Description of the attenuation of invariant ultrasonic beams after formation of the shock fronts in a cubic nonlinear medium in the absence of dissipation

Abstract

In the framework of a three-dimensional generalized Khokhlov–Zabolotskaya–Kuznetsov model of nonlinear hydroacoustics in a cubic nonlinear medium in the absence of dissipation, the attenuation of ultrasonic beams after the formation of shock fronts is investigated. For the main group of the equation of the investigated model, an optimal system of three-parameter subgroups that do not contain the time transport transformation is constructed. This system contains 68 subgroups. As a result, it is possible to construct 68 nonstationary submodels that are invariant with respect to these subgroups. We have built and studied five submodels. For three submodels, equations are obtained that implicitly determine the acoustic pressure in ultrasonic beams. Namely, for two of them, the pressure is implicitly determined from transcendental equations, and for one submodel, the pressure is implicitly determined from an algebraic equation. Among these five sub-models, two sub-models describe axisymmetric ultrasonic beams, two others describe planar ultrasonic beams, and one describes a one-dimensional ultrasonic beam. Using these submodels, the propagation of ultrasonic beams is investigated, for which either the acoustic pressure and the rate of its change, or the acoustic pressure and its gradient are set at the initial moment of time at a fixed point. The existence and uniqueness of solutions to boundary value problems describing these processes is established. For some values of the parameters characterizing these processes, using a numerical solution, graphs of the distribution of the dynamic component of the acoustic pressure are obtained. From these graphs it follows that the ultrasonic beams monotonically weaken with a time and completely disappear in a finite time.