Invariant submodels describing the nonlinear attenuation of high-power ultrasonic beams in a cubically nonlinear medium

Abstract

A generalization of the Khokhlov-Zabolotskaya-Kuznetsov model in a cubically nonlinear medium with the special nonlinearity coefficient describes the nonlinear attenuation of high-power ultrasonic beams. The invariant submodels of this model are described by the invariant solutions of its equation. We have studied two solutions of rank 1 of this equation. With a help of the first solution we researched a process of the extinction during an infinite period of time the traveling in the space powerful ultrasonic wave, whose level lines for the pressure are a family of parallel planes. For particular values of the parameters defining this wave, we managed to obtain this solution in an explicit form. With a help of the second solution we researched a process of the extinction during a finite period of time of the axisymmetric ultrasonic wave beam. The presence of the arbitrary constants in the integro-differential equation of this submodel provided us to research the nonlinear attenuation of high-power ultrasonic acoustic waves for which the acoustic pressure, speed and acceleration of its change are known at the initial moment of the time at a fixed point. Under the certain additional conditions, we established the existence and the uniqueness of the solution of boundary value problem, describing this wave processes. This allows us to correctly carry out numerical calculations in the study of this process. The graph of the pressure distribution was obtained as a result of numerical solution of this boundary problem.