Nonlinear Propagation of Sound Beam in Nonuniform Bubbly Liquids

Nonlinear propagation of weakly diffracted sound beams in a weakly nonuniform bubbly liquid is analytically studied based on the method of multiple scales and scaling relations of some physical parameters. The system of basic equations consists of the conservation equations of mass and momentum for gas and liquid in a two-flui model, the Keller equation for bubble wall, the state equations for gas and liquid, and so on. The compressibility of liquid is taken into account and this leads to the wave attenuation due to bubble oscillations. It is assumed that the spatial distribution of the number density of bubbles in an initial state at rest is a slowly varying function of space coordinates and the amplitude of its variation is small compared with a mean number density. As a result, a Khokhlov–Zabolotskaya–Kuznetsov (KZK) equation with dispersion and nonuniform effects for a low frequency and long wavelength case and a nonlinear Schro¨dinger (NLS) equation with dissipation, diffraction, and nonuniform effects are derived from the basic equations.