Fluid Dynamics

, Volume 18, Issue 5, pp 751–756 | Cite as

Nonlinear waves on shallow water in the presence of a horizontal magnetic field

  • K. D. Danov
  • M. S. Ruderman
Article

Abstract

Benjamin [1] and Davis and Acrivos [2] derived an equation for long steady nonlinear internal waves in an infinitely deep stratified fluid when the density varies only in a layer whose thickness is small compared with the characteristic perturbation length. Ono [3] generalized this equation to the unsteady case. The resulting equation was subsequently called the Benjamin—Ono equation. Steady solutions of this equation were found by Benjamin and Ono in the form of solitons and periodic waves. In the present paper it is shown that long nonlinear waves on shallow water in the presence of a horizontal magnetic field can also be described by the Benjamin—Ono equation, and not the Korteweg—de Vries equation [4], as in the case when there is no field. Moreover, in contrast to a soliton in a stratified fluid a soliton on shallow water in a horizontal magnetic field moves with a velocity less than the velocity of infinitely long perturbations of small amplitude. The dependence of the parameters of a soliton and a periodic wave on the intensity and direction of the unperturbed magnetic field is investigated.

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Literature cited

  1. 1.
    T. B. Benjamin, “Internal waves of permanent form in fluids of great depth,” J. Fluid Mech.,29, 559 (1967).Google Scholar
  2. 2.
    R. E. Davis and A. Acrivos, “Solitary internal waves in deep water,” J. Fluid Mech.,29, 593 (1967).Google Scholar
  3. 3.
    H. Ono, “Algebraic solitary waves in stratified fluids,” J. Phys. Soc. Jpn.,39, 1082 (1975).Google Scholar
  4. 4.
    D. J. Korteweg and G. de Vries, “On the change of form of long waves advancing in a rectangular channel, and on a new type of long stationary waves,” Philos. Mag.,39, 422 (1895).Google Scholar
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    H. Lamb, Hydrodynamics, Cambridge University Press (1932), p. 738.Google Scholar
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    V. S. Vladimirov, Equations of Mathematical Physics [in Russian], Nauka, Moscow (1971), p. 512.Google Scholar

Copyright information

© Plenum Publishing Corporation 1984

Authors and Affiliations

  • K. D. Danov
    • 1
  • M. S. Ruderman
    • 1
  1. 1.Moscow

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