Nonlinear Schrödinger equation for a fast pressure propagation in bubbly liquids

Abstract

The present study theoretically deals with a finite amplitude propagation of pressure waves in an initially quiescent liquid containing many small spherical gas bubbles. A fast propagation mode induced by the incorporation of compressibility in the liquid phase is focused. The wave frequency is larger than the eigenfrequency of single bubble oscillations and the wavelength is comparable to the initial bubble radius. Then, a nonlinear Schrödinger (NLS) equation with some correction terms for envelope wave of the carrier wave is derived from a set of averaged equations in a two-fluid model (Egashira et al., Fluid Dyn. Res., 34, 2004) via a general and systematic way based on the asymptotic expansion method of multiple scales (Kanagawa, J. Acoust. Soc. Am., 137, 2015). The NLS equation describes propagation properties in a far field.