Derivation of an Amplitude Equation for Weakly Nonlinear Pressure Waves of a Very High Frequency in a Compressible Liquid Containing Many Microbubbles

Abstract

The present paper theoretically treats weakly nonlinear propagation of plane progressive waves in an initially quiescent compressible liquid containing a tremendously large number of spherical gas bubbles, focusing on the derivation of an amplitude evolution equation (i.e., nonlinear wave equation). We emphasize the following points: (i) the compressibility of the liquid phase, which has long been neglected, is considered; (ii) the wave propagates with a large phase velocity exceeding the speed of sound in pure water; (iii) bubbles are not created or annihilated. From the method of multiple scales with an appropriate scaling of three nondimensional parameters, we can derive an attenuated nonlinear Schrödinger (NLS) equation, where the phase velocity is larger than the speed of sound in a pure liquid.