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Finite amplitude steady-state wave groups with multiple near resonances in deep water

Published online by Cambridge University Press:  27 November 2017

Z. Liu Open the ORCID record for Z. Liu [Opens in a new window] ,
D. L. Xu  and
S. J. Liao Open the ORCID record for S. J. Liao [Opens in a new window]
Z. Liu*
Affiliation:
School of Naval Architecture and Ocean Engineering, Huazhong University of Science and Technology, Wuhan 430074, China Hubei Key Laboratory of Naval Architecture and Ocean Engineering Hydrodynamics, Huazhong University of Science and Technology, Hubei 430074, China Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China
D. L. Xu
Affiliation:
College of Ocean Science and Engineering, Shanghai Maritime University, Shanghai 201306, China
S. J. Liao
Affiliation:
Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Shanghai 200240, China School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai 200240, China State Key Laboratory of Ocean Engineering, Shanghai Jiao Tong University, Shanghai 200240, China
*
†Email address for correspondence: z_liu@hust.edu.cn
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Abstract

In this paper, finite amplitude steady-state wave groups with multiple nearly resonant interactions in deep water are investigated theoretically. The nonlinear water wave equations are solved by the homotopy analysis method (HAM), which imposes no constraint on either the number or the amplitude of the wave components, to resolve the small-divisor problems caused by near resonances. A new kind of auxiliary linear operator in the framework of the HAM is proposed to transform the small divisors associated with the non-trivial nearly resonant components to singularities associated with the exactly resonant ones. Primary components, exactly resonant components together with nearly resonant components are considered as the initial non-trivial components, since all of them are homogeneous solutions to the auxiliary linear operator. For wave groups with weak nonlinearity, the energy transfer between nearby nearly resonant components is remarkable. As the nonlinearity increases, the number of steady-state wave groups increases as more components join the near resonance. This indicates that the probability of existence of steady-state resonant waves increases with the nonlinearity of wave groups. The frequency band broadens and spectral asymmetry becomes more and more pronounced. The amplitude of each component may either increase or decrease with the nonlinearity of wave groups, while the amplitude of the whole wave group increases continuously and finite amplitude wave groups are obtained. This work shows the wide existence of steady-state waves when multiple nearly resonant interactions are considered.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Waves/Free-surface Flows
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 835 , 25 January 2018 , pp. 624 - 653
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Copyright
© 2017 Cambridge University Press 

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