Experiments and direct numerical simulations (DNS) were conducted on shear layers at subcritical to supercritical pressures. The experiments were performed on single-component coaxial jets, where the slower inner flow was liquid nitrogen, the faster outer flow was cold gaseous nitrogen, and the environment was room temperature gaseous nitrogen. The experiments were performed with and without the effects of transverse acoustic waves. Careful attention was given to measuring all the initial and boundary conditions, including the initial jet temperatures. Automatic image processing allowed a large number of shadowgraph images to be processed, giving statistically significant measurements of the mean and root mean square (RMS) variations of the observed dark core lengths. It was found that acoustic waves could have an appreciable effect on the jets, causing them to develop wavy structures having mean core lengths that were always shorter than when the acoustic waves were not present. However, the magnitude of the effect depended strongly on the velocity ratio and on whether the jets were supercritical or subcritical. In particular, it was found that the jets were relatively insensitive to the acoustic waves at large outer-to-inner velocity ratios. It was postulated that this effect could explain the mechanism behind the temperature ramping method of assessing the combustion stability margin in liquid rocket engines. The numerical work consisted of assessing the suitability of existing numerical schemes for conducting DNS of supercritical shear layers obeying real fluid equations of state and characterized by steep density gradients, at Mach numbers as low as 10-5. It was found that existing schemes could not be extended to Mach numbers having such a low value.