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The high wavenumber instabilities of a Stokes wave

Published online by Cambridge University Press:  18 April 2017

Dieter E. Hasselmann
Dieter E. Hasselmann*
Affiliation:
Meteorologisches Institut der Universität Hamburg, Germany
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Extract

A stability analysis for high wavenumber perturbations of a Stokes wave of wavenumber k1 and slope ϵ is presented. Except for a correction term the governing equation is shown to be of Hill's type. The analysis predicts instability at wavenumbers k2 = ¼(m + 1)2k1. The two lowest and strongest instabilities are the Benjamin- Feir instability at m = 1, and the quartet resonance at m = 2. Both are incorrectly treated by the present method. For m ≥ 3 the analysis should be asymptotically (ϵ → 0) correct, yielding instability O(ϵm) due to m-fold Bragg-scattering. The non-resonant perturbations behave as predicted by WKBJ theory. The instability is too weak for experimental detection; numerical tests should be possible, but are not available at present.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 93 , Issue 3 , August 1979 , pp. 491 - 499
Copyright
Copyright © Cambridge University Press 1979

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