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On the stability of weakly nonlinear short waves on finite-amplitude long gravity waves

Published online by Cambridge University Press:  26 April 2006

Jun Zhang  and
W. K. Melville
Jun Zhang
Affiliation:
Ocean Engineering Program. Department of Civil Engineering. Texas A & M University, College Station, TX 77843, USA
W. K. Melville
Affiliation:
R. M. Parsons Laboratory. Massachusetts Institute of Technology, Cambridge, MA 02139, USA Present address: Scripps Institution of Oceanography, University of California, San Diego, La Jolla, CA 92093-0213, USA.
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Abstract

The modulated nonlinear Schrödinger equation (Zhang & Melville 1990), describing the evolution of a weakly nonlinear short-gravity-wave train riding on a longer finite-amplitude gravity-wave train is used to study the stability of steady envelope solutions of the short-wave train. The formulation of the stability problem reduces to the solution of a pair of coupled equations for the disturbance amplitude and (relative) phase. Approximate analytical solutions and numerical solutions show that the conventional sideband (Benjamin-Feir) instability is just the first in a series of resonantly unstable regions which increase in number with increasing perturbation wavenumber. The first of these new instabilities is the result of a quintet resonance between four short waves and one long wave. Subsequent unstable regions correspond to sextet or higher-order resonances. The results presented here suggest that steady envelope solutions for unforced irrotational short waves on longer irrotational gravity waves may be unstable for a wide range of conditions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 243 , October 1992 , pp. 51 - 72
Copyright
© 1992 Cambridge University Press

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