It is well known that sound propagation in liquid media is strongly affected by the presence of gas bubbles that interact with sound and in turn affect the medium. An explicit form of a wave equation in a bubbly liquid medium was obtained in this study. Using the linearized wave equation and the Keller-Miksis equation for bubble wall motion, a dispersion relation for the linear pressure wave propagation in bubbly liquids was obtained. It was found that attenuation of the waves in bubbly liquid occurs due to the viscosity and the heat transfer from/to the bubble. In particular, at the lower frequency region, the thermal diffusion has a considerable affect on the frequency-dependent attenuation coefficients. The phase velocity and the attenuation coefficient obtained from the dispersion relation are in good agreement with the observed values in all sound frequency ranges from kHz to MHz. Shock wave propagation in bubbly mixtures was also considered with the solution of the wave equation, whose particular solution represents the interaction between bubbles. The calculated pressure profiles are in close agreement with those obtained in shock tube experiments for a uniform bubbly flow. Heat exchange between the gas bubbles and the liquid and the interaction between bubbles were found to be very important factor to affect the relaxation oscillation behind the the shock front.
Equation for the three-dimensional nonlinear waves in liquid with gas bubbles