Submodels of the three-dimentional Westervelt model in the absence of dissipation

Abstract

A general form of all invariant submodels of rank 1 of the three-dimensional model of the Westervelt model of nonlinear hydroacoustics is obtained in the absence of dissipation. Some submodels, described by invariant solutions of rank 1 which are found either explicitly or their search is reduced to the solution of nonlinear integral equations. With the help of these submodels, various wave processes in nonstationary ultrasonic fields were investigated. Under certain conditions, the existence and uniqueness of the solutions of the boundary value problems describing these wave processes are established. Among the submodels studied, in particular, the following submodels are contained: a submodel that describes a "conical ultrasonic field"; a submodel, describing the "spiral ultrasonic field"; a submodel describing a "pulsating ultrasonic beam" initiated by singular directional sources; a submodel that describes a spherically symmetric ultrasonic field for which the rate of change of the acoustic pressure has a singularity at the center, although the acoustic pressure does not have of the singularity.