Invariant solutions of the Westervelt model of nonlinear hydroacoustics without dissipation |
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| Institution: | 1. Novosibirsk State Technical University, 20 Marks Avenue, 630073 Novosibirsk, Russia;2. Novosibirsk State University of Architecture and Civil Engineering (Sibstrin), 113 Leningradskaya Street, 630008 Novosibirsk, Russia;1. Department of Physics, Calicut University, Malappuram, Kerala 673635, India;2. Department of Physics, Farook College, Calicut University, Kerala 673632, India;3. Laboratoire Interdisciplinaire Carnot de Bourgogne, UMR 6303 CNRS, Univ. Bourgogne Franche-Comté, 9 Av. A. Savary, B.P. 47870, 21078 Dijon Cedex, France;4. CNRS/Universite Joseph Fourier, Laboratoire Interdisciplinaire de Physique (LIPHY), 38402 Saint-Martin-d’Héres, Grenoble, France;1. Department of Mechanical Engineering, B.I.T., KEC-201007, Gaziabad, U.P. India.;2. Faculty of engineering at HCT, Ras Al Khaimah, Ra''S Al Khaymah, United Arab Emirates;3. Departement of Applied Mechanics, MNNIT, Allahabad,211004, UP, India;4. Department of Mechanical Engineering, B.I.T., Mesra, Ranchi-835215, Jharkhand, India.;1. Department of Engineering Mechanics, Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Ivana Lučića 5, 10000 Zagreb, Croatia;2. LMT-Cachan, ENS Cachan/CNRS/University Paris-Saclay, 61 avenue du Président Wilson, 94235 Cachan Cedex, France |
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| Abstract: | We study three-dimensional Westervelt model of a nonlinear hydroacoustics without dissipation. We received all of its invariant submodels. We studied all invariant submodels described by the invariant solutions of rank 0 and 1. All invariant solutions of rank 0 and 1 are found either explicitly, or their search is reduced to the solution of the nonlinear integral equations. With a help of these invariant solutions we researched: (1) a propagation of the intensive acoustic waves (self-similar, axisymmetric, planar and one-dimensional) for which the acoustic pressure and a speed of its change, or the acoustic pressure and its derivative in the direction of one of the axes are specified at the initial moment of the time at a fixed point , (2) a spherically symmetric ultrasonic field for which the acoustic pressure and a speed of its change, or the acoustic pressure and its radial derivative are specified at the initial moment of the time at a fixed point. Solving of the boundary value problems describing these processes is reduced to the solving of nonlinear integral equations. We are established the existence and uniqueness of solutions of these boundary value problems under some additional conditions. Mechanical relevance of the obtained solutions is as follows: (1) these solutions describe 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. We found all the conservation laws of the first order for the Westerveld equation written in dimensionless variables. |
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| Keywords: | Nonlinear Westervelt model of hydroacoustics Intensive acoustic waves Ultrasonic field Invariant submodels |
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