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Steady-state resonance of multiple wave interactions in deep water

Published online by Cambridge University Press:  24 February 2014

Zeng Liu  and
Shi-Jun Liao
Zeng Liu
Affiliation:
State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China
Shi-Jun Liao*
Affiliation:
State Key Laboratory of Ocean Engineering, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China Key Laboratory of Education-Ministry in Scientific Computing, Department of Mathematics, Shanghai Jiaotong University, Shanghai 200240, PR China Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University (KAU), Jeddah, Saudi Arabia
*
†Email address for correspondence: sjliao@sjtu.edu.cn
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Abstract

The steady-state resonance of multiple surface gravity waves in deep water was investigated in detail to extend the existing results due to Liao (Commun. Nonlinear Sci. Numer. Simul., vol. 16, 2011, pp. 1274–1303) and Xu et al. (J. Fluid Mech., vol. 710, 2012, pp. 379–418) on steady-state resonance from a quartet to more general and coupled resonant quartets, together with higher-order resonant interactions. The exact nonlinear wave equations are solved without assumptions on the existence of small physical parameters. Multiple steady-state resonant waves are obtained for all the considered cases, and it is found that the number of multiple solutions tends to increase when more wave components are involved in the resonance sets. The topology of wave energy distribution in the parameter space is analysed, and it is found that the steady-state resonant waves indeed form a continuum in the parameter space. The significant roles of the near-resonance and nonlinearity were also revealed. It is found that all of the near-resonant components as a whole contain more and more wave energy, as the wave patterns tend from two dimensions to one dimension, or as the nonlinearity of the steady-state resonant wave system increases. In addition, the linear stability of the steady-state resonant waves is analysed. It is found that the steady-state resonant waves are stable, as long as the disturbance does not resonate with any components of the basic wave. All of these findings are helpful to enrich and deepen our understanding about resonant gravity waves.

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 742 , 10 March 2014 , pp. 664 - 700
Copyright
© 2014 Cambridge University Press 

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Liu supplementary movie

Movie1: Perspective view of surface elevation of group 1 when $k_{2,x}=0.9$ and $varepsilon=1.001$. With angles closed to collinear limit, the wave pattern behaviors like long-crested waves, which is warped by the wave group node.

Download Liu supplementary movie(Video)
Video 5 MB

Liu supplementary movie

Movie 2: Perspective view of surface elevation of group 2 when $k_{2,x}=0.9$ and $varepsilon=1.001$. With angles closed to collinear limit, the wave pattern behaviors like long-crested waves, which is warped by the wave group node.

Download Liu supplementary movie(Video)
Video 4.9 MB

Liu supplementary movie

Movie 3: Perspective view of surface elevation of group 3 when $k_{2,x}=0.9$ and $varepsilon=1.001$. With angles closed to collinear limit, the wave pattern behaviors like long-crested waves, which is warped by the wave group node.

Download Liu supplementary movie(Video)
Video 5 MB