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On the steady-state fully resonant progressive waves in water of finite depth

Published online by Cambridge University Press:  07 September 2012

Dali Xu ,
Zhiliang Lin ,
Shijun Liao  and
Michael Stiassnie
Dali Xu
Affiliation:
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Zhiliang Lin
Affiliation:
School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Shijun Liao*
Affiliation:
State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
Michael Stiassnie
Affiliation:
Faculty of Civil and Environmental Engineering, Technion IIT, Haifa 32000, Israel
*
†Email address for correspondence: sjliao@sjtu.edu.cn
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Abstract

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The steady-state fully resonant wave system, consisting of two progressive primary waves in finite water depth and all components due to nonlinear interaction, is investigated in detail by means of analytically solving the fully nonlinear wave equations as a nonlinear boundary-value problem. It is found that multiple steady-state fully resonant waves exist in some cases which have no exchange of wave energy at all, so that the energy spectrum is time-independent. Further, the steady-state resonant wave component may contain only a small proportion of the wave energy. However, even in these cases, there usually exist time-dependent periodic exchanges of wave energy around the time-independent energy spectrum corresponding to such a steady-state fully resonant wave, since it is hard to be exactly in such a balanced state in practice. This view serves to deepen and enrich our understanding of the resonance of gravity waves.

JFM classification

Waves/Free-surface Flows: Surface gravity waves
Type
Papers
Information
Journal of Fluid Mechanics , Volume 710 , 10 November 2012 , pp. 379 - 418
Copyright
Copyright © Cambridge University Press 2012

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