Theoretical Analysis on a Nonlinear Modulation of High Speed Ultrasound in Compressible Liquids Containing Many Microbubbles

摘要

This paper investigates theoretically an weakly nonlinear propagation of high frequency ultrasonic wave in an initially quiescent liquid containing uniformly many small spherical microbubbles. The incident wave frequency is greater than an eigenfrequency of single bubble oscillations and the phase velocity is greater than the speed of sound in a pure liquid. Remarking that the compressibility of the liquid phase is incorporated. The thermal dissipation effect is discarded. The basic set of equations is composed of the conservation laws of mass and momentum in a two-fluid model, equation of bubble dynamics, equations of state for each phase, and so on. By the use of the method of multiple scales with an appropriate choice of scaling relations composed of three nondimensional parameters in terms of nondimensional wave amplitude, two types of the nonlinear Schroedinger (NLS) equations with some correction terms are derived from the basic set. Then, both the NLS equations govern the competition of nonlinear, dissipation, and dispersion effects. Finally, we discuss the difference between two types of NLS equations derived here.