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Parametrically excited solitary waves

Published online by Cambridge University Press:  20 April 2006

John W. Miles
John W. Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093
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Abstract

A modulated cross-wave of resonant frequencyω1, carrier frequencyω =ω1 {1 + O(ε)}, slowly varying complex amplitude O(ε½b), longitudinal scale b/ε½ and timescale 1/εω is induced in a long channel of breadth b that contains water of depth d and is subjected to a vertical oscillation of amplitude O(εb) and frequency 2ω, where 0 < ε [Lt ] 1. The complex amplitude satisfies a cubic Schrödinger equation, generalized to incorporate weak damping and the parametric excitation. A solution is obtained that describes the standing solitary wave observed by Wu, Keolian & Rudnick (1984). The results depend on both d/b and l*/b, where l* is the capillary length (l* = 2.7 mm for clean water), and solitary waves are impossible if d/b < 0.325 for l*/b = 0 or if l*/b > 0.045 for d/b [gsim ] 1. The corresponding cnoidal waves (of which the solitary wave is a limiting case) are considered in an appendix.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 148 , November 1984 , pp. 451 - 460
Copyright
© 1984 Cambridge University Press

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