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7Математические модели физики атмосферы, океана и окружающей среды 101
References
1. Yu. A. Chirkunov. Nonlinear longitudinal oscillations of viscoelastic rod in Kelvin model // J. Appl. Math. and Mech. 79. (2015). 506-513.
A study 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
Yu. A. Chirkunov
Novosibirsk State University of Architecture and Civil Engineering (Sibstrin)
Email: chr101@mail.ru
DOI: 10.24411/9999-017A-2020-10373
A generalization of the three-dimensional model Khokhlov-Zabolotskaya-Kuznetsov model in a cubically nonlinear medium with the special nonlinear coefficients describes an effect of attenuation of powerful ultrasonic beams due to the formation of shock waves. We got all invariant submodels of this model. The submodels of this model are described by the invariant solutions of the three-dimensional partial differential 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. 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.
The study was carried out with the financial support of RFBR and the Government of the Novosibirsk region in the framework of the Project № 19-41-540004.
References
1. Rudenko O. V. On the 40th anniversary of the Khokhlov - Zabolotskaya equation. Acoustic. Phys. 2010. V. 56. No 4. P. 452-462.
Submodels of the three-dimensional model of Khokhlov - Zabolotskaya - Kuznetsov nonlinear hydroacoustics in a cubically nonlinear medium, describing an increasing pressure in ultrasonic beams due to inertialess self-focusing
Yu. A. Chirkunov
Novosibirsk State University of Architecture and Civil Engineering (Sibstrin)
Email: chr101@mail.ru
DOI: 10.24411/9999-017A-2020-10372
We got all invariant submodels of the generalization of the Khokhlov - Zabolotskaya - Kuznetsov model (3KZK) of nonlinear hydroacoustics in a cubic-nonlinear medium in the presence of dissipation. In particular, we obtained the submodels that we called by "Ultrasonic needles" that 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. Also we obtained the submodels that we called by "Ultrasonic knifes" that 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 submodels 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. 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 study was carried out with the financial support of RFBR and the Government of the Novosibirsk region in the framework of the Project № 19-41-540004.
