight of 
booster is only a small fraction of tha total. 

4q Radius Measurements 


Because of ths relatively niirraw angle of view of the high-speed cameras 
uqed, tbs bubble at maxUniu slse ordinarily CBSe very close to the edgoa of 
the photograph. In the‘ very shallow shots using 25 gm charges, the narrower 
dlmenslou of the pioturs was too small to Inoluds an entire bubble dlsmeter. 

The possibility of optical distorMon due to the lens was first 
Investigated. Photc^aphe of a grid taken witli the Fastax oeuoera showed no 
appreciable diatartion over the whole usable field. Calciilatlon^V 

of the diistortlou introduced by ths glass-air interface before the lens 
showsd that in the worst ease the correction would be entirely negligible. 

Measurements of the radius of the bubble were made on photo|pfaphlo prints. 
Dlamsters were measured in at least three diJrectlouB end averaged. Care was 
taken to avoid including irregularities outiside what was believed to be the 
true surface of the bubble of gases. As the minimum is approached, the 
bubble is progressively obscured by opaque streamers of explosion products. 





•1 


so tMt) closM to tiae mlnltmx<t^ uoaauremsnts ware not at all reliable. The 
overall precision of the bubble radius meaeurenent ia eatimat-ad to be 
about 2>. Ses9 improvement in theee meaRuremanta eotlld have been made b 7 
proviulng afcsBdards of length of noarljr the aaisie size iSS the bubble diaoetsr, 
and by veiy sharp foouuslng. 


III. RESULTS 


5. £jja$ ^ ££M la:^ 

These results constitute what are believed to be reliable measurements 
of the first bubble periods for TNT, tetryl, and torpex-2 in the absence 
of any interfering awfeoes, and at such depths that migration effects can 
be considered negligible. All charges were in the form of cylinders, of 
height slightly greater than the diameter. A few values are given for 
pentcllte and blasting gelatin, but these cannot be considered as giving a 
definitive value of the period. Some of the results included are for shots 
made in water of such depth that the aurfase oorreotlona would be small, but 
not negligible. 

Table 1 shows the results at several depths for cast TNT charges of 
density 1,5. Depths were aflesured a Boiardon depth gauged), but In a few 
casoB the meter wheal reading of the length of cable let out wcus used. Periods 
were read from pieaoeleotrio records.'*^ The period constant Kq ivas defined bv 

the equation^) 




where 


the period (aeo) 

H+33. where H is the initial charge depth (ft) 
charge weight (lb) 

refers to the first, second, or third bubble period. 


The TNT In each chargo weighed silghtly mort than a half-)jound} to that 
weight was added the TNT equivalent of the 44 gu tetryl booster and 1 gm 
for the detonator. As feir as bubble energies go, pressed tetryl appears to 
be about 3^ sore energetic than TNT, jo the weight of tetryl used was con- 
vif-rted to the equivalent weight of by multiplying by 1,03. 

The average value for the periv/ ’ constant for oast TNT obtained in this 
way is 4.36. 'fhe standard dvviatiou of the mean is shown together ’with the 
average Ijj the taole. 

« In a few eases, indicated In Table Z, idien plesoeleotric records were not 
available, the periods were read from radius-tlma cm*ve8 plotted from the 
F&stax pictures of the bubble cycle. These values are probably accurate to 
2 % or better, as indicated by the correlation whan both measurements were 
available. 




• 1 / :: 




560 


Two othor mlu«s have baea obtaiaed.iinder carefully controlled coa- 
dltlone* SUohtar» Sohnalder, and Cole^^ report a value of 4>36 for 
295 Ih ®BS ohargei in relatively shallow water, ^tor correcting their 
perl«4 asasursaentB for the effects of surfaces^) « eordsa and Arena®/ 
havs siisilatly oorraoted the period values f^ 2/2 I'b charges in shallow 
water fotaad by Arons, Bord^, anu Sttllsr?) and fress th^ir results the 
constant appears to be 4*32. These autbor;;;, however, hav<s asiaUi&ed a 
slightly higher energy factor for the tetri'l booster. If the factor of 
1.03 used In this report is applied to their data, their* TNT period 
constant becomes 4*35o Other values may be found In Table X. 


TABLE 1 

FIRST PERIOD CCfflSTAHTS FOR CAST TNT IN FR)SE WATES 


Shot He, 

Charge 

Weight 

(lb) 

Depth 

H 

(ft) 

Period 

Ti 

(msec) 


GIF 

0.651 

343 

27.4 

4.43 

Q2F 

.660 

304 

29.8 

4.38 

Q4F 

.660 

304 

30,0 •* 

4.a 

G5P 

.662 

305 

29.89 

4.40 

g6F 

.658 

304 

29.92 

4.41 

G7F 

.669 

298 

30.23* 

4.36 

G8F 

.663 

304 

29.82* 

4.38 

Q9F 

.651 

304 

29.64* 

4.38 

G17F 

.660 

305 

29.62* 

4.38 

Q18P 

.660 

302 

29.61* 

4.32 

QaCF 

.660 

538 

19.64* 

4.47 

G21F 

.651 

593 

19.10* 

4.73 

G23F 

.658 

567 

18.14* 

4.32 

Q7C^ 

.660 

503 

19.9 

4.30 

G7IF 

« 

0“ 

tJl 

CO 

463 

21,0 ** 

4.26 

G72F 

.660 

586 

17.85 

4.36 

G73F 

.655 

576 

18,16* 

4.33 

Q74F 

.660 

556 

18,1 ** 

4.% 

Q76F 

.660 

587 

17.7 

4.33 

Average (excluding G21F} 



4.3^.014 


w Period from pictures agrees to + 2%» 

** Period from, pictures— no piezooleetrio record. 


Table 11 presents the data obtained for torpex-2 charges. The depth 
measurement in this esse la perhaps Slightly loss reliable than for the TNT, 
Since it was found from meter wheel readings alone in four of the eight 
shots. Three of the eight periods were read from radiua-time curves made 
from the pictures. Since in the other five cases the dlfferenoa between 
this period and the electronic measurement was at worst 1.39K, this may be 
considered aufficlently good. The booster correction was made by multi- 
plying the v;eight of tetryl (44 0»} by 0.65* 


561 


Th« rtRultlag xjerlod oonstAnt^ ■ 5 a 065; nia^ ho compar&d with the 
$>10; ohtaisuid and Araos (rsferMioes Indicated ttnd(^r th« 

dleouailoa of tJXT ) . 

ilioro roaontly a aeries of 280 ga torpex>2 charges fired at this 
labcrAtcrjr tsithia 5 ft of the svtrfaoe gives «b avemge value of 5,00 for Xj 
after eorrecting for the of facts of surfaces®). 


table: 11 

FIHST PMUC© eONSTASTS FOR TOHPIS-2 IN FRF.E WATER*«^^ 


Shot No, 

CViarge Weight 

W 

(Ih) 

Depth 

H 

(ft) 

Period 

Tl 

(maeo) 

Kl 

0119? 

0.582 

597* 

19,5 ** 

5.017 

0120? 

.578 

600» 

19.46 

5.046 

0121? 

.620 

603* 

19.9 ** 

5.065 

0122F' 

.602 

611 

19.65 

5.096 

0123? 

•602 

613 

19.8 ** 

5.135 

0124? 

.576 

■603* 

19.34 

5.044 

0125? 

.574 

59 $ 

•10 '50 

5 01 ' 

0126? 

.572 

603 

19.5^’ _ 

5.100 

Ivdrage. 




5.065+0.015 


« Depth from n^tar irtiee<% 

** Period oalouiatod from film 

lief • 25 for composition of torpex>2. 


Tahla III gives the results obtained for 1/2 lb pressed tetryl charges; 
denslty^ 1.5; in free eater. The shots labelled ”GC" were recorded when 
the oharge eas in the vicinity of a. small steel oyllnderv); however, a 
orltioal comparison of these shots with the rest showed no systematic 
dlsorapanoy. Thia seems reasonable because of the relatively omall sise 
of the oyiindar. Oooealonally a large dlscrepatio* in the period appeared, 

Sa In sl^t GC4QE idiara the deviation from the msec, io about 30^*. This 
error eas pocasioned by the eleotr^io gear, and was not a scatter in the 
bropsrty being measured, ma Indirect evidence from the collapse of the 
cylinder, discussed In Ref. (9), shows the period to bo actually About 
20 maeo in this case, resulting in a ferlod constant conalstent with the 
other yalues. 

Ad entirely independent aeries of shots Is shown in Table IV, These 
wars carried but with a different sat of equipment and in water sufficiently 

* See alao 0217 in Table X. 


- 5 - 


562 


TlBLg in 

PIPS® PmOD CONSTAHTS FOR PRESSED T5TRXL IN FREE Wi^R 
ClMupgB W.ighti 0,500 lb 


Shot No. 

Depth 

Period 

Kl 


H 

h 


(ft) 

(msec) 


am 

300 

27.73* 

4>41 

G12F 

300 

27.52* 

4.38 

aisp 

302 

27.62* 

4.42 

Q14P 

303 

27,53* 

4o42 

G15F 

305 

27.16* 

4.38 

G16P 

302 

27.4 ** 

4.38 

G22P 

607 

16.02* 

4.40 

G31F 

547 

17,31* 

4.38 

GC39E 

712 

U.30 

4.45 

GC40B 

471 

24.55 

5.68 

GC41E 

471 

19,52 

4.38 

004^ 

237 

32.92 

4.40 

QC43E 

232 

32.57 

4.27 

QC47B 

474 

19.40 

4.39 

QC49E 

481 

19.47 

4.47 

QC79F 

704 

U.23. 

4.39 

CiCSCF 

695 

14,22 

4.36 

(icioup 

695 

U.21* 

4.35 

GC102F 

493 

19.24* 

4*49 

QC103F 

230 

33.06* 

4.34 

GC106E 

689 

U.35 

4.36 

QC107E 

240 

32.83 

4.44 

goiose 

462 

19.76 

4.39 

Average (excluding QG40E) 


4.39*0.01 

* Poriodi from ploturee iLgreos to + 2Jt 



Period obtained from p^otures—no piesoel^otrio record. 



TASLE I? 



FIRSl* PERIOD OONSTANTS FOR PRESSED TEIRU. 



Charge Weight! 
H • 39 f t 

0.500 lb 


Shot No* 

Total Water Depth 

Period 

*1 

(meeo) 



(ft) 


BE462 

72 

99.4 

4.40 

RE466 

108 

98.6 

4.41 

HE470 

117 

98.1 

4.41 

HE472 

117 

99.3 

4.47 

HE474 

117 

97.9 

4.41 

HE476 

89 

97.9 

4.40 

RE478 

84 

97.5 

4.38. 

Average 



4.41+0,01 


6 - 




desp that th» eo^reottoa whioh waa reade for surfaces asounted to only 
about rns avsraged result for 4.41* is in good agraemeftt with 

the value of Table lUf 4.39. 

Further results of work with pressed tetryl cshargsa are shewn in 
Tsibles 7 asad VI, These eharge# waighed slightly less than an ounce, 
end may bs eenaJdsred tw bei^ la free water (*i?rfaoe oorrsctiea Bruch 
less than ij) . Bote that both series of shots give a value of the 
period constant which averegas about IJt lower than that for the 1/2 lb 
chargOw 9 

The results of a slightly less reliable aeries of measurements in 
free water are shown in TaMa VII. The depth in this case was measured 
from the known length of cable let out, multiplied by the cosine of tiw 
sngle the cable made with the vertical at the swfaoe. As long as the 
angle is snalJ. tbl« gives a good measure of the depthi however, einoe 
the angle in tki series ranged vtp to 32», tlie uncertainty In the depth 
may bo as much as 5%* The period oonatant la, as a result, somewhat 
less woelaely known in this series, the standard deviation of a single 
observation being about 2%. The average value, 4.31, is also in this 
case found to be lower than that for the l/2 lb obarges. 

TABLE V 

PiaST PERIOD CONSTANTS FOR PRESSED TETRYL IN FREE WATER 

Charge Welghtt 0.0558 lb 
Total Depth of Wr.terj Sj-100 ft 
H • 55 ft 


Shot Humber 



4,354+0.007 


* The correction for surfaces was made by using the equation in Ref. (5) 
page 5. for oonvenience, this was transformed toi 


, 1.179 KWV3 J(y) 


by using the relationship 5.78 k 3 - rQ where K is the value of the period 
oonstjmt, A value of S is put into tho correction term as a first approxi- 
mation, and if the v»ilue of K found by solving the equation differs greatly 
from this, a second approximtlon should be made. The values of K reported' 
in Tables IV, VllI, land IE were found by this method. The ntjiaerieal oon- 
stant 5.78 is not a laiivcrsal constant, but may bo used In all cases con- 
sidered here without Inourrlrg much error* 






564 


TABLE VI 


7IBST PERIOD CONSTANTS FOR PR^ISO TETm 

Oha£^ Wcigbti O.OSSS lb 
Total Dapta of Water SS-lOO ft 
H - 39 ft 


eil9« «v« 

Charge Weight 

V 

<«e) 

Total Water Depth 
(ft) 

■p«i*lod 

*1 

(maae) 

Kl 

as46t 

25»3 

68 

47.3 

4.32 

RE465 

25.3 

106 

47.1 

4.30 

RE467 

25.5 

109 

47.5 

4.33 

BE4.73 

25.5 

114 

47.2 

4.35 

RE47S 

25.5 

93 

47.0 

4.33 

RS477 

25.6 

88 

47.1 

4.33 

Ri460 

25.6 

80 

47.2 

4.35 

BE481 

25:8 

81 

46.9 

4.30 

RE4S2 

25.7 

82 

46.9 

4.31 

Avsrs^ 




4.32410.006 


TABLE VII 

FIRST PERIOIO OOKSTAMfS FOR PRESSED TETHTL IN FREE WATER 
Charga Weights 0,05SS lb 


Shot No. 

Period, 

^1 

(maao) 

Depth 

H 

(ft) 

«1 

E15 

‘ “““lOT 

3^0” 

4.27 

Sl6 

10.79 

380 

4.26 

S17 

17.90 

196 

4.34 

SIS 

12.4S 

334 

4.47 

E19 

n.55 

344 

4.24 

120 

11.22 

347 

4.16 

S22 

17=45 

205 

4.37 

E23 

17.49 

207 

4.40 

E39 

7.20 

661 

4.39 

141 

7.50 

535 

4.15 

£42 

7.05 

672 

4.36 

143 

8.08 

545 

4.24 

845 

7.96 

578 

4.37 

146 

10.86 

380 

4.31 

AVaraga 



4.3Jajt).025 


) 




565 

No data wers obtained for loose tatrjrl oharges in free water but a 
few shots were snada with \/2 lb charges in water deep enough sc that the 
surface correction amounted to only about 1^. fhese values are given In 
Table VIll. It will be noted that the value of the peri<^ constat is 
about 3^ higher tlian for the pressed charges. Borden and Arcns^) also 
Sovasd a higher '^luo for loose tatryl ohargea of various sizes than tho 
value reported here for preasod tetryl. 

Because of the aecuracy* of the e^qaerlnental methods and the oon- 
firmatl'On ox tlie results usdar differing oondltione, it is felt that 
this difference between loose and pressed tetryl 1s a real one* There 
also may bo a real discrepant between the 1 oa and 1/2 lb pressed 
charges j although th® evideses Ig not entirely concluslvo. 


TiBLB Vm 

FIRST PERIOD CX)NST/JT3 FOR LOOSE TETRIL 

Charge Welghti 0.552 3,b 
H - 39 ft 


Shot Number 

Total Water Depth 
(ft) 

Period 

»1 

(meeo) 

Kl 

BB452 

72 

104.2 

4.52 

BE453 

63 

105.0 

4.55 

RE454 

69 

104.0 

4.51 

BE456 

69 

104.2 

4.52 

BS458 

69 

103,8 

4.50 

BE4^ 

64 

105,6 

4.58 

Average 



4.53+0,ca2 


* Depth aov accurately known due to rox;gh seas. 


Table IX gives the results of several shots made with various 
l/Z lb pentolite charges* These values were obtained incidentally to 
other studies, and' the experimental conditions are not strictly 
ooaparable* The data are not sufficient in number to give a f i n al 
valtis f or tbs period constat, nor la it felt that they are as accurate 
as the scatter wouM indicate. The average falls between the averages 
for the TNT and pressed tetryl charges of comparable weight. It may be 
slgoifloant that the periods for the two long cylindrical oliarges are 
somewhat lower than for the other charges. 

Finally, for comparison, a nuabar of period constants calculated 
from period values a^lmhle in the literature are given in Table X. 


- 9 - 


TABLi£ lie 


FlHSi? PmOB CONSTiijraS for 50-50 pentolite 


shot Husioeir Charge Ksight .uepth aoz%Q& Depth Period 


AYsrags 


26.07^ 

4.36 

22.9 ® 
22.4 ^ 

4.39 

4.2b 

23.47 

4.32 


4.37+0.023 


a. Long cylinder - 6 in. by 1.? In. 

b. Lower half cased in brass, setae shape as Q33P 

o. 44 gxi tetryl booster used. If eight added direotly to weight of pento 
Ute and total reported here. 

d. Period from pictures - no pioaoeleetrlc meaeuremant 

e. Correction made for surfaces. 


TABLE £ 

UISCELLANBOUS VALUES OF PERIOD CONSIAEl' 


Explosive 


Reference 


Cast TNT 

4.28 

10 

Cast TNT 

4.36 

4 

Cast TNT 

4 . 35 * 

6 

Cast TNT 

4.36 


Pressed Tetryl 

4.7 

11 

Pressed Tetryl 

4.31 

•ini' 

Pressed Tetryl 

4.39 


Loose Tetryl 

4 . 4 S 

6 

Loose Tetryl 

4.4i' 

6 

L(sese Tetryl 

4.53 

** 

T6rpsx-2 

5.10 

6 

Torpex-? 

5.065 


lltnol 

4.6 

12 

iSlij^ol 

4.91 

4 

lOOSt Blasting Gelatin 4.9 

** 

100^ Blasting Ge3.atin 4.74 

13 

30-50 Pentollte 

4.37 



# See comments in Sso, 5 
** Values from this report. 


Remarks 

CorrectJ.on for surface made 
Correction for surface mads 
Correotj.oc for surface made 
Free water 

Extrapolated value. 1 oz charges 
1 oa cliarges} Free water 
l/Z lb cljarges} Px'ea water 

120 gm; Correction for surface made 
300 gm| Correction for surface made 
250 gmt Correction for surface made 

Correction for siarfaoe mads 
Free water 

Correction for surface made 

Free water 

Free water 
Depth uncertain 

Conditions variable 


• 10 - 








587 


6e Saeond SSA Third Bubble Periods ^ Free Water 

Ho piazoeleots'io records of the bubble pulses after the first ware 
obielndd; kowever, it was found possible to get a period record from the 
motion pioturss taken of the bubbl? In free water. The raluos were 
obtained from radius-time curves which are probably not as preoiss as 
plezoeleotrio reoords. Comparisons between piasoelactrio and photographic 
records of ths first perl^d show differences of 1-2^; the later periods 
are mors difficult to maaaura and somewhat greater diserepanoies would bo 
expeoted. 

The resulto for second period measuremants are shown in Table XI. 

The period constant for torpax-2 is about 8^ higher than that for TNT» and 
those for tetryl and pentollte about 1-2^ higher. This order is essentially 
the same as for ths first period. The ratios T2/Tj^ wero also calculated 
and in Table XXI are oompared with values obtained from other sources. It 
might be noted that the aluniniaed explosives show a smaller value of this 
ratio in each series. 

Table XIII gives the values obtained for the third bubble periods 
from measurements on the phctographs. The bubble period constants cal- 
culated from thase values show greater scatter. This would be anticipated 
because of the difficulty of discerning the outline of the bubble after the 
first maximuB, 

7« jgf y . stj Bubble Perieds Close a Free Surface 

Aa .part of a study of ths interaction of bubbles with ths surface » the 
sjqplosions of a number of 25 gm tetryl charges wore ihotogi'aphed very close 
to the surface!) , Charge depths ranged from 15 in, to 5 ft* irtdch distances 
correspond approximately to 0,75 sad 3 bubble radii. Thus, in the shallowest 
case, the bubble always vented; at intermediate depths )> the behavior was 
not exactly reproducible, presumably because of uncontrolled variations in 
the surface or in the depth. At 21 in. ~ a distance only very slightly 
greater than venting depth -w the bubble appealjred to drag in air from the 
atmosphere os it collapsed. This resvilted in an increaise in ths appareiit 
minimum else and a deanping or ths osolUation. As a result, no plesoelectrlo 
record o£ the bubble pulse was obtained in these cetses. 

Ths periods were measured on the high-spead photographs by counting 
ths number of 1000 cycle timing marks between the freune showing detonation 
sixi that shewing ths minimum. Such an es;jblmatlon of the period is based 
on a subjective judgment os to thb times of detonation and of the mj. rtim uia 
relative to the f ranee and the timing marks. The error Incurred may easily 
smount to as much as a millisecond, and is probably more at the 21 In. 
depth where the minimum is Indefinite. The Tialuea of the periods so obtained 
are showi in Table XIV, the values for loose end pressed tetryl being shown 
separately. 

To obtain period constants comparable to the values obtained in free 
water, corrections for the effect of the free surface of the ocean and for 
the bottom must be applied. This was dons by tha method indicated in the 


-n- 


TABI£ n 


SECOND PERIOD CONSTANTS IN FME WATER 


Explosive 

Shot No. 

Charge Weight 
W 

(lb) 

Depth 

H 

Period 

T2 

(msec) 

Vti 

K2 

Cast TNT 

Q7F 

0,669 

298 

22.6 

0,75 

1-26 


Q8F 

.663 

304 

22,7 

.76 

3.33 


Q9F 

.651 

304 

22.3 

.75 

3.30 


C17F 

.660 

305 

23.0 

.78 

3.40 


G18F 

.660 

302 

21.6 

.73 

3.16 


Q2QP 

.660 

538 

13.9 

.71 

3.16 


G21F 

.651 

,65S 

593 

u.i 

.74 

3.49 


G25P 

567 

13.3 

.73 

3=17 


Q71F 

.658 

463 

14.9 

.71 

3.01 


G72F 

.660 

586 

13.0 

.73 

3.17 


G76P 

.660 

587 

12.2 

.69 

2.94 

Average 





0.73 

3.22+0,05 

Torpex-2 

G118P 

0.589 

595 


0.67* 



G119F 

.582 

597 

13.5 

.69 

3.47 


G120F 

.578 

600 

13.9 

.71 

3.60 


G121F 

.620 

603 

13.6 

.68 

3.46 


G122F 

.602 

611 

13.7 

.70 

3.51 


G123F 

.602 

611 

13.4 

,68 

3.48 


Q124F 

.576 

603 

13.3 

.69 

3.47 


G125F 

,574 

598 

13.2 

.68 

3.42 


G126F 

.572 

603 

13.3 

.68 

3.48 

Average 





0.69 

3.47+0.02 

Pressed 

GUF 

0.500 

303 

21.4 

0.78 

3.43 

Tetryl 

Q15F 

.500 

305 

20.2 

.74 

3.26 

G16F 

.500 

302 

21.0 

.77 

3.36 


G22F 

.500 

607 

11.8 

.74 

3.24 


G31F 

.500 

547 

12.8 

,74 

3.24 


GCIOIF 

.500 

695 

10,2 

.72 

3.13 


GC102F 

.500 

493 

14.0 

.73 

3.26 


GC105' 

.500 

230 

24.7 

.75 

3.23 

Average 





0.75 

3,27+0.03 

Pentollte 

G19F 

0.551 

335 

19.6 

0.75 

3.28 

50-50 

G33F 

.548 

396 

17,1 

.76 

3.25 


* No timing] from filmy assuming constant film spaed# 


- 12 ' 




.Ik* 


569 


TABLE ni 

••JUMiiART CF VALUES OF Tg/Tj 


Exploeiire 

This Report 

Ref. (7)« 

Ref. (4) 

TNT 

0.73 

0.77 


Preaaed Tatryl 

0.73 

— 

— 

Loose Tetryl 

— 

0.77 

— 

Pentollte 

0.75 

— 

— 

Torpev-2 

0.69 

0.72 


Llinol 

e»ee 

— 

0.70 


* Data obtalnad near sm'faosB. 


TABLE XI£1 

THIHD PERIOD CONSTANTS IN FREE WATER 
Charge Weights ^ ^ Ih 


Shot Number 

Explosive 

Period 

^^3 . 
(msec) 

ryij 

% 

GIRT 

TNT 

19.0 

0.88 

2.77 

020? 


11.9 

0.85 

2.94 

G23P 


10.7 

0.80 

2.54 

iverage 




2.7510.U 

G15F 

Pressed Tetryl 

16.9 

0.84 

2.72 

Q16F 


17.4 

0.83 

2.78 

G22? 


10.2 

0.87 

2.81 

GCIOIF 


3.6 

0.84 

2.63 

GC102F 


. 11.8 . 

0.84 

2.76 

Average 




2.7210.03 

G19F 

Peiitollte 

14.0 

0.71 

2.34 

0119? 

Torpax-2 

11.9 

0.88 

3.06 

0122? 


11.5 

0,85 

2.93 

G123P 


10,6 

0,79 

2.75 

0124? 


9.9 

0,75 

2.58 

0125? 


10,2 

0.77 

2.64 

Average 




2.79+0.09 


- 13 ' 


570 


footnotd m page 7 of this raport, the oos?seotlon9 amountisig to as much 
as 15%* Tha total water depth was 20 ft. The corrected period constants 
are given In the last colmm of Table ZXV. 

Sxeludisg the values at 1*75 ft (21 in,) for the reasons Indicated 
above, the average value of the period constKTxt for prossed tetrjrl 
is and for loose tetr^l is 4*57* These values show only a slight 

scatter and are only about 1$ higher tiian the corresponding ones In free 
wfiterW (see Tables III and Till), This higher result would itidlcate some 
effect of the surface, not taken into account in the theory, which 
beoomea of impori^ee mtsn the bubble is within 2>3 radii of the surface* 
However, considering that the total oorrectlon for the surface may be as 
much as 15^, i^e agreemeut is remarkably good* 

Hots also that the difference between loose aiui pressed tetryl found 
in the deeper shots Is preserved here almost quantitatively* 


TiELE nV 

FIRST PERIOD CONSTANTS FOR 25 QU TETRYL CHAROES 
CLOSE TO THE SNliS'ACE 



Shot No. 

Depth 

H 

(ft) 

Period 

Tl 

(msec) 


Pressed Tetryl 

Q99F 

5 

'77.8 

4.47 


Q135F 

5 

77.5 

4.45 


Q134F 

3 

77.5 

4.45 


Q133!r 

2 

74 

4.42 


G136F 

2 

73.5 

4.39 


GG139F 

2 

74.7 

4.46 


G132F 

1.75 

74.5 

4.55 

Average 




4.4^0,02 

Loose Tstryl 

GC15QP 

5 

79.8 

4.59 


Q147F 

5 

79.4 

4.57 


G146F 

3 

80.2 

4*60 


G145F 

2.5 

79 

4.58 


G144F 

2 

75 

4.50 


GUSF 

1.75 

75 

4.61 


G151F 

1.75 

78 

4.80 

Average 




4.63*0.04 

* Corrected for the effect of surface emd bottom. 


** The values for l/Z lb charges . Those for 25 gm chw’ges in free water 
appear to be lower, but the evidence is not conclusive. 


U- 




571 


8. iteximua Bubble Radii 

A» Fyaa Watey . Th* high-spsad motloa ptotures^^ of th« bubble mvA 
used for these measurements The actual size of the babble was calculated 

measureaents of its apparent dlaseter and of the apparent length of a 
knosB scale on each print. This tms chocked bjr calculating the sine of the 
bubble f]ron the image size on the flln and the known optical characteristics 
of the system (see Ref. (2}» Appendix I). Object sizes calculated by the 
two methods agree to about 2SEg Since the scale comparison was less 
laborlousj this method was used to find the bubble radii reported here. 

In the case of the 25 ffn chargee (Table Tfl) the area In the plct’oro 
covareii by the bubble was found by planimetarlng, and the radius was then 
calevilited from the area. Ho cheoks ware made in this series. 

The preoislon of the radius measurement appoaire to be about 2%, This 
degree of ju^eoision is possible, however, only up to the first maximum. 

After this point, as tiu bubble ooUapsea, its outline is obscured by the 
streamers In the water. This makes an exact measurement of the size of 
the bubble very difficult, and at the minimum the measurements may be 
considerably in error. Sane attempt to compensate for the masking of the 
bubble was made by measuring the radius of the sphere at the base of the 
streamers, assuming that the atroanara themselves contained none of the 
gas. This esnnot be dona very well at the Jttdnimum, since there they form a 
oompact blur completely masking the bubble. 

After the first minimum, the bubble, while nearly spherical, is 
never as smooth as d\n'lng the first expansion. It is thus apparent that 
the difficulty of estiaatJjig the position of Its outline becomes greater 
in the second and succeeding cycles, and th® validity of conclusions based 
on thV estimated radii becomes more doubtful. 

From the measured radii, smoothed radius-time curves were drawn for 
each charge photographed. In some oases through the third minimum. As 
can.be seen in Fig. 1, a typical curve is extremely steep right after 
detonation and near the minima. Little reliance can be put on the va7.ues . 
in these regionsj however, near the first maximua, the radius con be 
estimated rather accurately because of the flatness of the curve . As a 
eipnseaunncB , the oaleui^tlons carried out were based on the meudLmum radii, 
aiod no attempt was made to use the observed minijaum radii other than as 
u^per limits. 

Tables IV acid ivl show the results obtevinsd for the first maximum. 

The shot numbers refer to the same charges discussed under bubble periods 
and the exaot weight of caoh ohai'ge oan be found there. From the valuea 
of the measured radii wars oaloulated values of the proportionality 
constant in the equation 


* It should be noted that the standard procedure was to photograph the 
cylindrical charge with its axis normal to the optical axis of the recording 
system. Any exoaptlons to thle charge orientation will be specifically 
pointed out. 


- 15 - 






573 


TABLE XV 

FIRST MAIBIUM BUBBIJfi RADIUS 
Charge Walghti l/2 lb 


Explosive 

Shot Number 

Depth 

H 

(ft) 

(in.) 

Jl 

Caist Tl]T 

G2F 

304 

18.8 

12.5 


G4P 

304 

18.8 

12.5 


G5F- 

305 

18.8 

12.5 


OeB’ 

304 

18.8 

12.5 


Q7F 

298 

13.9 

12.4 


G8F 

304 

18,9 

12.5 


G9P 

304 

18.7 

12.5 


G17F 

305 

19.4 

12.9 


QISF 

302 

19.4 

12.8 


G2QF 

538 

15.9 

12.6 


G23J- 

593 

15.7 

12.9 


Q23F 

567 

15.6 

12.6 


Q71F 

463 

16.6 

12.6 


G72F- 

586 

15.4 

12.5 


G73F 

576 

15.6 

12.7 


G74F 

556 

15.7 

12.6 


Q76F 

587 

15.1 

12.3 

Avarnga 




12.6+0.04 

Pressed Tetrpl 

GllF 

300 

17.3 

12.6 


Q12F 

300 

17.2 

12.5 


G13F 

302 

17.3 

12.6 


G14F 

303 

17.5 

12,8 


Giy 

305 

17.4 

12.7 


G16P 

302 

17.8 

13.0 


G22F 

607 

14.3 

12.9 


G3lF 

347 

24.9 

13.0 

Average 




12.8+0.07 

Torpex-|2 

G119P 

597 

16.9 

U.5 


G12QF 

600 

16.9 

14.5 


G121F 

603 

17.1 

14.4 


G122F 

611 

17,1 

14.6 


G123F 

611 

16.8 

U.3 


G124F 

603 

17.1 

U.7 


G125P 

598 

16.6 

14.2 


G126F 

603 

16.7 

U.U 

Average 




14.5+0.07 

Pentollte 

Q19F 

335 

17.5 

12.7 


G33F 

396 

15.9* 

12.1 


^ Average of two radii of the ell^tly e].llptlcal bubble. 


- 17 - 


574 


( 8 . 1 ) 


*lia 


'-it) 


V3 


where - laaxinnjiii raSiua for r.-ih oscillation (ft) 

W “ charge weight (lb) 

Zo “ H + 33 t where H ie the depth of the charge in ft. 
The values of are shcim in the last column of the tables. 


TABLE m 

FIBST ULXXUUU BUBBLE RADIUS, PRESSED TSTRYL 



Charge Weight] O.OS^ lb 


Shot Number 

Depth 

H 

(ft) 

(in.) 

Jl 

£15 

380 

7.75 

12.6 

El6 

320 

7.3 

11.9 

E17 

196 

8.7 

11,6 

E13 

334 

8.7 

13.6 

E19 

344 

8.35 

13.2 

E20 

347 

8.2 

12.9 

E22 

205 

9.6 

13.0 

E23 

207 

9.8 

13.3 

Average 



12.8+0.25 


B« Radii Close to a Free Surface . Photographs of the bubbles from 
25 gm tetxyl ch^ges described under period meaeurements (Sec, 5) were 
measured, and radius^time curves drawn. The resulting maximum radii are 
given in Table XVII together with the values of J;;^. 

The precision of measurement is less for these shots than for the 
previous ones despite the superior definition of the bubble surface 
occnaioned by back lighting in this Instance. The scale used was a 
12 J.n* tj^'anaparent ruler, with the 1 in. squares on the ends blackened. 

This was rather indistinct in the photographs, and checks of the 12 in. 
against the 10 In, distance often disagreed by as much as 2-45f. Another 
Indication of possible errors appeared when a check of the bubble radius 
was mode from the optical eliaracteristlos of the pystem and differed from 
the above maiksuremant by 2-3^. The overall accuracy in the radius neasure- 
ment Is probably no better than 3-4%. The greatest difficulty comes from 
the fact that only 3 >art of the circumference of the bubble is visible in 




- 18 - 


575 


tha photogysphs*. 3lQca the bubble m&y very well ba ovoid when ao close 
to the aurfaoet the nearly vertical radii (whioh were the only ones that 
could be measured) i:aay not accurately represent the radius of tha 
equiviilant spherical volume. 


TABLE XVII 

FIRST MAniroii BDBBLB RADIUS FOR TSTRYI. GUmm 
CLOSE TO TKS SURFACE 

Charge Velghti O.O 55 R lb 


Charge 

Shot No. 

Depth 

H 

(ft) 

% 

(in.) 

^'1 

Pressed Tetxyl 

Q132F 

1.75 

19.9 

U.O 


097P 

2 

21.0 

U.8 


G133F 

2 

19.7 

13.9 


0136F 

2 

20.0 

U.l 


GC139F 

2 

20.0 

14.1 


G9SF 

3 

19.7 

U.2 


GW 

3 

19.4 

13.9 


G9<?F 

5 

19.3 

U.2 


G1:J5F 

£ 

19.7 

U.4 


G137F 

5 

18.6 

13,7 

Average 




U.1 i 0.13 

Loo(ie Tetryl 

0U8P 

1.75 

20.6 

U.5 


G151F 

1.75 

20.8 

U.7 


G144F 

2 

20.4 

U.4 


GU5F 

2.5 

20.2 

U.3 


GI 46 F 

3 

20,5 

U.8 


GC15QF 

5 

20.5 

15.0 


Q147F 

5 

19.7 

U.4 

AvSriSige 




U.6 1 0.08 


It will bo noted tliat the average value of for loose tetryl la aboui» 
above that for ths pxssa^cl tetryl. The period constant for loose tstryl 
was ajLso found to be about this same percentage higher than that for pressed 
tetryl, not only in free water (l/2 lb charges, S^o, 5) but also In this 
Me aeries of shallow shots (Sec. 7). Sines both J;]^ and (the period 

constant) artt; directly relat^ to the energy of the bubble, the fact that 
the difference between loose and pressed charges Is maintained here seenia 
significant, since equal syeteraatic errors in the measurement of J and K are 
exceedingly unlikely. The results are most plausibly interpreted as 
indicating a real difference in bubble enei’gy, due either to variation of 

See Ref. (1), Fige. 36-39, ~ ~ 


876 


detonation ensrgy vdth density of loading or to a difference in the partition 
of a given amount of detonation energy batwDon the shook wave and bubble 
phenomena for the two types of chai'ge. 

Tha values of the proportionality constant, J, , are higher here than for 
the shots in free water. The values of K. (Table XIV) are similarly higher 
than the free water values. This would argue that thera may well be a 
systematic difference between the deep and shallow shots. (See Seo. 10.) 

In Table SVIIl are shown for comparison some results from an earlier 
reports) on larger charges. The first cliarge was within 3 bubble radii 
of tho surface and is thus roughly comparable to the 25 gm charges at 5 ft. 
The other two are relatively deeper. Tha values of are slightly higher 
than for the 1/2 lb charges in free water (Table XV), but this is within 
the limits of error of Table IVIII and may not be significant. 


TABLE XVIII 

MAXIMUM BUBBLE HADIUS, LABGB TMT CKAHQES 


Shot No. 

Weight 

W 

Depth 

H 


Jl 


(lb) 

(ft)« 

(ft) 


aii2J 

' 56 ' 

36 

12.3 

13.2 

G.H4J 

56 

83 

10.0 

12.7 

G117J 

295 

95 

17.5 

13.3 

Average 




3=3 


* Calculated from period. 


C, Radii for Second and Third Periods in Frsa Water . These were 
found as described above and are summarized in Table XIX. No values for the 
very shallow shots were measured because of the uncertainty in the depth 
after the first oscillation. The scatter in J is an indjcatlon of the 
decrease in precieion of these valuss as compared to the first bubble cycle. 

One value for a 56 lb charge is included. The exterior of the bubble 
appeared to b® very rough during tho aeepnd oscillation and the radius 
G-niipt b© estimated at all precisely* To calculate Jo in this case, the 
bubble was assumed to migrate upwards 15 ft between the first and second 
bubble maxima. 


IV. COMPARISON OF EXPERIlffiNTAL RESULTS WITH THEORY 

9. Raesms S£ ^ Theory t T]ia Bslg 2£, SiM Adiabatic Parameters 

The theory of the oscillating motion of the bubble of gases produced by 
an undei*^ter explosion has been developed in recent years principally by 
DoringH), Zollar^5), Henoss^^), Kell and Wunderltchl' ) in Germeny; Taylor^^^ 


- 20 - 


577 


TABLE XIX 

SECOND AND THIHD MAXIMUM BUBPIE RADII TW fflM WATIR 
Charge Weight* 1/2 lb 


Ejcploaive 

Shot Nymbor 

(in.) 


^3 

(in.) 

•^3 

TNT 

06P 

13,2 

8.B 




Q7P 

12,7 

8=4 




G8P 

12.8 

8.5 




Q9F 

11.4 

7.6 

10.0 

6.7 


QITP 

13.4 

8.9 




G18F 

12.1 

8.0 

10.6 

7.0 


G-2QF 

10.6 

8.4 

8.2 

6.5 


G2'.iF 

9.9 

8.1 

7.9 

6.5 


G23F 

9.9 

8.0 

7.8 

6.3 


G71F 

11.0 

8.3 




G72F 

11.6 

9.5 




Q76F 

11.6 

9.5 



Average 



8. 5+0.2 


6, 6+0.1 

Pressed Tetryl 

GUi' 

12.1 

8.3 




G15F 

n.s 

8.6 

9.1 

6.7 


G16P 

12.2 

3.9 

9.5 

6.9 


Q22F 

9.0 

3.1 

7.1 

6..< 

Average 



■^^0„2 


6.7+0.15 

Torpex-2 

G118P 

10.2 

8.7 




C-119F 

10.9 

9.3 

8,2 

7.0 


G12QF 

10.5 

9.0 




G121F 

10.8 

9.0 

8.6 

7.2 


G122F 

11.1 

9.5 




G123P 

10.7 

9.1 

8.1 

6.9 


G124F 

11.1 

9.6 

3.3 

7.2 


G125F 

10,4 

8.9 

8.S 

7.6 


G126P . 

10,2 

8.8 

8.3 

7.2 

Average 



9. 1+0.1 


7.;^+o.i 

P-'sutoitt-e 

G19F 

12,7 

9,3 




G3^ 

10.9' 

8.4 



56 lb TNT 

G109J 

9.3 ft 

9.1 




- 21 - 


578 


Temparl«jjl9) in Great Britain, and Herring^), Kennard'^^), Shiffman and 
Friedman*^? in the United Statsa. In this section, the s:cpsriffisntal resultn 
desoribad prsvioiiely in this report will be oompared idth the theory 
subjeot to the following eus«mptiona» 

:l. the llqiuid is regarded as an Invleold, inocnpreasible madlm of 
unlimited extant, 

11. the vertical motion of the qenter of gravity of the bubble Is 
considered to bs negligible, 
lii. the Icinetio eiaergy of the gas Is negleoted, 
iv. the eiQjassion and oontraotion of the gas is assumed to be adiabatic 
and to f oilow the equation 


w'*' 

pv “0 

(9.1) 

eq>proprlata for an ideal gas. 


Under l^hese conditions, it is convssiient to introduce 
daflnltions of the radius and time, in aoooi'danoe with Ref. 
have the formt 

non^dimoncional 
, (21), which 

A ■> La 

(9.2) 

T = Ct 

(9.3) 

in which A and T are the aotpal radius and time respectively, a and t are 
the non-dimensional radius and time respoetively, and the length and time 
Boallng factors are defined by the following equations i 

L . JJW- » 1.729 (xQ)^^ 

^ Po h 

fQ.A'i 

\ r 

3 0 3/2 ^ w, 

C - ^ L - 0.374 (rQ)^3 

fa 


where W is the weight of the charge, Q the detonation energy per unit weight, 
r the fraction of the detonation energy remaining In the bubble motion after 
passage of the shook wave, the initial, hydrostatic pressure at the depth Zq 
(which Includes the eqxilvalsnt depth of the atmosphere) where the bubble is 
formed, and jo the density of the liquid medium; the numerical factors in the 
third members of Sqs. (9.4) and (9«5; are appropriate for L and Zq in feet, 

C in aeconds, rQ in calories per gram, W in pounds, with ^ - 1.025 (sea 
water). 


- 22 - 


579 

The U30 of the non-dimenalonal variables thua defined allows the 
expression fo? the conservation of energy to be written in the forms 

.3 . 1) . k .3a-»-) . 1 (,.5) 


in which the dot slgnlfiee differeatiatloa with respect to non-dlmenaionaJ. 
time. The parameter k is defined by the expressions 


k 


c Po 

(rQ)^ ('T-l) 


2Lf.&) r 

(rQ)^ 


1 r, >'-l 

... £a 


(TT-l) 


(9.7) 


The following oonslderatlons give the theoretical relation between 
the observed period sod maximcim radius and the value of rQ, under different 
assumptions regarding the parameters k and 'V in Eq. (9.6)t At maximum or 
minimum radius. , a vanishes in Eq. (9.6) leading to an equation 


s3 a3(l-«^) .= 1 


(9.8) 


whose greatest and least roots should be the non-diaenslonal maximum and 
minimum radii} note that if k vani»b^s (internal energy negligible) the 
maxiiauiD radius becomes simply % ■ i or Ajj ■ L. The roots for various 
values of ^ and k appear in Table IS, whese a^j is the non-dimensional 
maximum re^ius and the non-dimensional minimum radliis. 

The time for one oscillation is computed by obtaining twice the time 
end maximum llldil.* In the case where k is neglected. 

Sq. (9.6) oaii be integrated In. terms of the incomplete beta function^/ and 
the numerioal result is. 1.492. On the other hand, when k ie not neglected, 
Shiffman and Friedman^) liave shown that the period can be approximated 
very accurately by means of a quadrature formula, involving the Tchebyoheff 
poljmomiels . Such calculations were parforraad by Shiffman and Friedman 
for E few values of %' and k using fifth-order Tchebyoheff polynomials: 
those authors remark that the fifth-order approximation agrees witldu IjS 
with tbs third-order approximation* In the present report Shiffman' s and 
Friedman" B calculations have been extended to Include a wider range of 
values O.T the paraiosters 'ir' and k for reaeons described below. A sample 
caioulation was also performed comparing the fifth- and seventh-order 
approximations which were found to agree within 0.025^. Tlis results are 
reported in Table XO, end are also exhibited graphically In Fig. 2. 

* Examination of experimental radius-time curves reveals that they are 
essentially symmetrical about the maxiniuffl radius. 


- 23 ' 


580 


4M * M trxt 

XJUMiEl AA. 

lUXBiUM AND ICmiUDTf NOK-OBOSNSIONAL MDll FOR VARIOUS r AND k VALUES 


r 

k 

•H 

% 

Isa 

0.1 

0.9647 

0.02154 


0.2 

0,9246 

0.06844 


0.3 

0.8774 

0.1350 


0.4 

0.81B8 

0.2211 


0.5 

0.7363 

0.3359 

1.25 

0.1 

0.9645 

0.04642 


0.2 

0.9235 

0.1172 


0.3 

0.8742 

0.2031 


0.4 

0.8100 

0.3064 

1.3 

0.1 

0.9643 

0.07747 


0.2 

0,9225 

O.I681I 

1.4 

0.05 

0,9827 

0.08242 


0.1 

0.9639 

0.U72 


0.2 

0.9201 

0.2657 

mmm 

0 

1.0 



TAELS El 

KOH’DIUEKSIONAL PERIOD OF BUBBLE OSCILLATION FOR VARIOUS IT AND k VALUES 





OJMENSIONAL PERIOD vs PARAMETER 












582 


s£ ^ §Qg£BZ 



Qoiabiaatiea of Eqa. (S*l)» (9«3)f and (9«5) gives an etjuatioa for 
rQ in ittrms of the experimental period ocnetamt 


sQ 



( 10 . 1 ) 


while a giallar combination of Sqa. (8.1), (9.2), and (9.4) gives an 
equation for xQ in terma of the experimental radius constant 


r« . 0.1% (i)’ 


( 10 , 2 ) 



Waving the natural assumption that rQ is ijEidependent of the depth and 
noting that t and 8|j are sllghtl 7 depth dependant through tlie pui’ameter k, 
it is expected that the proporiiionallty "oonstants” -J and K sill vary 
directly with ajj and t. If such variaid.on of J and K with depth had been 
measurable with aufflolant precision over a wide enough reng», it would 
have been posalble. In principle, to determine uniquely the parameters 
nr and k as well, as rQ. Since, however, such variation was maaked by 
the experimental error in J and K for the anall range of depths adtually 
observed^, it has not proved feasible to make a unique calculation of 
these parameters. Thsrefore the two Sqs. (10.1) and (10.2) ware first 
used to eliminate rQ with the result 


^ - 4.62 ^ (10.3) 

■« ^ 


The observed values of the period and radius proportionality constants ware 
use4 to compute the apparent value of t/ay-. Referenos to Fig; 3 (whioh 
was bbtainsd ifrom tits flguirss jh Tables XX and XXI) then allows the 
dstormisation of k for an arbitrarily ohossn nr * The non-dimensional 
period and radixis then beocme deteiminate, and a value of rQ may be 
obtained which is uonalstent with both the radius and period data. In 
Table XXII, these energy values are given for the first period of 
oscillation of several of the sjqplosives dsscribsd previously in this 
report. 


* litbough the depths varied by about two-fold, k depends upon depth to the 
l/2 power or lees, and Sy and t are not very sensitive to k. 



-26- 






585 


la aacih caaa the value of the bubble energy is also given neglecting 
the internal energy of the gas (k i* 0), which fixes the -vsilueB aj| « 1 and 
t - 1.492* hare it is to be expected that the radius ca}.cillation will 
give a lower value than that obtained from the period constant, since t is 
relatively insensitive to k, whereas Sjj, when the internal energy is taken 
into ecco’jat, may decrease lOjS by comparison with its val\se for no 
internal, energy. This la found to be true in all eases except the two 
involving tatiyl charges vary near the stirfaoe. Consistent oaloulatlons 
of the energy cannot be muds in these two cases, since the energy cal-> 
oulated from the naxlnnus radius is apparently higher than that obtained 
from the period of oscillation (this leads to values of t/e^ lower than 
833y found in Fig. .1) . 

No complete explanation of this anomaly can be given at present. 
However, it la interesting so® "tti® theory predicts for a case in 
which the adiabatic law, ;Eq, (9.1), is replaced by the simple assumption 
that the gas pressure is a constant, pg. This assumption may be a rough 
representation of the facts, if, during a liwge part of the motion, ths 
condition of the gas is largely determined by the eondensatioa of water 
■wpoTf a condition which might be true vdien the explosion occurs at the 
shallowest depths described in this report. 

7?he resultant expressions for non-dimensional radius and period are, 
respactlvalyt 



( 10 . 4 ) 



( 10 . 5 ) 


Although the ratio t/i|| is never less th^ 1.492 for positive pressure 
ratios, p»/p , and thus cannot explain the low value t/ajj ■ 1.4® found 
ej^erlmsnlolJy, It seems noteworthy that both the radius ““^period 
constants should be larger on the basis of Eqs. (10, and (IP.5) than 
in the ease where the adiabatic law holdfe and that the oba,erved donstahta 
are indeed found to be larger near the surface than fit grSater depths. 

CaloiUatlons for the second period of bubble oscillation are given 
in fable mil. 

From this discussion it is seen that suitable data are not available 
at ipresent to deteiwiinw the most appropriate value of the specific heat . 
ratio. It should be mentioned that the theoretical calculations of Jones**^ 
indicate an effective value of T - 1,25 for TNT of the density used in 
these experiments. 





587 


Using the data ahomi above, It is now possibla to nroposs a 
partition of the energy released in the explosion. This ia shown in 
Table ZH7. For the ^bble energy, a value of 'k’ ■ 1«30 was assumed. 
The values for the total energy are obtained from Ref, (23) • They were 
calculated by assuming that the 03^gan was consumed in the following 
sequence s 


i. 

Hg + •|' Oo 


H2O 

il. 

C ♦ -J- O2 


CO 

iii. 

CO O2 

— fc. 

C02 


The shock-wave energy values have been computed f^’om empirloal equations 
for the energy flux as determined experimentally at Idle Underwater 
Explosives Researoh Laboratory. 


u'sm xnv 

ENERQY P^LRTXTXOR XU URDERNATER EXPLOSION 


Exploaiva 

Total. Snerssr 
(oaVci) 

Shcok=Wave Energy 
(caVgn) 

Bubble Enargv 
(First Period) 

(eal/g“) 

TNT 

977 ft 

244 ^ 

467 

Pressed Tetryl 

1035 * 

2630 

500 

Bleating Qelatin 

1509 * 

331 ° 

626 

Torpox-2 


349 ° 

728 


a. See Ref. (23), "Assumption 11" 

b. See Ref. (24) 

c. See Ref. ( 13 ) • 


While these values seem rcastmsble, it must be emphasised that there 
are uncertainties end a^aii^iptions in all of the calculations involved and 
that further rsfinpments iii „calcu3atioh and. aeasurement in the future' will 
recessltste some rbvlslon. 


- 31 - 


iSP^'SDlX 1 


EliFlRlC&L MDHJS-TIiySi GURVH! FOR OSCILLATING O&S BUSSLS3 IN FRE3S WATER 

\sy 

M. Ar«ova 


In asy study of th« phenomena assooiatad with the hubhlo fonsad by the 
gaseous pit'cducts of ^ underwater explosion^ it is essential that one know 
the else and shape of the bubble as a function of time* Asauinlng that the 
bubble la lilfitys spherical In shapes the bubble phenomena are complstely 
detenained if the bubble radius is known as a function of tlma* High-speed 
notion pictures of the bubble formed by oharges which are stoi greatly 
elongated snow that this is a good approximation for at least the fiirst 
bubble osciUatlon. 

To date it haa been necessary to integrate nunerioally th« theoretical 
equatlona of motion derived by H#Tring3), Sblff^nan and Prisdman^l), TaylorlS), 
and others in order to obtain a radiua-tlne curve* If one has an analytlo 
expresaion for the radlita as a function of tlma, one can avoid the laborious 
numerical integrationa required in order to predict such phenomena aa 
algrationn bubble-pulse, etc . 

The purpose of this appendix ia to find an analytic expression for the 
experimentally deteniined radius-time curves in free water end to indicate 
possible uses for this expression. 

In’ attempting to find such an analytic expression, it is convenient to 
reduce the experimental data (for 'the raditis aa a function of tine) to non- 
dimensional form. The units of tlna ,a^ length, used in this appendix are 
the first babble period (Tj,) and first naxlisum babble radius (A)q,) , 
respsotivwly. These units of time and length were chosen beoauaa they oen 
bm‘ computed from the equations in 'the body of this report* and because they 
reduce the data to a fobs which is easily analysed; 

The radius-time curves used in this analysis are those mentlMied in 
Sec ; 8 of' this rejiort. The composite oux^s (Figs . 4-9) In which A/Agy has 
been plotted against tAi tor TST at depths of 300 and StO ft| tainryl at 
300, »ind 600 ft| torpex-2 at 600 tty blasting gelatin at 500 ft contain 
7* 7» 9» 4» 8* and 3 individual radius-time ourveS respectively. 

Figures Ut 5, 7, 8, and 9 show that there io. relatively little soatter In 
the dans from shot to shot when a given weight and type of explosive is 
observed with the standard charge orientation. There la, however, oonsidereble 
scatter In the neighborhood of the minimum due to the dlffloultiaa mentioned 
previously in Sec. 8. The data for blasting gelatin (Fig. 9) are of 
particular interest since they olaarly indicate the nOB-spherlolty present 
in the early stages of the osoiliatlon of a bubble formed iy a cylindrical 
charge whose leng'th and diameter are approxlttately equal. There la con- 
aiderabl# doubt as to the values of the radlua for tha contracting phasa of 
the oscillation alnos the bubble was unstable and apparently formed four 


* It should be pointed o\it that in preparing the conpoalte curves the observed 
values of Ayn and T], were used to reduce each record 'to non-dimensional form. 













I 



I 


) 


) 



FI& • 

COMPOSITE, NON-DIMENSIONAL RADIUS-TIME 
CURVE FOR TETRTL 


CH«R«C WCIOHT: QBO LI • 

0.0 an LI • 


CMRIE OCFTH! 300 FT 
CURVE MRMKTEN, « > 0190 






rw.T 

COMP(»ITE. NON-IXMENSIOMAL RA04US-TIMC 
CaJRVE FOR TETRVL 


CHARM HKIRHT 0.80 LB CSSWe PeRTH «90 FT 

CURVC MRRMITIR, «>O.IM 



0.4 


0 .( 


0.1 


1.0 


T 

T, 


ri& s 

□OMPOS1TE, NON-DIMENSIONAL RAOlUS-TIME 
CURVE FOR TOHPEX-2 

OHAIiW WeiOMT: 0.*O LS CHAROE KPTM; COO FT 
CURVE WR4METER, V • O2S0 







595 


aftiall bubbles at the minimum. The radius values used in Fig. 9 were obtained 
by estimating the 1;otal volume of gas present at any instant and then 
computing the radi\ia of an equivalent sphere. Figure 6, which includes 
data from two sizes of charge, shows that for a scale range of approximately 
2tl it is possible to represent the radius -time curve for a given explosive 
at a fixed depth in non-dimensional form. Although data from a wide range 
of oxplosivs weights at a given depth ware not available, it is believed 
tha.t these non-dimensional curves will represent any size of charge in free 
water. 

The shape of the composite ciirves resembles that of a curtate cycloid^ 
(the curve traced out by a point on the spoke of a wheel which is rolling 
along a straight line in a plane). The aquation of a curtate cycloid in 
rectangular, parametric form 1st 


A « m - n cos 


T ■ m^ “ n Bln m > n > 0 (1-1) 


The constants m and n In this equation must be chosen so that A A yi and 
T ■ Ti for y equal to TT and i*esp9otively, and so that fits the 

composite curves at ^ equal to 2tK It should be noted that for ^ equal 
to 2tf, A/Ajii is idsntie^ly A mi/ Aai • If this ratio is represented by the 
symbol q, Eq. (I-l) hacomest 


iJl' <c|> t] 


»i« y]. (i-s) 


If one plots £lq, (1-2) with q taken as 0.25 along with the oomposito 
curve for TNT at 300 ft, it o<in be seen that the equation has been made to 
fit the data at f eqvial to TT* and Zft\ but that it predicts a value of k/ knf\ 
which is too low for intermediate values of ® . This situation can be 
remedied if one writes the equations in id-ie formt 


* Various authorities disagree as to the distinction between a prolate end 
a curtate cycloid. The author prefers the definition given above. 


- 39 - 





^ ■ (^) |l - (^) OO. ^ 1 

ii'aV [f 


«h«ro £ is a constant which in general should depend on q. If Eq. (1-3) is 
to supply physically admiasahla radius-time cvirvas, (1-q/l+q) + £, must not 
exceed unity. The hast fit is obtained, however, when; 


+ e - 1 

1+q * ^ 


Assuming that Eq, (l-4) is generally valid, the parametric form of the empirical 
radius-time curve becomes t 


itr • f 1 COB 1 1 

•s-iAL -i- ^ j 


l?- 


(1-5) 


Figures 4"9 show the data mentioned previously along with curves oalotilated 
from Eq. (I'-S). These curves show that, with a suitable choice of q, E^ (1-5) 
give a good analytic representation of the radlua-time curve for the portions 
of the bubble oscillation which are of interest; namely, for values of T/Ti 
greater. thAh 0,05, This follows from the fact that bubble phenomena such aa 
the Kig>ration to the time of the first minimum in the radius (A^j), the presswe 
pulse emitted by the bubble, etc., are Independent of the ejmets^ne of the 
radius-time curve in the early stages of the period. 

It should be noted that the empirical radius- time curve in the form of 
equation (1-5) involves only one aurbltrary parameter, q, which has a direct 
physical significance, aa has been pointed out previously. Hence there are 
t’so rs&xti iiam to rhi.sh ths aquations developed above can be put. One can 



-JV, 


oomputs q on the b&sii of the existing theory provided one has a laiowledge 
of the pl^sloal properties of the gas In the bubble. Using this value of q, 
one can compare the predloted radius-time curve with the known experimental 
radius-time curve ea a check ofl theory. In this connection it should be 
mentioned that the experimental data probably provide an upper limit to q . 
sinae the bubble ia ubsciu'sd at the tims of the mlninram a mass of material 
which, is more likely solid than gaseous. Thlo difficulty has been discussed 
in detail In Sec. B of the present report. 7he other line of attack would 
be to use the value of q necessary to givui a good fit with experimental radius- 
time curves as an aid in arriving at suitable values for the average physical 
properties of the gas in the bubble. 

Another of the uses to which the empirical radius-time curve has been 
put is the following} Using the bubble theories mentioned previously, one 
can compute the radius of the biubble for the case of a sero excess pressure 
field surrounding the bubble. Expressing this radius as a fraction of the 
maxionxm radius of the bubble, one owj determine, from the empirical radius- 
time curve, the fraetiwa, of the periodic time at which the excess pressure 
vanishes. It is found that this time agrees excellently with similar times 
determined from piemoelaetric gauge measu}.*ements of tho pressure field 
surrounding the bubhla^^). 

Due to the paucity of photographic data for oharges fired at small 
non-dimonslonal depths, no attempt has been made to develop an empirical 
rapressatation of tho radius-tios curve for this very Important case. 

However, It can be stated with aBs\a*ance that Eq. (l-5) does not hold when 
the praaenoe of free or rigid surfaces becomes Important. 


598 


LIST OP REFEREHGES 


Is Photography of Uadarwater Bxpioslona, II: High speed Photographs of 

Bubble Phenomena^ by S» Sijift, Jr.» F. 13. Fy«, J» 0, Dsoiua. sad 
R. S. Price, HavOrd 95“4&, Daoember 194^. 

2. Photography of Undarirater ibqplosioiw. III: Servlss i?6aponaij by 

J. C, Deeius, P. M. Fye, R. S. Prioa, V7. S. Shultz, and E. Swift, Jr., 

HiiVOrd 96 - 46 , February 194? • 

3. Theory of the Pulsation of the Gaa Bubble Froducsd by an Underwater 
Ejqploaion, Conyers Herring, HDRC C4“sr2C~010, October 1941. 

4 . Measurement of Bubble-Pulse Phenomena, It Mark 6 Depth Charges TNT 
Loaded, by C. P. Slichter, W. 0. Schneider, and R. H. Cole, OSRD 6242, 

NDRC A-364, March 1946. 

5 . Private Communication, B» Friedman, Applied Mathematics Group, 

New York University. 

6 . Private Conimunieatloii, A. Borden, Taylor ilodol Basin. 

?. Measurement cif Bubble Pulse Phenomena, II: Small Charges, by A. Arons, 

A. Borden, and B. Stiller, OSRD 6578, NDRC A-470, March 1946. 

8 . Measurement of Bubbl© Puls® Phenomena, Vi Studies of the Anomalous 
Pulse Using Small Charges, by E, Swift, Jr.. W. M. Plook, and 

R. S. Price, NairOrd 409, {not yet published). 

9. Daiaage to Thin Steel Cylindrical Shells by Underwater SaploBiens, 11, 
by J, C. Deeiuii and 0. Gever, NavOrd IO 6 - 46 , November 1946. 

10. Measurementa of Growth and Oscillation of the Bubble Formed by an 
Underwater Explosion, by Mins Design Department, Soientlflc Section, 

Undex 35, Key 1943. 

11. Vertical Displacements of the Gait Bubble Formed by an Underwater 
Explosion, by H, F. Willis and R, T. Aokroyd, May 1943. 

12. The E]q>erlmental Evidence on the Behavior of the Gas Bubble Due to the 
Explosion of a Mk VII Depth Charge, by Mine Design Department, September 1943. 

13. Comparison of Underwater Explosives Having Varying Oxygen Balance, by 
C. R. Niffenegger, J. P, Sllfko, E. A. Oh-rlstian, A. H. Oerter, end 

J. S. Coles, NavOrd 402, June 1947. 

14 . Die Soiwlaguagen der bel einer UntsrwasserBp'engis^ ©netefceaden^Oasblase 
inai die Druckverteilung in Iher Umgebung, W. Dorlng, Instltut fur 
theoretlsoh Pbyslk dor Unlversitat Gottingen, January 1943. 

15. Adiabatlsoh pulslerende Gaskugel in unendllob ausgedehnter Flussltfcalt, 

K. ZoUer, Forsohnngswtalt Graf Zeppelin. 

16 . Bemerkugen sum Gasblaaenaufstelg, by Hermea, CPVA G1B7/44. 


- 42 - 


599 


17. Osoiliatlon of Gas Glotes in Undarimter E:cplofilona, by A. Kail and 
W. Wtmderlioh, TMB Translation 209, Februiury 1946. 

18. Vertical Uotlon of a Spherical Bubble and the Pressure Surrounding It, 
by G. I, Taylor, S. W. 19. 

19. Critical Survey of Bubble Phenomena based on Iriformation Available up 
to August 1943, by H. N. V. Temperlevj Undex 64 , November 1943 » 

20. Uigratlon of Undertvater Gas Globes jDue to Gravity Neighboring 
Surfaees, Isy E, H. Kennard, TMB R-182, 1943. 

21. Studies on the Gas Bubble Resulting from Undemater Explosions] On the 
Boot location of a wine Near the Sea Bed, by Su. Shif^oan B. Friedman, 

AME 37 .3R. 

22. The Pressure Volume Relations and the Chemical Constitution of the 
Products of Detonation in TNT during Adiabatic Expansion, by H. Jones, 

P.. G. 212, July 1941 . 

23 . Report on Study of Pure Explosive Compounds, lit Correlation of Thermal 
Quantities with Explosive Properties, by Arthm* D. Little, Xne., Report 
to Office of tho Chief of Ordnance (Contract W‘‘19~'020-0Iu)~6i436). 

24 . NDRC Div. 2 Interim Report on Underwater Explosives and Sxploslona, 

CE- 23 , p. 20, June 15-July 15, 1944. 

25 . Preparatiion of Gbargea for the Stiidy of Eteploaian Phenomena at the 
Undertater Explosives Rassmrch Laboratory, by P, Nawnark and B. L. Patterson, 
OSBD 6259, NDRC A-381. 

26 . Energy Partition in Underwater Es^losion Phenomana, by A« B. Arons aiid 
D. R. Yennie, NavOrd Report 406 . 


- 43 - 


ANALYSIS OF THE SPKAY IX)ME 


G. W. Walker 

Mine Design Department, Admiralty 


British Contribution 


601 





ANALYSIS OF THE SPRAY DOME 

0. g. talker 


I 



mno JtieKign Dapartisent) 
AdBlraAtv 

mi 

a « « • « 


Obj*et. 


Tiie oftjeet ot this rssearch Is te jiroylfle sufficient flaia ffom smieh It »!1! bo oosslsle 
to torn tn accurate cstlmto of tne amount anb ulstrlbution of explosive materlai rei|Ulreu to 
croouco oastruetlon of a given structure at a given rang* of btitance. 

The reaonreh might erccaeb on entirely tmolrlcal linos. Aoart from iho onorsnaua ecst of 
mttrlal anb tine Involved, such a msthco Is not In accordance alth scientific method, which alms 
at co-ordinating Informtlon In such a nejt that a reasonable eatimste nay be fonnoo of the effect 
of varying any particular set of circumstances. 

Thus the research will follow two main llhssi- 

(t) full seal* soa experiments In which the oanuge (lone on certain known structures 
py given oxolcsions Is quantitatively determined, 

(a) Tnc detemUnntlon of those ohysical quantities In nn sxuloslon which dotsrains 

n Ship Is aanugod as the rssult of the eressurcs SAerleO on It, and which are orocagateo 
thfdugh She watir as s result of too oxptcslon. Thus a comoletu kn-owledje of the time develooment 
of oreiiora In the water at any olstanee from a given mine If, a vital link In the chain' which 
cunnseti the oxoloslon with tnc oimage done. 

It "111 nrob«hly h* some time nefore an accurate solution of this difficult crcbled) Is 
obtalmid, so thst meatwhlle we croceed to an account of an Investigation of part of the rroolem 
wnlch has been carried out In Mining School Curing the oast year, and which has thrown consloerahlo 
tight on the matter. 

The exolcsion of a mlnct In water, is fuiloweo by nr. uuhruvei of jm nud itupoi. 

Oeneral aoo6t •Suomnrlne nines'' oIsCjiH'S tnn question or a study of tns tn"nc'neho hi a ttr;ans 

of quantitative meesursment. His view, oxcressoo oago ui. Is that the phenomena are too varlanlo 
to give results of groat value. 

Now the onenomena uoservoo occur In distinct chases of wnicn there arc two or cossibly 

thre* oecenelnq on elrcurastancOs. it nwasrs that njneral Aebnt Is thinking aoout the very fine 

seectaeular offect of the clacharge of gas eic., and that ha had net observed or nao not 
tionaloereo tne remarkably sharp sonsy phenomenon ehaso (1) which alimst liwwolsttly follows tnv 
firing of tne mine end crpcedes tho uoheavel of gal, etc. 

This ibray rises almost Instantaneously In the torn or a oorno as shown In “lato 1. The 
onwr cage Is yty;' s^srol/ alvldob from tho still water surface ana every part of thi oorno seems 
to rist vtrilcelly. 


Ilumerous experiments have neon maOi. and shew that the soray phenomnon Is a definite one 
and that It loans to Imoortsnt conclusions as tc tho nature or tns explosive wave orooagotpd 
from ths nine. 


Adtumotlenc have tw t>e mags Pefore we can Interpret the spray phenomenon and the 
Justification for the assumptions consists In th') runner In which the conclusions agree witn tne 
ixeerlBentSi 


Thr 




'/ 


- (« ' 


602 


- 2 - 


Vhs BSSiiiTtDtUns •■.roi- 

(j.) That ths axalasivh irasaures am orc^egatea »itft the namal vtlcclty 'C 

souao wsvos In wator. That vs1.‘.nltj> la 1*50 m»tr«s ior secana, anil In r.'una 
nunhcrs we ahnil cat) It 1100 netres cor acc.'nl. henjureaenta ;f ths 
ccmcrsaalbit Ity ot v»tar snati that sxcoit »lthln a f<w Toot :jf tiio extlcsbn 
thero Is n; rsnson t,' excect nny scrljua dccarturc fr;m the aOCM smoj. 


(2) That the navo Is scharlcdlty symmetrical about the source. 

From this It fallals that the pressure at any oolnt while OeoenOent on time varies Inversely 
as the distance r from the source, that the velocity of dlsolacement Is entirely radial ana IT 


0 ^ ■ density of water 
C > soeed of orooojatlon 


D « oressure of any point 
V u radial velocity at same oolnt 


then 0 = Pn ^ 


with very great accuracy. 

Wi nuiy now oxolaln tne soray ohenc-menon In genera) terms. 

Let X y be the free water surface of doss wotcr. 

M a submsrgod mlnsat decth a 

K* an Imago nine at the ootical IiMge of N In X T. 

The vXiote effect at any oolnt » Is due to the orlnary effect from m ccirblned with the 
effect reflected from the surface x y. fseo Figure t), 

How the surface la a free water surface ard thus tile resultant oressure there must be 
opnisnently aei'o. This can Os secured and ths effect of the reflection at any point In the eater 
Is reoresentea oy an Imaginary exoloslon at M* of equal magnitude but In npobilte sign to that at 
M and starting at the same Instant. 

This secures that at any oolnt O on the Surface the pressure le permanently zero, but the 
resultant velioelty at O Is cemcounoad of two nqual effects arriving at creelsely the sane Instant, 
one dlroetso from H to J and the other«from g to M*. , The resultant Is no horizontal eomccnent but 
a eurtly vsrtlcal voloelty. Calculation ahows that this velMlty 


Hpw Ojj Is varying with time and wa can further show that If the negative pressure which the 
'Water wl.l suosort Is neglljlble then each point of the water aurfoco In succession must, part 
company with tha jwnervl hnwu pf water at the Instant umen u^ aliains a ma.xiirum. and proeesds 
upwards with the eorreaooiulng Initial velocity. In oartlevlar at 0 vertically noovs ths mine 
the Initial velocity of the soray v^ « 2 whtre 3^ Is the maximum oresiura oeveloeea at 


dl.stanee d. Conversely If we can measure the initial velocity we can calculate o “ 1 p C v tlio 
maxlniufil oressuro develooco at distance d. 'osslblt corrections to this slmolo result will Pe 
coneiderud lat-sr. 

The sdfay ohendmcndn Is so raclo that a hlntme rtcora Is requlrpo to show Its devwIppeNnt 
and enable us to estimate the initial votocily of proiastlon of tne spray. 

Our hlnema normally takes from 16 to 20 pictures Per second. Th* times are at prttsnt 
oetwminoo by Including at the corner of each picture a photograph ot the raololy isoving nand of 
s chconograoh fixtd in front of the camera. This hand moves through UO” In l steons. 

The distaftcu scale is oci&rmined by Including tbs photograph of a mmsures base line of 
60 feet marked at one end by an empty mine case from which the mine proper Is suipendsd snd at the 
other end py a flshermsn's blob. 


M thus ..... 


W8 thui obtfiin a sarlas of o!ctui*es *i\lch are meajurefi so as to glvs th# attained by 

the spray at r aarlus of Known tlrnosi ana from these the Initial velocity U obtained by analysis* 

It aofioarcd at the very outset that the ssray panicles wera not proctidlng with a 
retardation oqual to gravity, Out with a eonsldefiUily graatar rctardatlnn; Th« fiabt*rui Inference 
woe that the soray eartlclrs were so Sfnall that consluerabU viscous roslstancs was In pla/t 

Aceorfllj^ly Instead of the equation of motion of th* particles boing 

whar* S It the height ht cine t, tnO g la th* acctlarstlen of gravity. It It ottumtd that th* 
aquation It 


^ f K - A 

at’ at 

whtrs k Is ths unknown vltcous conitant. 
Th« tolutlon Is 


V bilng the Initial velocity. 


Our meaeureieentt give a series of values of S at certain tlMs, The Initial tine Is, 
however, unknown, and we have to aatenslne three unknown quantities v^^, e and the Initial time 
fron t'ae ooiervat Iona, 

Three otservatlons are tneorctically sufficient out to rusuce casual error we take d points 
end combine In oalrs to at to got averaga valuta, 

a soecinen rtesrd it reproduced, elate 3, It shows the exolcilon of a Toroedc war Head 
too Ibt. of T.H.T. 

The following le a copy of the measur^nts and the analytia. 


Torpedo War Heao SQO ins, T.H.T, at depth to feet 
In Hater 100 foot doep. 

Distance between Blobs •• to fcot 
On picture » t.O mm. 

Hence scale Is 1 mn. • so feel ■ got cms. 

TlMe aCcle t second - iao° 


Ho, Index Hand, 


ucandi 


1 




.06 

2 

a,s.. 


,0*00 

.4 

S 

S»2 


s0e33 

s7 

U 

0*1 


• 1286 

1*0 

s 

8.3 


.17 CO 

1.3 

0 

8.3 


.2162 

1.8 

7 - 

5.K 


.2806 

1.7 

Frcit thees obaervatlons drawn on squared 

aupsr th* follcrwiitg Spiuothed values are taken 

as tht bastd of calculation 






1 

a 

Cal cul clad. by. Fonaila 

(») 


‘l 

.os 

,0d 

(2) 



.♦7 

.HU 

(J) 


.to 

.Of 

,8} 

{«; 


• IS 

1,16 

1.16 

(s) 


.to 

Ltd 

1.49 





from 





. ; .-tJU.- ■ • -T. . ■ •• 




- " 

■: ij _ 

• 'it ''v 



'rti'vne'*" ‘ 




604 


4 


From (l), (i), (3) Rna (4) »• a«t H • ».s» 
From (3), (4), (S) and (6) «a gat k » i,Cd 

man k ■ 1 < M 


\ i 


Saegnis B.iM/atCand 
an flletura - 

From (l) and (4) wr got * *0097 ■ a.Mt 

(3) anfl (-5) !<#• got tj " -dost - 3*0S4 

(3) and (6) we gat t, » .0H« • s,ase 


t, < «097 • a,»14 


3720 CkSi gar saaorid. 


Thud taking /O q ■ 1; C > 16 x M* Cfu. par eoeand. 

u gat p nix > * /O^ Cv • 20« atmt. • 3000 loit /iguara Inch 
• It 34 tons oar square Inch. 

as the maxinun pronsiire prasucod Oy 900 10s. T.k.T, at distance ot so feat. 

From the deduced values For k and t^^ the vatuai o7 S are calculated and entered aOouot 
The coRoarattve raeulti are shoxn In the adjoining diagram Figure l. 

ir Stokes viscous law is appllcaolo the aOwf value of k would load to the conclutlon tiiat 
the particles ore aOout 0.4 nn. In diameter, 

A numoer of sxcerimerita have bean made with nlnei> oecth ehargsi, and iwr headi, Vhe 
explosive, the weight used, the death, the primer, nave all been varleo, Tho iMthoa of excerinenting 
hSa been graJually tivrovdd where excarience showed the naeeeelty, Dlssrecanclea occur from tine 
to time, so that until ths reaacn for thaso nna been fully Investigated, It la net conalotrM 
dttslra|)le to give final rosulta. At the eamc tint It Is coat|b1e to esy with rcasanable certainty 
that the reaulte to far obtqinea cinflnti the view that for ths tens wslghl of excicrive the maxlnum 
pressure la Inversely proportional to. the distance, while for the sane distance the nexinue preeiur* 
appears to oe nearly proportional to the square root of ths might of the exolcelve. The dlfferenca 
between dlfforant uxploilves such as oure r.k.T.. ard different gnulea of Amatol do not appear v«i‘y 
definite as far as noxlmun pressure Is conesmeO, but a final declilcn nut wait f;r further 
experinenis. As regards magnitude It soesars titat a 3dO. lb. ss£th charge ;f 90/90 Ahatol crimed 
with 3 discs of e,£, glvos a maximum srsssure of i,l9 tons car aquhre Inch at a olatance of 90 fast. 

The maximum pressures obtsineo by this method ere In thpmselvpe V4i> useful, but the method 
gives no Infonmtlon as to ths time element In the oxolulve wave for which a very olfferent mathed 
of orocedure Is new being Invostlgatcij. But the known specs of detonation, aay 9000 metres car 
second, in rcund nixnbers, and the general dimensions of the mine, sty i utre, sugjitt that tho 
order of nv^gnltudc of the time from Inltlstlon to nexlsaim ortsaure la of oropr eoconos. It 
wilt vary of huurse, with the exolcsive, the anount of it, ana tne manner of Detonation, but this 
figure and th* known' value of theniiixlffiupi oreiiura arc most uaeful as a 2 «<«a In >h( n»s!;ih n? 
ths apparatus for maasurling the time pressure ssqutnce. 

The cofflosrativsiy slmely character of tf|a orinary wave from (he explosion, as outllnud, nay 
appear to be at varlapcp with the oOqsrved fact that tha daaags to a vhsj*! depend: srestly on thf 
cositicn relative to tho charge ana to the Surface, 


Lef, us rcyert to Figure 1 and ctinilder the sffeete at a point » In ths eater, 

Thn dliturcance from tne Image mine U oooaelte In sign to thd direst effect. Is leas in 
the oroeirties Tj end and arrives later by the tiaa taken to traverse the distance (r^- r,j at 

tha rate 1900 metros psr ascorfd (the spccc of ,,ihd wave In water). ».'.w at any caint the crlemry 

amP the reflictcd'ilelunancei each rlit to thhir fflnxlnum In a very short ties, ahlen m have seen 
is of croer .}/to,goo th sf a lecbnq, IT then r, - r^ Is imatrs or say 3 feet, tha IntarvnI between 

the twd .a||7tl& ^ i:/i,pucan .of a SKond and tna two effaett art practically seoaratad, lut 

'if r^ - q is-;|sa Wu as 3 ttu, tha Intervsi'l Is ’only l/eo,owth cf a second nno thus aeall cenpareo 
wlth'i/lP.OOotji of a aecand aS above,. Tns two afftets would, thus tend to eahcsl each ather, and 
ws thus SM how to .reconcile the upoarent panoex thst the resultant action may b» small near tha 
aurfaea, white at some deoth there are two stearate tarsc shocks. 

It may 


‘V 






- 5 " 


60S 


It mtiy D9 Shown that at 200 yarOs tiorliontal dtstanca tro<s a mins SO feat Heap, 

Pj * waiilo Oe equal to l matro at a depth of IS foot Oenoath tno surfnco, ond equal to 3 cms. 
at a depth of i Inches henoath the surface. 

This Is shown alagr.mniHtlcally In figures 3 and >. 

In the same way figure l shows now rooucing the onoth of the mine tends to lower Its 
damaging efficiency sideways, hj tends to oecomo more nearly equal tc r^ as the mine nooroachos 
the surfnee. so that the ooocslng actions are more nearly equal in mag.-ltuoe, nm more nearly 
equal as rsgaros time of arrlual, sad hsneo thsy tans to neutrniiao onen oinor. 

Those Inferences are In agrecnicnt with alrect ohserwatlcn. For If we cOserve at say 
iOO yaras tram an explosion only n vary slight snxh Is exocricncoe In a rtwlag boat or a shallcw 
draugnt mc-tcr launch, wnareas In a deep draught trawler the shoch Is very marked. 

In order to preserve simplicity in tn» preceding argument, we have not referred to a 
fflodlfylng clrcuiraiancc which must now be considered. 

The original surface layer breaks away Trom the lower water. Out successive layars follow 
as the reflected wavs spreads so that a saucon shaped daprssslon should rapidly dovelop dewnwaras. 
Between this saucer surface ard the acme above there will be a continuous succession of soray, 
moving however with smaller speed, so that It does not break through, and we can learn nothing 
about It by dIrKt observation. 

The dovoloping saucer 'boundary forms at eoch Instant the effective reflecting surface fur 
the prliK'.ry wave and tnus the refloeteo negative wave undergoes olst. rtlon In Its loter stages 
without affecting anything that occurred earlier. The not result Is that a body In the water will 
axoarlcncc the orltaary crushing pressure, which will bo followed at the aoDroortote time by the 
erection of a vacuum on the side nearest ine tmsgc mine and which will spread to the remote side. 
The destructive effect of this on a subiMrIne already crumbled oy the primary wave nay be very 
Important. 

Although the priitury velocity acquired by the water at seme oseth may bo very high, 
e.g. It metres ocr sreond In the esse of a 300 la, charge at 50 feet yet the time Is so short that 
the water .loes not lavp more than about 4 cm. 

Although we have discussed only the reflection from tne surface, the effect of the bottem 
m»y also be examlnod by the niethoo of imojes. If o.g, tne odttom Is rigid rock the corresponding 
Image would n,:vo the seme sign os tne trum mine since the condition Is now zero resultant normal 
velcelty Initueo .-.f zoro rosultunt orossurc. Frcb-ioly, nowuver, the bottem Is s;ft and yields 

at first and at greater depth becomes rigid so that the cofcposite action may not be nearly so 

slmble as that of a free water surface, ckberlnents are, however, in gregress and further 

discussion of the effect of the bottom Is reserved until these have been analysed. 

Wo have new t: consider possible eorreotl.'ns to tno simple formula c. = i /O Cv for the 
msxirum oroisuiv. 

(l) ws novo noglectcd a term Involving the square of v„. It is .-.f order v,/C compared 

with the main tenm. At 00 feet from s 3<C Ip. charge v^ Is about 23 metres per second, 
while C Is tiOO metres oor second. Thus the correctlon'is of order i* In this pace 
and iroy 6C neglected,’ For larger charges nho shorter distances the effect might rise 
to n few Pur cent and would havu to oe allowed fpr If tns mensurements ever attain such 

n oagree of accuracy. The ccrractlon Is negative In sign, 

(z) we nave assumed that the water cannot suooort nogalivo pressure. 

Now Osoo'ne kcynolds found that sln-froo water might In faveurabU conditions support a 
negative pressurj of 72 lbs. t--.* square inen. 5lnce sea water nas plenty pf air olssolved In It, 

It would scorn doubtful If It Coulo suooort anything like this amount. On the othor'hand certain 
foatures In the stray ontnoiTiunon, . ,g. the share boundary suggest that the voter tray suooort 
nog'.tivo ur..''eore as nuch ss a ton ;or square lr;h without oreaking. The evIdonCe Is, however, 
still amolgu.us, out will Oe .xomlncd later. * 

If If Shoulo prove that the negative pressure must exceed say P tons ter square Inch 
befors rupture takes dace, then the maximum pressure as calculated above Is an under ostinate 
of tne correct value of i f aooruxlmately. The result of a finite value of » Is that the soray 


Instead ,,,,, 


808 


~ t .. 


instead ot crfnkinj off continuously in indefinitely thin layers would not break Immediately but 
the first It^yer would have a finite but snvjll thlckness< Thus taking the 3®> lb. dapth charge 
at 50 feet, If ® = 0,2 tons cer square Inch the thickness of this If.yer would probably be about 
6 citis. and the corrocted maxlraum orussure would be 0,1 tons more. At present, however, this Is 
quite speculative, 

As already stated an opoaratus tor the direct recording of the time pressure sequence Is 
being constructed in Mining School, 


U) 

( 2 ) 

(3) 


Experiments to this have boon in orogress elsewhere. So far as is known these aret- 
Hookinson's Pressure Bar method In use at Woolwich Arsenal. 

Hlllier's experiments at Troon by means of crushpr gauges. 


A method devised by Sir j.j. Thomsen which makes use of the olezo electric properties 
of quartz. 


It Is hoped that In course of time the different methods may be compared, as the problem Is 
such a difficult and delicate one that It would be moat unwise to trust to any one method entirely 
until satisfactory agreement of two irjepondent metiioos can bo estabi Ishao.. 

















A METHOD OF DETERMINING THE DEPTH OF 
UNDERWATER EXPLOSIONS 


R. A. Shaw 

Marine Aircraft Experimental Establishment 
Scotland 


British Contribution 


December 1941 


A METHOD OF OETERM'HIKG THE DEPTH OF 
UNDERWATER EXPLOSIONS 


611 


R. A. Shaw 


Mnrine Aircraft ExperiRiente:! EstHblishment 
Scotl gnd 

December 1941 

«*«*•*** 


Suiwnarv . 

A of aetermlnlng th> oectn of detonation of n boiao oxoiooing unosP water from 

(neasorcmsnts of the dletrlbutlon of velocity In the spray do* Is descrlbeo. The msthod Is 
oaseo on the assomptlon that the detonation wave aporoxlfflstes to a sound wave* This leads to 
the relaiiorishlc that the Initial velocity at each colnt in the dome la Inversely proportional tc 
thu square of Its distance from the centre of octonatlon, so that tho deoth of detonation can 
bo found from the ratios of tne Initial velocities at different radii, Tne method haw been 
aoolled to the analysis of tests with deoth Cliarjos, nnO reasonably consistent results cbtnlnca. 

In t roduction , 

Mien CopOs are Intended to be droeced Inin water, It is Important to know st what depth 
detonation takes olace* 

A method of obtaining the oeoth Is described which oepsnis on an analysis of the velocity 
distribution In the soray dome wnlch iopears on tnt surface of tne walur above the exploolnq charge 
Immediately »ftor aptonT'tlon. 

Tho nolhod depends on an aoeroxlMtlon, which will have to bn justified by results, but 
os a methoc It hnii soveril marked mdvantogesi standard bombs are used, the tests can be done 
anywhere orovloco calm deep water Id available, and the only eqiilomsnt necessary Is a cine ;amera 
operated from a surface craft. 

Method , 

A typical axamcle of the soray dome Is shown In Figure i»wMch la made up of enlargemonts 
from tho elno film of « Inst with a deoth charge. Ths growth of the some dome Is Shown graphically 
In Figure J, which gives the nolght of tne dome at various radii, plotted on a cmtion time base. 

Tne assumption mado In the msthoo Is tnot, to a sufficient .icoroximatlon, the Initial 
velocity of the soray ,\t any oolnt In the ocnie Is Inversely oroportlonal to tho square of tho 
distance of that point from the centre of dotonafon. This Is equivalent to saying that 

V » Vj. eos^ 0 

when V ■ Initial velocity at P 
Vji " * ■ at 0 

and since the deoth of oatonation, 
d. Is given by 

0 ■= OF cot 0 , 

the ratio Inolcates tne deoth In terms of the radius 0", at which thif maasurar.ents have 
Deep taken. In oractice, the depth Is calculated by taking the a.n'age iraii several natlil 



choseti .,.*, 


612 


. 2 .. 


cnosen generally Detwcen IC4 and 6Qi of the naxtfnum radius of ihe doffis* The fact tnai tne none 
Is llmiteo tfi diamotor Is Itself an indication that the vaflallcn of velocity assumeo Is only 
aporoxlfTBte. 

Notes on the practical oatalls of tns method an> given in the aooendix. 


Basis of method . 

The basic assumotion In the method is that tne oeionation «av8 oxcesi in ihe InsTnaiiatv 
neighbourhood of the exolc5lon» behaves aoproxirretely as a sonerlcal sound wave. For a soherical 
sound wave the strain energy in the wave is constant* i»o. 


,2 


constant 


It fcfllo'i where the wave Icnyth* u Is constant ano small compared with the radius r 
a* a constant 

and tho oreaaure Is therefore inversoly orosortlonal to tho radius. 

At the surfecii tho detonation cressure wave h reflecteo os a suction wave and the 
Particle velocity at the surface which Is assumed to be the initial velocity of the spray Is 
given by 


p VC » 2o CCS d at 

p v^c a 20^ at 0. 

wherep is the density and c Is the velocity of sound in sea water* 

It fotlowa that ^ 

^0 ''a 

« '» CCS^^ 

0 

Results obtained sc far with the methods 


The method has been aooiled to tho analysis of results cbtalneo with depth charges In 
tests during Seotumberi mi« The depth charges were dropoed trar aircraft In a wide range of 
height and sceed conditions* 

Particulars of these teats and tho results of the analysis are given In TaU)« i, Thefo 
is no direct check on the cilculatod aeoths of dotonetioh, but sorft« indication of their sccurrxy 
can be obtained, in Flgares 3 am u caiculviteo oeocn is olottej against two time intervals 
which are relaceo to the oeoth and yet are measured quite Indecenoently pr tne oesth calculation. 

It will be seen that with one excrutlon the depths stained are all within to foet of a 
mean curve and tho majerity are within 3 fdct, goth o* the time scales used nrs Influenced 
by other factors which cannot be allowed for with any certainty* The time underwater before 
detonation (see Figure 3) must deoono on inv imoset conditions which vary widely and the time 

from letonation to the aopoarance of the olumi' (In Figure :;) Is bound tc be semewhat erratic 

If inly jn account jf the Irregularity cf the olumo Itself* It will os seen, hewever* that In 
Figure A In which the time base Is Ifnst oeoendent the external conoitiens of the test, 
tfv^ro Is tho closest grousing of tho points. There is ine P3lnt In Figure 3 which be notco, 

in one drop the depth charge bouncou on imeact so that by the sfccno Impact It had lost tho 
greater cart of Its Initial velocity. The time underwater from the second Impact to detonation 
was 5,7 seconds arij the calculated 000th was 57 feet* This corresponds to a mean sinking 
sooeJ of 10 f.c.s. which Is tht* nccertsd figure for these Jeoth charges. 

It Is ru^ispnable to suoposn therefore, that tne varlotlcns shown In Figure 3 and u are 

real variations in ths oecth .f detonation nno tnat the metncd In practice gives tne aesth to 

within a very few feet. For conclusive ;r:of control tests with Je;th charges cetoiated at 
known deoths ar*' required, 


- 3 - 


613 


it tiua I'Ot:; fii3i!=st<sa that for a given weight of charge the product of the Initial 
velocity nt the centre of the dome ana tho deoth of detonation Is constant. For a 300 To, weight 
of Charge; a value of v^a of l|S00 square feet/second oassa or. results obta'neu by HllHar was 
used to decide the depth of detonation of 50C lb. A/S oombs. The results cf the oresent analysis 
give varl.-atbns of v^a from 2O00 to 6O00 with an average of 4000 square feet/sccona. There Is 
evidence to support variations ;f this order frem tests with mines so that the assumption cf a 
uofistani vnluo dses not appear to be justified* it will be seen that t; assume v,3 Is constant 
for a given Charge weight is equivalent to assunlng that the maximum oressure in the detonation 
wavs Is conotant at a given distance from tho Charge; Since It Is known that the detonation 
wave Is charactorlzcd by Its nvaxlinum oreesure being reached suddenly In a sharp front. It does 
not saem likely that the nexlmum pressure will Itself bs constant. It would be expected to vary 
with temperature .and with manufacturing conditions, and in this regard It Is Interesting to note 
that the varlaticns In the value of v^d cn Individual oays (In table 1) are rather smaller than 
th«y are In general. 


There Is one other point In the analysis to which reference should be made. Within the 
dome area the surface water Is shattered by the suction Oevelooed In the reflected wave, but at 
the boundary, where no soray aooears. It must be assumed that the water Is caoable of withstanding 
the suddenly hddI led suction. This boundary suction has been calculated In tens of the velocity 
at the centre of the dome as 

0 . ^ '' o ‘ ‘^"1 

s 2 

“ “ 34 Vjj cos Ib./sq.ln. 

where 9 ^ Is the angle subtended by the dome boundary, I.e. 


The values obtained for this suction vary from 700 to i,800 Ib./sq.ln, but the greater nimoer 
are In the neignbeurhood of i,000 lb,/sq.in. 

Conctusions . 


So far as can be judged the proooseo metnod of oeriving the deoth of detonation from the 
distribution of velocity In tne dome gives the acotn accurately to within & few feet. 

The accuracy obtained can only be oecldea with certainty from control tests In which 
depth Charges are aetonatei at known depths,- 

sufficient reliance can be clacad on the iiiethud of analysis to show that the velocity at 
the centre of the dome is not by Itself a reliable Indication cf the depth. 

















“ 3 - 


615 


A°° £ HDIX 

V Ofg-S OK PRACTICAL DETAILS OF METHOD 

It A stanoara 35 n.rri, Sinclair or Vinton camera has Bean used In the tests at an average 
film sceoa of 25 franws/secoridt As large a aome as oosslole Is required tor satisfactory 
analysis so a Drocor choice of lens ana distance is Imoortantt With a 6 incn lens, the camen 
shoulfl 09 aoout 700 to 1,000 ynras from the Iraoact, for a dome olameter of 200 to 300 feet. 

2. The scale of the oomc has to be ablalned from the scale of the asroolane. Corrections 
have to os assUsa for the olfferent ncsltlons of the aeroplane and the Impact, and tor high 
aroPSi When the icrcolana Is cut r.f tnc picture seme time bet re iicpact cecurs, these corrections 
are Imooriant. To Keep the corrections srnall, It Is advisable tc use as long a focal length 
lens ns possible, as this will enable the ground atstanca to be as large as oossible by 
eomoarlson with the height and forward travel during the tall of the bomb, 

3, The raessured doirie heights arc corrected for obliquity. This correction Isi ^ 

at the centre of the done, whore D Is the dome olameter, x Is the height of the camera above 
water level, and y Is tha distance between camera aiic dome. The correction Is proportionately 
less at radii away from tho contra, 

1 *. Roth sloes of the acme are measured and the mean Is tjken In thu depth calculations. 

From Figure 2 It will be seen that the variations are at times considerable. 

5, The table of reduction used to obtain the Oeoth from the measurements Is shown in Figure 5. 
The example won fd put In tho table Is tho same as that shown In Figures 1 and 2 , In practice 
Instead of taxing the ratios of the Initial velocities at the various radii It has been found 
more convenient to take the ratios of tho aeme heights corrected for the gravity tall. 

Equating these height uno valoelty ratios is only strictly justified when the time Intervals are 
small ahU tho aocoluustion of the soray Is proportional to its velocity. Tho aporcximatloii 
amounts to neglecting terms Involving t^ and higher powers, where t is the time Interval from 
detonation, It Is reallsod that th? deceleration of the soray Is more nearly proportional to 
the sqqars of Its velocity, but tho effect of both apbroxlmatlons has been found to be negligible 
In oraetlce, The analysis of tne two arocs which gave the shallowest detonation has been 
reooatoo using velocity ratios In place of the corrected height ratios; agreement was obtalnao 
witnin less tnan 0.6 foot. 


U 


616 


]>Van« 

No.1. 

0i>037 3C03» 

ran first 
iipneai-aiioe of donss 




N 0 . 3 . 


t s 0*111 seos* 


H 0 . 5 . 


t IS 0*185 BSCS. 


K0.7. 

t ts 0*26 secs* 





K0.15. 


t s 0.55 secs. 


no.25. 

t s 0.53 sees. 




Soale 


0 50 100 150 200 ft. 

■ ■ » ■ . . - 


yiC, 1 Typicau SPiiPtv &cmE 

cm*v(>»rto oa^H oftiwwiofif av Peer. 













Raow*, Fr. 


Local datit kft 




Sm • A • 90 - S 


Sl 

I WSBUIBHiBBES I 




oM igi‘ « f03/t 

A 


l B3^ffl EB33!3l 
- 1 EE3 B3BfflBW | 
13353129 53531 





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\ 

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\ 

N 


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mtau atkmimtmd tletti __ _ 

y thtoiutMm Zf ^ 


tMbsi itttuif St — jn^n. 

(fiwmffstt/ 


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FiG.5. TYPiCftL PtHfULYBtS . 














A FURTHER INVESTIGATION INTO THE DOME ANALYSIS METHOD 
OF DETERMINING THE DEPTH OF UNDERWATER EXPLOSIONS 


R. A. Shaw 

Marine Aircraft Experimental Establishment 
Scotland 


British Contribution 




April 1942 


621 


A FURTHER INVEST I 6AT0N INTO THE DOME ANALYSIS 
METHOD OF DETERHiNINQ THE DEPTH OF 
UMDERWATER EXPLOS.ONS 

Marine Alrereft Sxparlaental Satabllahnant 


Siiawarv . 

Th* d«M «ntly>lt Mihoo of dttth dotomlratlon h«> M»n cnteliid by tb* ontlydi of roeordt 
oOtointO wIVh ehorgot it kncwn dietht. It Nu btiA found In aeelying tha iMthsd that wMla tiva 
caleutatid daoth la gintnilly In fair igraanant «'th tha trua depth, the asriMMint la liwrgvad If 
anowanca la mda for the dong inapt and the initial p1uia charactaristic of the particular auoloalon. 
Mhtn a eorrietlon far thiae fietori la apollad, the depth ditirmlneo by the dome anatyala nathod 
Ihduld not be In error by norg than iOi and will generally be vary much cibier to tha true depth. 

Tha relation which hai been found between dona and oluim suggiats that the fonaatlon of both Is 
dependent on the cliaracterUtIca of the pressuro pulpo sot up by tha asploalon. This should appear 
In the oreeauro racordt and an attempt wilt bo nada to relate the oraieura rntuUs to the dam and 
alum character I at tea. 

Introduction. 


A fflethud of detenelning the depth of a cnarge eapioding underwater frcn an analyals of Um 
spray done which adodere on the surfece over the exploding dherge was deiorloed |n a eravlous 
rtport. The method has ms been cheekeo by an analysis of sons eino rocoros of unserwatsr explosions 
et known oeoths. very e’loao ig.’eemsnt Is found between the eetculetad and the known depth Jn many 
Inatances Put a fow axemptes where the method le as much no IP - 2 Sf In orror nave occurred. 

It was Observed In the first report on tho mothoO that very targe vorlatlons In Initial oow 

velocity could oe cPtsIneo under apparently similar conditions. This Is confirmed py tne present 

rosults but It has now oetn observed also '.hat these vsriatlon! ore associated with varlatljns in 
the Initial velocity of the plume. Further, the oome Itself, oven In calm water, Is nol to rogular 
In form as was at first thought. Somo domst arc found to Pe higher at ihr contra, end tone higher 
at tho rlrn than mule follew If they were generatao In atrlct orooortlon to the (coslnn)* law uiod 
In the dome nnilyilt method. 

It has now bean found that when tho original dens analysis nethw Is In error In calculating 

the deoth, tne error can bo ralatao to the surfeto ohenomena «e n whole. This has been osne by 

assigning Oorm »n.l uiwmi fnciofa In owch InaUnCu, The Jan, faOi.T is the gi'noianl of iho talculaied 
depth ulsng thv 3;ime radius, sn) the olum factor Is ilmoly tho rntic :f the mean Initial velocity 
of the dame nt Its centre* A correction Involving these two factors has been found one when this 
Is applitc to the deoth calculated by the original oome analysis method, vory close agreanant with 
tho known oeoths Is ••btalnea. 


- *1 

ft ( 


The effect of this Is that whtn the true oopth is unknown the oenth esn oe calculated with 
fair accuracy by tha dome analysis method If a eorreetlon In term of these feet its Is aoPliM. 

The overall range of charge weight In the results contlderco Is from ISO to l.OOO 1b. ana 
of dooth from 15 to UO feet.. The tests Include some In which the charges were flreo on the sen 
bottom. 

The symOols used In the text sm ocflneo betowi- 
(I) CljaryP 

Weight w lb. 

depth /it cetonaticn ^ ft. 

oeoth Cctlculate^ oy origlnril 
soMi anal >9 is metnoi z ft. 

calculAtF^i sft^r ® 

correction for 3on»e ana otufif factors cc ft. 


:l ■ - 


622 


00 


- 2 - 


Ooine 


mitU.i va’.ccity v^x e«ntre rt«/sec« 

acwe factor ni 


NVitfn tnc dcms analysis mothoo Is socUsd tho oeotri Is catculatod by tqklng the 0 (xne hoipht 
St diffofent dai^ raoll ofooorllontnj than to the dome height at the centre* This Is done 
St ihrsc sm*.ll Intervals frur<« tha first Aooo.'.raoce o* the d<vw?. it Is found that the caUulnte 
r^eoth varlus Doth along live raolus and with tltw, Vhc variation Is an Indication of the distortion 
of t!« dorsa and as a measure of this the donio factor^ n* has oeon oef iiw «si-“ 


100 


(»s!< (isoth ca)culAted at d flora roolud aqual to 3lZ 3^ 

''0 

mean aeoth calculated at \ done raolus equal to A d. ) 


in practice It Is found that th^‘ of the dctitt bctuieen raoli equal too.b and l*p x the 
depth of detonation Is most tatlsfactory for measuramont ana calculation* 


The factor n ms a oosltlvc value when the calculated dopth tends to Increase outwards from 
the centre and Is assoclntec with a soms which la relatively high at the rim. When n Is negative 
the grestdCt deoths are calculated at the Inner n.^li ana the oow Is relatively high near the centre 

{lit) Mume 

Time frem detonation to first scoearance 
of the plume through the dome 
height of doma when plume aCDbers 
moan initial velocity of the olume 

*0 


tp seconds 

h feet 
n ♦ 

■f? ■> — - — * ft./secon4. 


Discussion of rtsults. 


TM results oototned from thu onolysls of tn« olnt rscorOs are given In Teble 1. The 
range of ileotM ecveriso !o the reauUa examlnt<i Is equlvelsnt to a vorlntlco of from J ust over to 
to nearly to fset with a 3M in, charge If the deoths are corrected In Diooortlon to Vchargo' weight, 

on the seme bssis of n JM lo, ehnrgo the vortntions found In v„o are frew under t.ooo to 
e,000 square feet ear second for chargee in deeo WAter and fron a,OCO to 8,000 square feet per socond 
far charges on the Pet tern,. ' 

The urcao relation founa octween dome vetxlty and olutno velocity which was the first result 
In thiu latest analysis Is shaim In Figure t, included In this figure arc Mark VII deotii charge 
lesults aescrlocC in the original I'goort on the doin! analysis mothod. It will se seen at once that 
for all weights and dsoths of charges, and for chargee both cn end off the bottom there Is a general 
tendency for v^ and V to Increase together. In the range considered v Is on the nvorage equal to 
0.75 Vg (lUhough the scsttoring cf the oolnts la equal to a i ,101 variation on this. Despite the 
wloe vsrlotlcn there Is clear evlsonec from the figure that a relation exists between the seme and 
the olwe. 

tne next step In the analysis was to discover If the error round In the depth determination 
were also related to the clur* characTorlstles. The oerceniage error In the dootn calcointlon 
was plotted on the basis of the plune-dcwe velocity ratio V/Vy but the points were found to be 
widely scattered and no rolatlonshio was aoearent. This suggested that If a relation existed It 
Ihvolvod a second factor. It has bean observed that the aeius were not always unifoini in slube and 
that III acme Inttancea Inrge variations occurred In the booth calbulnted at different done radlU 
It seemed ooaalble tnst these variations would tncmanlvsc Influence the accuracy of the mean oeplh 
figuro, txamoles of the variation In isetn oslculntoo across the done radius are shown In Figure 2 , 
The throe results llluitrsted are those wnicn give the highest uosltlve and negative values of the 
dome factor which were obtained In those tests and a zero value. 

In Figure la the percenloge error In the dcoth calculated by tho original done analysis 
mothdo la clotted on Ine basin of the Dime factor 7/v^ and each result Is lobelled with the value 
of the dome factor n found In Its c.nlculation. It will be seen that without tho oonx> factor n 
there Is. no rolatlonshlb aoearent but that with the oome factors shown the results «t once align 
themselves In banos across the figure. This Is e.TtphasIseO by the cross lines which have baen drawn 
In the figure to cerresoorti to the average relation between percentage error, n and V/v shown bv 
tho results. ® 



623 


- 3 - 


Tnis relattoi^shlo celwe^n thfr frtof in g«etn calculatco &y the original dome analysis 
msthao and tho v?.lun«. o# n »na has been reaucco to a formula for a correction factor, /i, 
where 


true oeotrt 


l!a 


Ana « 0.5 ♦ 0.78 V/v^ * O.OU n - 0.0«H2 n* 

Tns vsliie of /I has Boon CAlculateo tor OAcn of tna rasjits shown In Figure 3h ana a fewlsea 
figure for the CdleuKteo ooath, 

0 c„ 
uc *■ — ^ 
u 


obthlneO In each Inst Vice. Tho error In oc ms then hern elottea In Figure 3h ana It will be seen 
at once thst when ;^)l9wance for the acmi ana utwiw factors Is maae In this way, very goce agreement 
between the calculatea ana the true aeeths Is cbtalnea. The inajcrlty of the results art within 

± 51 of the truo eeeth ana n: rrsult Is irare than t0« In error. Whsn It Is rtallsea that all the 

results at kntwn .'oaths available tc ante have bean ane.lysaa ana are shown In this figure ana that 
tha rosuUs tncluoo charges oath an ana off the Dolton It will be eeen thht the nethoo pronises to 

orovloe a retinoic noeas of aoier.nlning the teeth -T charges ewaloaing unOsr water. Haw far the 

methoa can be ret lea ucon cutslOu tnu range coveraa by the results examined will be uncertain until 
further tests are done. The rnn,ie already covered (fren tlttto nore than 10 feet to over SO feet 
far u geo to. charge) Is wide enough for most oracticet ourcoses. 


Scad difficulty imy nrlic in practice with bonhs fltteo with a olstcl or rule which detonates 
at shallow aoetns. with these the cavity which the benfi fams at sntry nay not be closed when the 
hnwn cwal.naes, ii condltlcn not rehresentod In static tests, if the ptuna fact.or In these conaltians 
Is hasnd an tha olume which aaoenrs arermturely through the still sosn cavity the factor will os toe 
high. If It Is b-r.spo .n the oline which aaoenrs later through the unbrehen water It may ho too 
Uw because the force of the otuiie will have boon eoent In the open cavity. A niethed of correcting 
the calculated oeotn In these conditions couta bo tonsed on tests with air launched charges droooed 
In Shallow water of known Oepth. The bon^s waute only require to bo fitted with Imoact fuses so a. 
to detonate on the bottooi. *cndlnj the results of such tests It win probable be Ohst to negloci 
the orusent corructlon In uoolyinq the dw.!! analysis mthed to chnllow firing bombs when the records 
Shew that the cuvity Is wllll oevn’. The Jecths assocl.ttea with these conditions will be uf the 
order of 10 - is foot and oinec the uncarructea uons artslysls method nay be expeete: to give such 
depths In gonernl within two fent er loss. It will probably bn sufficiently accurate. 

Tht relationshit ba i uttn dame and ^twnip . 

llthough the foregoing analysis raw orovided a prectlcal uwthoci of determining the depth of 
detonation it does not exolain the relatlonshlos which have been found. 


It has generally bean issumaa that the aoray fomlng the done Is produesd by the action cf 
the detonation orepsura uulse. Rocords obt-alned with plesc olectrlc oresioro gauges have Inalcnteo 
that the diitcnotlon oraosuro pulse Is very uniform In the horlsontal claw at the level of tha 
charge. Thnt this aoes net aoeoac to conrerm with the large vnrlatkns which occur In the soray 
velocity wculo bo okolnlnec If ilmllor variations In ths oreesure pulse were found to occur In the 
region ebove the charge. This region dove not appear to have been exelorod so Tar with pressure 
gauges.. 


rnc valua of v^U for a JM 1b. cliafge which would be calculated from the peak pressure In 
the detonation bulso bosea on the results of tests In the olatM of the charge Is 3,000 square feet 
csr soeonO. In general therefore (see Table i) the Corey velocity Is higher than would be exeected 
from tho offset nf the detonation pulso frund In tho horUontal olanc although It Is leas In some 
Instances. 

The sresont nnnl/sls shows that the dene voloclty is related to ths Plume velocity. As 
might havs been esceeted there Is clao evidbnee that the dome shace, I.e, the Initial dUtribution 
of velocity BCfcaa the dome, U related to the initial alums Shace. The growth cf the olume fer 
each of the present results Is shown In Figure «. The result! have buen arranged In order frem 
tho hlghsst cosltlvo to tho lowest negative value of the dome factor, n. Irrespective of the other 
eonoltlons of the test. It wilt be seen at once that there Is a marlisd tendency for positive 

values ..... 


See also Rote 1 In Aerendlx. 


iv. 


values of n to oe assaciateo with olumes which fork ano break out strongly on either side of ths 
centre while ntjative values of n are founa when the clume develoDS as a central column. 

A more solid oasis for the eomoarlson than the general outline of the Dlumo will uO found 
Dy obssrvlnj in each examole the Jlroctions In which the alunss velocity Is hlghsst, cartlcularly 
In tne early stages of Jevolccmsnt. It will oe seen then that oositive values of n are assoclateO 
with slu.T»s which thrust out most strengiy to either side and are relatively weak at the centrsi 
Values of It near ecro ar'.- found with oli/nes which oeveioo evenly in all directions, negative values 
of n with olumes which thrust upwards most strengiy at the centre. 

Some exceotlons seem to occur out these may oe exolalneo by the fact that the surface 
ohenorner.a are three-oimensional and the dome outlines drawn in Figure 4 may not be In one olans. 
Two views at right angles are really required to olace the glume contours relative to the dome. 

It has been renvarked already that positive values of n are found with domes which are 
relatively high at the rim and negative values for domes which are relatively high near the centre. 
It seems clenr then that the Initial glume is vury closely relateo to the Initial dome Shape, 

The dome velocity tends to be increassd in the areas where the olume, which aooears latar, Is 
most concsntrr.tsc. I*. Is not Intended to suggest by this, that tho dome velocity Is simply 
reinforced by the addes glume velocity. Tns acme velocity Is Imoartea to the soray within a 
few mill IsecoryJs of detonation and the distribution cf velocity which Is usee In the osDth 
calculation Is determined from photograohs taken within the next one or two tenths of a second, 
Generally the plume dose nut auuuar until more tnan a secono later. 

To exDl'iln the relation between the dome and plume it seems necessary to find that a 
pressuri' dulse associated with the early aevelooment of the plume horns Is suoerlmoeseti on the 
general detonation oulso. If this Is found to be true It will indicate that the gas bubble may 
oe far from sonnrieal In Its early stages ana that Us Initial Irregularities tend to persist right 
uc to the time ;.f the acosafance of the olume, it seems probable that tne Irregularities In the 
gas bubble, will be much more irarkea above the charge; any Instability in this dlrectldn tttulo be 
eneeuraged by tho hydrostatic srossure gradient, as a corollary to this it aooears to follow 
that the destructive effect of an explosion may be most marked above the charge. 

Coyiclusions. 


Cine records obtained In tests with charges exploded at known depths have been analysed and 
have shown that tha original dome analysis method of depth determlnat ldf> will give the death fairly 
closely In most Instances, but tnat examples when tne netnoo Is 251 in error may occur. 

it has been found, however, test the errors Intrcouced In the dome analysis method are 
related to the snap? of the dome ano to the early behaviour of the Plume. By introducing factors 
into tne dome analysis to allow far dome shape and plume behaviour the depth of detonation can be 
determined with confidence to within ± 1D» ana will In general be much closer than this. 

Some difficulty In apolying the correction factor may arise In oractico with oonbs which 
are detonated at small oeotns wnllo the cavity formed at entry is still ooen. Special tests 
would be required to Investigate tnis condition but at the shallow depths Involved the uncorrected 
dome analysis method m(\y be expected to give the depths sufficiently accurately for most purposes. 

The broad relation of dome and pluitv velocity and the relation between Seme and plume 
Shade which nave been found in the analysis suggest that variations must occur In the pressure 
pulse which Influence both the dome and the clume. 




- 5 - 


62 o 




ffOTES 0!f THE APPLICATIOff OF T HE REVISED DOH E ANALYSIS KETIIOD 
T he stai?es oJ the method are : - 

1. Determine tne aooroximate fleoth, a , by the original flome analysis methoo, 

‘•0 

2> Calculate the initial dome veVecIty at the centre ft, /second. 

3, ejot the calculated deoth on the oasis of dome r?,oios (ns in Figure ? in this rsoort) 
and determine the dome factor n. This Is the mean value of the aeoth gradient Detween a 
radius equal to f a. and 3/2 d , exoressed as a oercentage. 

0 '•0 

4, Note the time t|^ from detonatlr,n to the first apoearance of the elume and the height, h, 
of the dome, at the same Instant. 

h + 

From this calculate ^5 = ■■■ ' ■ 


3. Evaluate the correction factor /u. where 

/I = fl,riO + 0.78 V/Vg + 0.01? n - 0,00012 n^. 
6. ohtaln the cot rectod deoth o^ as 



Ifotes uihich wav be heHful at the different gtaecs are be.lou ^ 

1. It Is found uest to worK with dome measurements taken at the centre and aPorOxinately 
Detween radii equal to 1 d^. and 3/2 d^ on either sloe of the centre, Measurements at four or 
five radii are sufficient. ® for Oorai's ‘‘ which detonate after only a short time delay (say one 
second or less) the cavity formed at entry will still De open. in thess c 1 rcumstancos It has 
Deon found satisfactory to Dase the oeoth calculation on dome measuranonts taken only on the 
side away from the cavity. The other side of tne dome tends to oe distorted Dy the cavity ana 
Is oDseures oy the splash from the Imcact of the Domb, 

2. in obtaining tho initial velocity, v^, ri-oni ine dome height It Is Imoortant to use the 
time from the first aooearanee of int spray. This may oe Dotwson two frames of the cine film 
but the exact Instant can usually be decided by clotting the curve of ooine height for successive 
f/arnes and noting ivhorc it cuts the axis. 


3. In general this method of obtaining n will be found to give a fair measure of the mean 
depth gradient. If there. should be any sudden distortion In the depth calculated close tc the 
measuring points this may be disregarded ana the moan gradient taken through the renalnlng oolnts. 
in the present analysis this was only found necessary in one examole. 

4. Experience in the annlysls has shown that it is the appearance of the main plume that must 
be looked for In determining t^ and that the fine scouts which sometimes orecede the plume can bo 
disregarded. 



























































MS.vlilMCM lO-b w»2. AmgCMM •kH’tC. HU 1. vt|i.m Wl -f .-WMI 1 !-«’ C’-«C ia:i: 

.'f.o ih w Ml »u.r *» le lut wuv4> 0: •< TjI iiri • * .ir X) i-J'.t sts nt ?o risr ^ iv. ui ai 







THE RELATION BETWEEN THE APPEARANCE OF THE PLUMES 
AND THE GAS GLOBE BEHAVIOR IN UNDERWATER EXPLOSIONS 


G. K. Hartmann 

j i Naval Ordnance Laboratory 

! > 



, ! 


American Conliributlon 



631 


"The Relation Between the Appearance of the Plumes and 
the Gas Globe Behavior in Underwater Explosions" * 

G. K. Hartmann 

U. S. Naval Ordnance Labors toi’y 


The appearance of the water surface above an underwater 
explosion changes after the explosion In what seems to be an 
irregular fashion as time goes on. Although the initial 
velocity of the water surface and the resultant shape of the 
dome in its early stages are extremely reproducible, the later 
stages of the water motion seem to occur in a random manner. 

The purpose of this note is to show that the later appearance 
of the surface and in particular the rather arbitrary behavior 
of the plumes are related in a fairly quantitative way to the 
oscillation and migration of the gas globe. The period of 
oscillation and the amount of migration are of course deter- 
mined by the weight of charge of a given kind and by the depth 
of the charge . 

The phenomena under consideration will now be described 
in more detail . 

Observation of the water surface reveals that for an ex- 
plosion at a moderate depth, say 300 lbs. at 30 ft., the water 
is broken into white spray over a delimited area (outside of 
which there is a distinctly dark region) and that this spray 
moves upward vertically with greatest velocity over the charge 
and decreasing velocities at greater distances. This spray 
dome, which can be shewn to be caused by the reflection of the 
shock wave at the free surface. Is actually freely falling 
v/ater and moves upwax'‘d under the influence of gravity and air 
drag until it reaches its maximum central height whereupon it 
falls back into the isui-faye, oont.tm».ously changing its shape as 
it does so . At some stage in this pi’oeess other fingers or 
Jets of spray and often explosion gases, identified by their 
carbon black, break through the dome with various velocities , 
Sometimes these plumes appear traveling radially from some point 
near the surface and sometimes they are limited to single high 
speed vertical Jets . The character of these plumes and the 
times at which they appear are related to the oscillation and 
migration of the explosion gas globe. 


* This note is taken from material presented by the author at 
the Washington meeting of the American Phy.slcal Society in the 
spring of 19^7- 


332 


Figure 1 made up from data taken at Woods Hole shows the 
time of appearance of the plumes as a function of the charge 
depth, taking the zero of time as the first appearance of the 
spray dome which is the same as the detonation time, save for 
the negligible travel time of the shock to the surface* The 
charges used were the equivalent of 400 lbs. of TNT. The blacked- 
in points represent the more pronounced velocity change in the 
case of two charges . The period, T;ilj first oscillation 

is calculated taking account of the Influence of the surface, 
and the period of the second oscillation, Tg, is taken, for 
these depths, as equal to that of the first".* In this depth 
range the decrease in period due to loss of energy la roughly 
compensated by the increase In period due to the rise of the 
gas globe. At the end of the first oscillation, at time t = Ti, 
the gas globe is small, compressed, and moving rapidly upward. 

It is probable therefore, that for charges originally at depths 
of 25 to 30 feet the columnar upheaval through the spray dome 
at about one second is canaed by the gas globe reaching the 
water siirface in a state of high pressure and high velocity. 

The known facts concerning upward migration are wholly consist- 
ent with this view. The spray dome at this time may be 30 or 
4o feet high and the columnar plume will require at least a 
tenth of a second to emerge from it traveling at a typical 
velocity of 250 feet per second. The initial velocity for the 
spray dome due to the shock wave reflection will be about 120 
feet per second for a 400 lb. charge at 25 feet. This indicates 
that the velocity of the plume is a mass motion of water driven 
by the gas globe and is not analogous to a spi»ay dome of tiny 
droplets. Hence, the potentiality of this plume for damaging 
a ship is probably high. 

For chargee at greater depths the plume emerges at later 
times with respect to the period of oscillation. There are two 
reasons for this; First the gas globe has further to travel 
and second, its velocity is less in the leas corapreesecl plisses. 
The blacked-ln points in Figure 1 for the depth region from 33 
to 45 feet show that the gas globe reaches the surface in the 
early stages of expansion after the first collapse. The addi- 
tional points to the right along the + Tg curve correspond 
to the expansion after the second collapse of the gas globe . 

This second collapse is necessary if the globe is at all large 
xflhen it nears the surface. According to this vlexv, there will 
be no surface events at the time the gas globe nears the surface 
if this happens at its maxlmxan slae. In this case there will 
be a delay of about one half period until the collapse and con- 
sequent rapid upward velocity occur which will result again in 
a columnar display as indicated by the heavy points on the 
Ti + Tq curve. For charges of this size at depths less than 25 
feet the dome and plume phenomena merge together. However, 
radial plumes associated with radial expansion of the gas globe 
are observed at about 20 feet whereas for very shallow explosions, 
in say 5 to 10 feet of water again a columnar formation is 
observed . 


2 



( 

: / 


I 

i* 



► 5 

I i 


833 

Figure 2 Is taken from a report by Schlichter, Schneider 
and Cole on the Measurement of Bubble=Pulae Phenomena. It 
shows the appearance of the dome, radial plumes and coliunnar 
plumes at different times and for different Initial charge 
depths . The charge weight used Is 290 lbs . for this series of 
shots . 

One of the difficulties in Interpreting pictures of the 
type shown In Figure 2 Is the fact that the top of the dome is 
used as a reference point rather than the watei’ surface which 
is necessarily obscured. If, however, the height of the pl\amo 
Is plotted against time. It is easy to extrapolate this to zero 
height and thus find the time of emergence of the plume from 
the surface, celled the plume time In Figure 3« Similarly, sur- 
face velocities can be found by extrapolating differentiated 
height va . time curves . The results have been plotted by the 
above mentioned authors and are shown in Figure 3. It is seen 
that vertical plume velocities are high for charges at depths 
for which the plume time coincides with the gas globe period, 
or sum of periods . This shows that hl^h plume velocities occur 
when the gas globe reaches the water surface In a collapsed 
state. Furthermore, the Jog in the plume time curve corres- 
ponds with the maximum expansion and the subsequent contracting 
phase of the gas globe . This means that for charges at these 
depths it is necessary to wait about an additional half period 
for the plumes to appear. 


It la seen, therefore, that a reasonably satisfactory 
account of some of the visible surface phenomena can be given 
by referring them to the behavior of the gas globe, and that 
plumes of a similar appearance occur if the explosion bubble 
breaks surface in the same phase of its oscillation after one 
or two or even three complete pex-’lods. Thus the occasional 
observation of a high columnar plume characteristic of a 
shallow explosion occurring in the case of a relatively deep 
explosion is explained by the critical combination of circum- 
stances which permits the gas globe to oscillate through two 
or three periods and arrive at a depth near the surface in the 
same phase as if the charge were statioaliy detonated at that 
point. 

It is also clear from Figure 1 that the time of appearance 
of the plumes is net very sensitive to the charge depth and 
Jumps dlscontlnuously fi'om short times to longer times as the 
depth is increased. Observation of these times therefore does 
not lead to a good indication of the actual depth of explosion. 


3 


DEPTH OF EXPLOSION iff) 






634 


I !' 

i 


'! t! 


70 


60 


no 


40 


30 


20 


10 


CHARGE WEIGHT EQUiVALENT 
TO 400 lbs TNT 

0 = DENOTES FIRST PLUMES 
DENOTES SECOND PLUMES 
Ri Ar.wpn-tw pniwTc iwnir.ATg 
THE MAJOR UPHEAVAL 






















L J 

ii 




\ 

_fi 






m 

SI 



■ 

m 

a 


1 

(CALCULATED) 

\ 


T, + 7^^ 
(ESTIMATED) 

\ 



a 






< 

o 

1 




H 

a 




... 



a 

m 


a 

a 



■ 

■ 

■ 




\ 

%. ^ 
A 

— A-J^ 

A 

StAwAAfiAA 

A 

A_ 

A 

^ A 
A 

I 



■ 


i 

a * 




1 

\ 



■ 


i 

■ 


m 

1 



■ 

a 



■ 

■ 

II 

m 



■ 

B 

a 

■ 

■ 



■1 

m 

n 


B 


a 

■ 

■ 

■ 


■ 

■ 

[| 

■ 

■ 

n 

a 



1 

■ 


1 




a 

a 


0 1.0 2.0 
TIME FROM EXPLOSION TO APPEARANCE OF PLUMES iMCJ 


FIG. I TIME OF APPEARANCE OF PLUMES AS A FUNCTION OF 

CHARGE DEPTH 













CHARGE WEIGHT 290 Ibs TNT 






THE DETERIvHNATION OF PEAK PRESSURE OF AN UNDERWATER EXPLOSION 
FROM A STUDY OF THE INITIAL DOME VELOCITY 


D. A. Wilson, B. A. Cotter, and R. S. Price 
Underwater Explosives Research Laboratory 
Woods Hole Oceanographic Institution 


American Contribution 


26 May 1947 


CONTiSNrJS 


637 


Abstract 


INTROD’JCTIOfi 

EXPERIMENTAL 

1» Pond Work 

2. Tank Expariaenta 

3» Critical Depth Esporiments 

expehijsental results 

4» Methods of Computation 

5. Nonomessett Pond Reeults 

6. Tank Reeulta 

7. Uathods for Calculating T 

DISCUSSION 

8. Accuracy of the Method 

u' rS“l^r Clarge efr„t. 

mu Tension Loweringf Problea 

12. The Tensile Strength of Water 


Appendix 

List of Haferencea 


t5 CO- O^SjTi Vi VI W IV 


LIST 0? TABLES 


1?A0 


Doma Velocity Raoulte for PentoUto — Pond Serlec 7 

Done Velocity Reaulta for Preeaed Tetryl 7 

Qoae Velocity Similarity Curra Raaults for Single 
Rngineer'e Special Cape — Tank Series 9 

Dame Velocity Syapathetlo Detonation Rssiuts — Tank. Series 9 

Dome Velocity Surface Tension Lowering Resulte — Tank Series 9 

Adding 3300 to Each ae Calculated for Tank Shots 10 

Approximate Value of T Obtained from Critical Depth 
Eatloatlona 10 

Apnroscimate Value of T Obtained from Doaa Periphery 
Calculation 11 

S SuBimary of Betimated Vsluea of T 12 

A-I Charge Data Pond Experiment Pentolite Series 16 

A-IX Charge Data Pond Experiment Tetryl Series 1? 



039 


LIST OP FIGURES 

FIGURE PAGE 

1 Eapsrinsntal S»t Up for Sees Velocity Shots Uoiag Ssall 

ChargAS 16 

2 Experlfflental Sot Up far Dome Velocity Shots in Steel Tank 19 

3 Cap HoUar for Tattk Siiots 20 

4 Critical Depth Experiment Using Caps 21 

5 Typical Example of Low Dome Velocity 22 

6 Influence of Surface Ripples on Dome Appearance 23 

7 Typical Examples of High Dome Velocities 24 

8 Similarity Ctu=7s for Faak Pressure vs* wV3/a 

for Pentollte 25 

9 Similarity Curve for Peak Freusura vs. sV3/a for 

Pressed Tetryl 26 

10 Typloal iixamplea of High and Low Done Valooitles 27 

11 Effect of Surface Active Agent on Dorns Velooil? 26 

12 Syripathetle Detonation Series 39 

13 Done Velocity Similarity Curve for Peak Pressure vs* 

for Engineer's Special Detonators 30 

13a Corrected Peak Pressure vs. for Engineer's Special 

Detonators 31 

14 Sympathetic Detonation from Dome Velocity Points Placed 

on Similarity Curve from Fig* 13 32 

15 Doua Velocity Results with Lowered Surface Tension Poiints 

Placed on Similarity Curve from Pig. 13 33 

15a Doss Velocity Results with Loworsd Surface Tension Points< 

Placed on Similarity Curve from Fig* 13 and Pig. 13a 34 

16 Calculation of T from Greatest Dome Diameter 35 


QU)ssm' 


l«lght of explosive charge (lb) 

Distance of cents? of explocive frcs tsurfaoe (ft) 

Peak pressure (ib/in.^) 

Done velocity (ft/asc) 

Velocity of pifO|.».s&t-ion of ahook mvc (ft/seo) 

Initial density of mter (lb/ft3) 

Indicates observations on Faatax notion picture oanera 
Indicates obaervattloof! on General Radio "atrealc" eaoera 
Number of obeervatiotis in a group 

Deviation of a single observation from the mean of the group 
Standard deviation of & single observation from the mean 


«r» «■ 

standard deviation of the mean of a group of observations 


CT< 



Tension nscsaaary to rupture mater (lb/i»«^) 
Surface tension (dynes/cn) 

Particle velocity (ft/seo) 


4 \ 




(n - 1) 


641 


ABSmCT 


A sjrstsmatic Inveatlgatlon has bsen made of the detenaination of 
peak prasaiii'o ft-oa the aeaBurad upsss4 rise of the spcm.y dcjne ehsn aa 
explosive is detonated at va 2 °yi.ng shallow dspthn. The experimeatal 
work has Included a series using aaall chergos of a typical explosive 
in a small fi*esh water pond and another series using Engineer's Special 
Detonators exploded In a three foot ouhic tank filled with fresh water. 
The latter series has been extesided to Investigate the effeot of 
varying surface tsnaloa on the dome veloelty. Measurement of tide 
velocity haa bean made photographically using oither a "streak” 
camera or a high speed motion picture camera operating at about three 
thousand frames per second with photoflaah illumination. 

General agreoment with theory has been observed. There is a 
dependence upon surface tauaion not hitherto reported. There is no 
avideneo of an initial velocity higher than that predicted theoretically. 
It is possibls to arrive at es upper limit for the value of the tension 
naeeasary for tha rupture of water at its surface. The method provides 
a satisfaotory approach to a measure of the absolute value of the peak 
pressure from an underwater a3cploel««Js 


t 


-vll- 


642 


THE DETERtaUATICB OF PEAK PRESSDRK OF AK UMDERWATER EXPLOSION 
FBDM A STUDY OF THE INITLAI.. DOME VELOCITY 


I. INTRODUCTION 


Whon an explosive obax>ge le detonated beneath the surf ace of water j 
provided the depth of the explosive ia not too groat, the resulting 
ahook wave hitting and being reflseted from the surface causae a mass 
of water to become detached and leave the surface « This detached mass 
of water is projected normal to the surface with an initial velocity 
exactly twice that of the p»‘tlole velocity of the shock wave itself* 

This doubling of the partiole velocity at the surface ia due to the 
reflection from the free water-air interface of a rarefaction wave of 
the same magnlttada but opposite in sign to the incident compression 
wave* The particle velocity in a shook wave is given by the expression^)*’ 


u 



U 


tmd the Initial velocity of the projected spray by twice that, or 


2 Pa 


U) 


where u ia the particle veloolty, the initial veloeity of tbs rising 
spray dome, P;^ the peak pressure of the shook wave, the initial density 
of the ttodiuB,and D the propagation velocity of the shock iravw. 

Although the surface effects resulting from an underwater e»Ioslon 
have been intensively studied for various reasons during the war,^) tha 
velocity of the rising spray dooM has not beoia seriously proposed 
heretofore as sn accurate method for the detarmination of the peak pressure 
of the ahook wave except by earlier workers, azwng them, IlilUar*?) Tha 
Underwater Exploslveii Research Laboratory became interested in this method 
because the Ineroesing dependence upon the piemoelectrio gauge asda 
desirable the development of an independent method for determining absoluto 
peak pressures as a oheolt: on the pie trie remvlts. 

Introducing a finite force T necessary to rupture wator at ite 
surface so that a spray dome may be formed, tha equation4) 


* All such numbers refer to the List of References at the end of thie report* 


- 1 - 


C2) 


e43 




l) 


whiob. Xa more rasiXistio tl3a& Eq* (1)* nay oasily ba deduced* T is exactly 
equal; ef eoursa; to the tensile strength of the wfttor and; under such oon- 
dltioiif that 1q baooaee just equal to sero, thie eqmtion provides a moan# 
of determining the force nsceseary to rupture water tinder the conditions of 
the esparlment. Any means of determining T ie interesting beeauss a know- 
ledge of thie Value is very desirable and not easily determined. 3) 

Although Eqs. (l) and (2) derive from ptire hydrodynaiaioB^»^) the 
derivations take no account of poesibla variations ef such ftinclansental 
properties ae eurface tenaion a^ viecosity# It would oeem daairable to 
investigate the effect on the dome velocity of varying thaoe properties. 

This ohould give us a better insight into the thermodynaaio fcshavior of the 
liquid medium \uider such conditiona. The present work includes a study of 
surface tension effects ae w>ll aa the effect of spraading oil slick on the 
surface. The Umltatione of time aM film consumption prevented eontlnoiag 
the work Into a program to study viscosity variation. It is hoped that this 
work will be continued* 


I IX. sxpesIUENTAL 



1. pgmui9rt 

The experiments were performed in the fresh water nxatid on Noneiieseett 
leland. The relative position of the various unite of the recording equip- 
ment lo best Been by reference to Fig. 1. For moat of the shots; two 
cameras wars usi»d to record the velocity of the rising done. To record the 
^streak** type of picture; a General Bsdlo type 651 AE osolllograph recorder 
^le ^n g 35 ns film was used on all shots. The oenera wee mounted on Its side 
(with the film moving horlssontaily) about a foot above the water surface. 

A vertical slit 0.1 in. wide was placed in front of the foceJ. plane in order 
to narrow the field of view and thus produce a more clearly deflaed record 
of the edge of the iiuga forned by the rlsivig dome.**®’ To understand the 
phenomena better, and to obtidn a different sort of record for ueamiremeut, 
a Fsistax mm camera was alec used for almost all the shots* This camera 
was a3.BO placed on 4i;» aide ho that the rising dome would be photegraphed 
along the long dimension of the frame. This introduced another simplification 
of measurement beoause lines could be drawn through a succession of frames 
to estsblleh a common bass line and slope of rise velocity. Timing marks 
were put en the edge of the film In the General Radio camera by means of a 
Bperk coil energised by an electronically amplified pules from a 100 cycle 
ttmlng fork. In the Faatax camera, 1000 cycle timing marks wore marked on 
the edge of the film by meanti of the timer described by Cole, Stacey, and 

Brown.W) 


■ 2 - 


644 


To reduce ambiguity in interpratetlon, the records of all exeej^t a few 
early shots were made with artlficiel light provided by firing ten photo- 
fleeh bul'bn almultaneoualy with the shot. The flash bulbs were fired in 
two specially*d 0 signed parabolic reflectors each holding five bulbs. 

Nuabar 21 or 22 bulbs were used for the high velocity shots; number 31 
bulbs were used for the low velocity shots to take advantage of the longer 
duration of the flash from -Ae focal piano tyoe of flash bulb. liluaiaatioa 
from the flaeh bulbs waa sufficiently intenao so that variations of natural 
lighting could bo neglected. The increase in illumination also allowed 
narrowing the vertical collimating allt in the General Radio camera from 
0.1 in. to 0,01 in. This greatly Increased the definition of the resulting 
record although it mads focussing and aiming a little more diffietilt. 

Synchronization of the photoflash bulbs, detonation of the charge, and 
remote control of the cameras were obtained by the use of the time delay 
circuit described in the Underwater Photography II report.^) Both cameras 
ran approximately 2.5 sec using 120 volts for the Fastax and 107 volts for 
the General Radio cameras. These applied voltages caused the General 
Radio camera to run 50 ft of film through at about 300 in ./sec aiad the 
Pas-fcax, 100 ft of film at about 3000 frames/sec. When the caiseraa were 
started, a switch in the primary of the timing spark coil wae closed so 
tSiat the timers were operating only during the time the cameras wore 
running. 

Those parts of the recording mechanism which had to be relatively 
close to the exploding charge, i.e., the two cameras and the timing 
mechanism, wore protected by being enclosed in a specially constructed 
reinforced concrete box with a safety plate glass window. 

2. Tank Experiments 

In order to determine the dependence of dome velocity upon surface 
tanalon it was necessary to construct an experiment in which a reasonable 
amount of water would be involved. Accordingly, a welded tank was oon- 
atrueted of 1/8 in. sheet iron to the dimonaions of a 3 cube with open 
top. This tank was supported on a heavy base constructed of angle iron 
such that the upper edge of the tank waa five fact above ground level. 

This placed the upper edge exactly in line with the optical axis of the 
lens of an Eatltman High Speed camera located inside the laboratory and 
focussed on the center of the tank through a plate glass window as 
indicated in Fig. 2. All the oquipmont was thue protected from the spray 
resulting from the explosion. With the exception of the camera, all the 
supplementary equipment was the same as that described under the 
preceding section. As with the Fastax camera, the High Speed camera waa 
placed on its side so that weaeui’ements of successive frames would be 
greatly simplified. Illumination was again obtained with synchronized 
flash but, because of the decreased distances involved, only five flash 
bulbs were used for each shot. 

The explosive used for all of the tank ahots wae the Engineer's 
Special Cap manufactwed hy the Hercules Powder Company of lllmJ.ngton, 
Delaware. It compriaes a meteil tube 1/4 in. in diameter and 3 in. long 
with the charge located in the end away from the detonator wires. The 




645 


oompoBitlon of the chargo* ae given by the Hercules Company, la 13*5 gT 2 dna 
of pentaerythritol tetranltrate plua 0.41 grams of a 75/25 mixture of 
diazoainitrophenol/potasaiuin ohlorate primer. In the calculation of 
similarity curves, it was asawad that this corresponded to 1.20 grams of 
a single explosive compovtnd. It may or may not be correct but since the 
sama lot of cape was usfsd throughout, one actual weight usac does not 
matter bacauso no attempt waa Duide to compare the abao].ute values of this 
e:qplosive with those of other explosives. 

The depth of the cap in each explosion was accurately located beneath 
iho surface of the water to > in. by means of the support shown 
in Fig. 3« Orientation of the cap was always the same with the axis of 
the cap pas’allel with the axis of the camera and with the wire end towards 
the camera. Water was allowed to flow slowly Into the tank at all times 
BO that the water level was always the same, being regulated by overflow 
over the edges of the tank. For the determination of the similarity 
curve for these caps, single caps were fired, alx at each of eight depths, 

1 / 4 , 1 / 2 , 1, 2, 3f 4, 5, and 6 in. 

The sympathetio detonation series was executed in the following maonor. 

One cap (donor) ms fired with its canter exactly 3/4 In. from the center 
of a second cap (acceptor), the axes of both caps being parallel with the 
optical axis of the camera. Six: shots were fired with the plane defined 
by the axes of the caps parallel with the eurxace of the water 
(oriantation A D and twelve with this plane perpendicular to the aurface 
of the water. Of these twelve, six were flrod with the donor charge below 
(orientation A) and six with the donor charge above the acceptor (oziautatlon D), 
D A 

The caps were separated 3/4 in. from center to center by taping them togethej? 
with aoourately made l/2 in. wooden separators and then taping both separator 
and cap to the cap support. The depth of ell these shots we# 5 in* to the 
center of gravity of the two caps which in the case of the vertical 
orientation placed the center of the upper cap 4 5/8 in. below ths sux'face. 

Surface active materials used were of three different types. For the 
first twelve shots fired at depths of 4 and 8 in,, the tank contained a 
solution made by dissolving 4 lb of a commercial synthetic detergent 
(markoied by the National Cooperative stores as "Synthetic Svado") in the 
contents of the tank. This is probably a eulphonated oil derivative similar 
to "Dreft", Mixed with the approximately 200 gallons of water in the tank, 
this produonu a solutiou whose surface tension was about 45.5 dynss/cm. 

The second experiment was with first 5 lb and then 10 lb of Elmer and 
Amend Company "Aerosol" 10^ solution added to the contents of the tank. 

Tide gave solutions with sm^face teaeions of about 41 and 35 dynss/cm 
respectively. To avoid excessive dilution of these solutions, the water 
supply was not allowed to flow continuously but was only sufficient to 
replace solution lost with ea