Weakly Nonlinear Theory on Two Types of High-Speed and High-Frequency Pressure Waves in Compressible Liquids Containing Many Microbubbles

Two types of weakly nonlinear propagations of plane progressive pressure waves in an initially quiescent compressible liquid uniformly containing many spherical gas bubbles are theoretically investigated. The treatment of two types of propagations corresponds to an extension of our previous result (Yoshimoto & Kanagawa, Jpn. J. Multiphase Flow, 33 (2019), 77) to a generic form. The main assumptions are as follows: (i) The incident wave frequency is much larger than an eigenfrequency of single bubble oscillations; (ii) The compressibility of the liquid phase, which has long been neglected and induces the high speed propagation mode, is considered; (iii) The wave propagates with a large phase velocity exceeding the speed of sound in pure water. From the method of multiple scales with two types of appropriate choices of three nondimensional parameters, we can systematically derive two types of nonlinear Schrödinger (NLS) equations with some correction terms in a unified way. These two equations can describe high-speed propagation of pressure waves in compressible bubbly liquids.