Submodels of the three-dimensional model of Khokhlov–Zabolotskaya–Kuznetsov nonlinear hydroacoustics in a cubically nonlinear medium, describing the nonlinear extinction of the ultrasonic beams in the presence of dissipation

Abstract

Not so long ago, the effect of increasing pressure in ultrasonic beams due to inertialess self-focusing and the effect of attenuation of powerful ultrasonic beams due to the formation of shock waves were experimentally discovered. Three-dimensional model of Khokhlov–Zabolotskaya–Kuznetsov of nonlinear hydroacoustics in a quadratically nonlinear medium in the presence of dissipation does not allow to take into account these effects. A generalization of the Khokhlov–Zabolotskaya–Kuznetsov model in a cubically nonlinear medium with the special nonlinearity coefficients describes these effects. This generalization with a special nonlinearity coefficient describes the nonlinear attenuation of high-power ultrasonic beams. Our article is devoted to the study of the submodels of this model. We obtained that an equation defined this model admits an eight parametric Lie group of the transformations. It is a main group of this equation. The submodels of this model are described by the invariant solutions of its equation. We have studied all essentially distinct, not linked by means of the point transformations, invariant solutions of rank 0 and rank 1 of this equation. These solutions are found either explicitly, or their search is reduced to the solution of the nonlinear integro-differential equations. In particular, we obtained exact solutions describing the extinction of very powerful ultrasonic beams: “ultrasonic needles” and “ultrasonic knives”, which we obtained in previous articles. The presence of the arbitrary constants in the integro-differential equations, that determine invariant solutions of rank 1 provides a new opportunities for analytical and numerical study of the boundary value problems for the received submodels, and, thus, for the original model. With a help of these invariant solutions we researched the nonlinear attenuation of high-power ultrasonic acoustic waves (one-dimensional, axisymmetric and planar) for which the acoustic pressure, speed and acceleration of its change are specified 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 solutions of boundary value problems, describing these wave processes. This allows us to correctly carry out numerical calculations in the study of these processes. The graphs of the pressure distribution obtained as a result of numerical solution of these boundary problems are given. From these graphs it follows that for the obtained submodels at each fixed point, the extinction of the ultrasonic beam occurs both during a finite period of time and during an infinite period of time.Application of the obtained formulas generating the new solutions for the found solutions gives the families of the solutions containing the arbitrary constants.A mechanical relevance of the obtained solutions is as follows: (1) these solutions describe in a cubically nonlinear medium with dissipation the nonlinear attenuation of high-power ultrasonic beams, (2) these solutions can be used as a test solutions in the numerical calculations performed at the studies of ultrasonic powerful beams in a cubically nonlinear medium with dissipation.