GHz ultrasound propagation with a high speed in compressible liquids containing many microbubbles

This paper theoretically deals with weakly nonlinear (i.e., neither linear nor strong nonlinear) propagation of plane progressive quasi-monochromatic waves in an initially quiescent compressible liquid containing a tremendously large number of spherical gas bubbles on the basis of the derivation of two types of amplitude evolution equations (nonlinear wave equations). The important points are as follows: (i) the compressibility of the liquid phase, which has long been neglected, is taken into account; (ii) The incident wave frequency is very larger than an eigenfrequency of single bubble oscillations; (iii) bubbles are neither created nor annihilated; and (iv) the thermal effect is not considered and wave dissipation are then owing to liquid compressibility and liquid viscosity. By the use of the method of multiple scales with an appropriate choice of scaling relations of threenondimensional parameters, we can derive two types of the complex Ginzburg-Landau equations (or nonlinear Schrodinger equations), where the phase velocity is larger than the speed of sound in a pure liquid.