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On transforms reducing one-dimensional systems of shallow-water to the wave equation with sound speed c 2 = x

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Abstract

We obtain point transformations for three one-dimensional systems: shallow-water equations on a flat and a sloping bottom and the system of linear equations obtained by formal linearization of shallow-water equations on a sloping bottom. The passage of these systems to the Carrier-Greenspan parametrization is also obtained. For linear shallow-water equations on a sloping bottom, we obtain the solution in the form of a traveling wave with variable velocity. We establish the relationship between the resulting solution and the solution of the two-dimensional wave equation.

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Authors and Affiliations

  1. Ishlinskii Institute for Problems in Mechanics, Russian Academy of Sciences Moscow Institute of Physics and Technology (State University), Moscow, Russia

    S. Yu. Dobrokhotov & D. S. Minenkov

  2. Institute of Computation Techniques, Siberian Branch, Russian Academy of Sciences, Novosibirsk, Russia

    S. B. Medvedev

  3. Moscow State University, Moscow, Russia

    S. B. Medvedev

Authors
  1. S. Yu. Dobrokhotov
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  2. S. B. Medvedev
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  3. D. S. Minenkov
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Correspondence to S. Yu. Dobrokhotov.

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Original Russian Text © S. Yu. Dobrokhotov, S. B. Medvedev, D. S. Minenkov, 2013, published in Matematicheskie Zametki, 2013, Vol. 93, No. 5, pp. 716–727.

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Dobrokhotov, S.Y., Medvedev, S.B. & Minenkov, D.S. On transforms reducing one-dimensional systems of shallow-water to the wave equation with sound speed c 2 = x . Math Notes 93, 704–714 (2013). https://doi.org/10.1134/S0001434613050064

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  • DOI: https://doi.org/10.1134/S0001434613050064

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