SELF-SIMILAR WAVES OF THREE-DIMENSIONAL MODEL OF WESTERVELT IN THE EXISTENCE OF DISSIPATION

Introduction: Many mathematical models of continuum mechanics are formulated as linear and quasilinear differential equations. Symmetric analysis of these models is one of the most effective ways to obtain the quantitative and qualitative characteristics of the physical processes described by them. The relevance of the study of the Westervelt model is due to the use of this model for calculating parametric sonar antennas and for calculating ultrasonic fields in medicine Materials and research methods: Research methods are: group (symmetry) analysis of differential equations and general methods of mathematical physics. Research results and their discussion: The three-dimensional Westervelt model of nonlinear hydroacoustics, which is important for applications, is investigated in the presence of dissipation. It has been established that among the invariant submodels of rank 1, this model has only three types of essentially different (not connected by point transformations) self-similar waves. These include the self-similar wave propagating along one of the axes of coordinates, the flat self-similar circular wave and the self-similar spherically symmetric wave. Integro-differential equations describing these self-similar waves are obtained. Under some conditions, the existence and uniqueness of such self-similar waves are established. Conclusions: In nonlinear hydroacoustics, the Westervelt model is used to study ultrasonic fields generated by high-power radiators. For the Westervelt three-dimensional model in the presence of dissipation, it was found that among the invariant submodels of rank 1 this model has only three types of essentially different (not connected by point transformations) self-similar waves. These waves are investigated. The presence of arbitrary constants in the obtained integro-differential equations describing these self-similar waves opens up new possibilities for studying other boundary value problems (other than those studied in the article) that have a physical meaning. The practical significance of this study is due to the use of the Westervelt model for calculating parametric sonar antennas and for calculating ultrasonic ields in medicine.

нелинейная гидроакустика, модель Вестервельта при наличии диссипации, автомодельные волны, nonlinear underwater acoustics, Westervelt model in the presence of dissipation, self-similar waves

1. Westervelt P. Parametric acoustic array // J. Acoustic Soc. Am., 1963, vol. 35(4), pp. 535-537.

2. Novikov B.K., Rudenko V. I., Timoshenko V. I. Nonlinear underwater acoustics. New York, AlP-Press. 1987. 262 p.

3. V. A.Voronin, S. P. Tarasov, V. I. Timoshenko. Nonlinear acoustics. New York, AlP-Press. 1995. 314 p.

4. Nonlinear acoustics. Ed. By M. Hamilton and D. Blackstock. London: Academic. 1998.

5. Ostashev V.E. Acoustic in moving inhomogeneous media. London: E&Fn Spon. 1997. 259 p.

6. Nachef S., Cathignol D., Tjotta J.N., Berg A.M., Tjotta S. Investigation of a high intensity sound beam from a plane transducer. Experimental and theoretical results // J. Acoust. Soc. Am., 1995, vol. 98, pp. 2303-2323.

7. Tavakkoli J., Cathignol D., Souchon R., Sapozhnikov O.A. Modeling of pulsed finiteamplitude focused sound beams in time domain // J. Acoust. Soc. Am., 1998, vol. 104, pp. 2061-2072.

8. Lee Y.S. and Hamilton M.F. Time-domain modeling of pulsed finite amplitude sound beams // J. Acoust. Soc. Amer., 1995, vol. 97(2), pp. 906-917.

9. Averkiou M.A., and Hamilton M.F. Nonlinear distortion of short pulses radiated by plane and focused circular pistons // J. Acoust. Soc. Amer., 1997, vol. 102(5), pp. 2539-2548.

10. Sokka S.D., King R., and Hynynen K. MRI-guided gas bubble enhanced ultrasound heating in in vivo rabbit thigh // Phys. Med. Biol., 2003, vol. 48, pp. 223-241.

11. Чиркунов Ю. А. Подмодели трехмерной модели Вестервельта при отсутствии диссипации // Ж.: Наука. Инновации. Технологии. 2018, вып 1. С. 81-94.

12. Чиркунов Ю. А., Хабиров С. В. Элементы симметрийного анализа дифференциальных уравнений механики сплошной среды. Новосибирск: НГТУ. 2012. 659 c.