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A note on modulated cross-waves

Published online by Cambridge University Press:  26 April 2006

John Miles
John Miles
Affiliation:
Institute of Geophysics and Planetary Physics, University of California, San Diego, La Jolla, CA 92093-0225, USA
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Abstract

Larraza & Putterman (1984) and Miles (1984b) derived a nonlinear Schrödinger (NLS) equation for the envelope r(X,τ) of a spatially and temporally modulated cross-wave for which the spatial mean square $\overline{|r|^2} $ vanishes (e.g. a solitary wave). Sasaki (1993) found that the accommodation of non-vanishing $\overline{|r|^2} $ (e.g. a cnoidal wave) introduces a non-local term proportional to $\overline{|r|^2} $ in the coefficient of r in the NLS equation. Sasaki's result is confirmed through an average-Lagrangian formulation, in which the functional $\overline{|r|^2} $ appears (after appropriate normalization) as the Lagrange multiplier associated with the constraint of conservation of mass for the envelope. This functional is a constant, and implies a quadratic (in amplitude) shift of the resonant frequency, for a periodic ($(\partial r/\partial\tau = 0) $) wave; but if the X and τ dependencies of r are not separable it implies the replacement of the NLS equation by a nonlinear integral-partial-differential equation.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 295 , 25 July 1995 , pp. 301 - 304
Copyright
© 1995 Cambridge University Press

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References

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