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Resonance theory of water waves in the long-wave limit

Published online by Cambridge University Press:  28 March 2013

Takeshi Kataoka
Takeshi Kataoka*
Affiliation:
Department of Mechanical Engineering, Graduate School of Engineering, Kobe University, Rokkodai, Nada, Kobe 657-8501, Japan
*
†Email address for correspondence: [email protected]
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Abstract

The instability due to resonant interactions of finite-amplitude water waves is examined in the long-wave limit. In contrast to the well-known case of a small-amplitude limit in which the resonance is considered for a flat surface, here we consider a periodic approximate of the finite-amplitude solitary wave which is the long-wave limit of the periodic wave. The resonance conditions for the corresponding perturbations yield a new family of resonance curves that are totally different from those of the small-amplitude limit obtained by Phillips and Mclean. Under these resonance conditions, we conduct a systematic asymptotic analysis for small wavenumbers to obtain the growth rates of the perturbations explicitly and clarify whether each resonance curve is associated with instability. These results are verified numerically by showing that the instability bands for finite-amplitude periodic waves in shallow water are located along these unstable resonance curves.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 722 , 10 May 2013 , pp. 461 - 495
Copyright
©2013 Cambridge University Press

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