A study of the generalization of the three-dimensional model of nonlinear hydroacoustics of Khokhlov–Zabolotskaya–Kuznetsov in a cubic-nonlinear medium in the presence of dissipation

Abstract

Abstract This article is devoted to the study of the generalization of the Khokhlov–Zabolotskaya–Kuznetsov model (3KZK) of nonlinear hydroacoustics in a cubic-nonlinear medium in the presence of dissipation. This model is described by third order quasilinear partial differential equation (3KZK). We obtained that the (3KZK) equation admits an eight parametric Lie group of the transformations. The submodels of the (3KZK) model are described by the invariant solutions of the (3KZK) equation. We have studied all essentially distinct, not linked by means of the point transformations, invariant solutions of rank 1 of this equation. The invariant solutions of rank 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integro-differential equations. For example, we obtained the invariant solutions that we called by “First ultrasonic needle” and “Second ultrasonic needle”. The submodels described by this solutions have the following property: at each fixed moment of the time in the field of the existence of the solutions, near some point, the pressure increases and becomes infinite in this point. These submodels can be used, in particular, in medicine as a test in preparing for the operations with a help of an ultrasound. Also we obtained the invariant solutions that we called by “First ultrasonic knife” and “Second ultrasonic knife”. The submodels described by these solutions have the following property: at each fixed moment of the time in the field of the existence of the solution, near some plane the pressure increases and becomes infinite on this plane. The same as Ultrasonic needles, these submodels can be used, in particular, in medicine as a test in preparing for the operations with a help of an ultrasound. 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 (3KZK) model. With a help of these invariant solutions we researched a propagation of the intensive 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. The graphs of the pressure distribution obtained as a result of numerical solution of these boundary problems are given. A mechanical relevance of the obtained solutions is as follows: 1) these solutions describe in a cubic-nonlinear medium with dissipation the nonlinear and diffraction effects in ultrasonic fields of a special kind, 2) these solutions can be used as a test solutions in the numerical calculations, performed in studies of ultrasonic fields generated by powerful emitters in a cubic-nonlinear medium with dissipation. Application of the obtained formulas generating the new solutions for the found solutions gives the families of the solutions containing the arbitrary constants.