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Solitary internal waves with oscillatory tails

Published online by Cambridge University Press:  26 April 2006

T. R. Akylas  and
R. H. J. Grimshaw
T. R. Akylas
Affiliation:
Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA
R. H. J. Grimshaw
Affiliation:
School of Mathematics, University of New South Wales, Kensington, NSW 2033, Australia
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Abstract

Solitary internal waves in a density-stratified fluid of shallow depth are considered. According to the classical weakly nonlinear long-wave theory, the propagation of each long-wave mode is governed by the Korteweg–de Vries equation to leading order, and locally confined solitary waves with a ‘sech’ profile are possible. Using a singular-perturbation procedure, it is shown that, in general, solitary waves of mode n > 1 actually develop oscillatory tails of infinite extent, consisting of lower-mode short waves. The amplitude of these tails is exponentially small with respect to an amplitude parameter, and lies beyond all orders of the usual long-wave expansion. To illustrate the theory, two special cases of stratification are discussed in detail, and the amplitude of the oscillations at the solitary-wave tails is determined explicitly. The theoretical predictions are supported by experimental observations.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 242 , September 1992 , pp. 279 - 298
Copyright
© 1992 Cambridge University Press

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