Abstract
Nonlinear propagation of pressure waves in an initially quiescent liquid containing non-uniformly distributed spherical microbubbles is theoretically studied on the basis of the method of multiple scales. We focus on almost unidirectional waves propagating mainly in one direction with a weak diffraction or weak focusing to the transverse direction (like a ultrasound beam). Based on a methodology recently proposed by the authors, the KZK type equation, in which the dispersion and the non-uniformity of number density of bubbles are considered, is derived from the basic equations of bubbly flows.
