Unified Theory Based on Parameter Scaling for Derivation of Nonlinear Wave Equations in Bubbly Liquids

Abstract

We propose a systematic derivation method of the Korteweg–de Vries–Burgers (KdVB) equation and nonlinear Schro ̈dinger (NLS) equation for nonlinear waves in bubbly liq- uids on the basis of appropriate choices of scaling relations of physical parameters. The basic equations are composed of a set of conservation equations for mass and momentum and the equation of bubble dynamics in a two-fluid model. The scaling of parameters is related to the wavelength; frequency; propagation speed; and ampli- tude of waves concerned. With the help of the method of multiple scales; appropriate choices of the parameter scaling allow us to derive various nonlinear wave equations systematically from a set of basic equations. The result shows that the one-dimensional nonlinear propagation of a long wave with a low frequency is described by the KdVB equation; and that of an envelope of a carrier wave with a high frequency by the NLS equation. Thus; we establish a unified theory of derivation of nonlinear wave equations in bubbly liquids.