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On the theory of long waves

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Abstract

A new method of studying plane steady wave motion of a gravity fluid is elucidated in this paper. This method succeeds in establishing the existence of a solitary wave, for example, and in giving the first complete foundation for the approximate Rayleigh theory [1], which concerns the theory of finite-amplitude long waves. Underlying the method are general boundary properties of univalent functions, used earlier by the author to construct a qualitative theory of jet fluid motions [2].

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Literature cited

  1. 1.

    A. I. Sretenskii, Theory of Wave Motions [in Russian], ONTI (1936).

  2. 2.

    M. A. Lavrent'ev, “On some properties of univalent functions with application to the theory of jets,” Mat. Sb.,4(46), No. 3 (1938).

  3. 3.

    Some Problems of Mathematics and Mechanics. Collection of Papers Honoring the Sixtieth Birthday of Academician M. A. Lavrent'ev [in Russian], Nauka, Novosibirsk (1961).

  4. 4.

    Some Problems of Mathematics and Mechanics. Collection of Papers Honoring the Seventieth Birthday of Academician M. A. Lavrent'ev [in Russian], Nauka, Leningrad (1970).

  5. 5.

    M. A. Lavrent'ev, “On the theory of longwaves,” in: Transactions of the Institute of Mathematics, Academy of Sciences of the Ukrainian SSR [in Russian], No. 8 (1947).

  6. 6.

    M. A. Lavrent'ev, “On the theory of longwaves,” Dokl. Akad. Nauk SSSR,41, No. 7 (1943).

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Additional information

The variational methods of conformai mapping developed by M. A. Lavrent'ev (the M. A. Lavrent'ev congruence theorems) have been applied richly in the papers of Mikhail Alekseev himself, his pupils, and followers in the theory of quasiconformal mapping, in problems of hydrodynamics with free boundaries, in filtration theory, in numerical methods of solving applied problems, and other branches of mathematics and mechanics. The survey of the results obtained here and references to appropriate papers can be found in [3, 4], for example. The translation from Ukrainian of a similarly titled paper of M. A. Lavrent'ev ([5] (1947)) is the first Russian publication of the complete proof of the classical theorem on the existence of a solitary wave announced by the author in 1943 in Doklady Akademii Nauk SSSR [6]. This article was translated by M. P. Shcherbyak and edited by V. N. Monakhov.

Translated from Zhurnal Prikladnoi Mechaniki i Tekhnicheskoi Fiziki, No. 5, pp. 3–46, September–October, 1975.

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Lavrent'ev, M.A. On the theory of long waves. J Appl Mech Tech Phys 16, 659–702 (1975). https://doi.org/10.1007/BF00854079

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Keywords

  • Mathematical Modeling
  • Mechanical Engineer
  • Industrial Mathematic
  • Solitary Wave
  • Univalent Function