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Steady-state resonant waves over periodic beds of infinite extent: the water wave Bloch problem

Published online by Cambridge University Press:  02 April 2025

Dali Xu Open the ORCID record for Dali Xu [Opens in a new window] ,
Haojie Li Open the ORCID record for Haojie Li [Opens in a new window]  and
Hongsheng Zhang Open the ORCID record for Hongsheng Zhang [Opens in a new window]
Dali Xu*
Affiliation:
College of Ocean Science and Engineering, Shanghai Maritime University, Shanghai 201306, PR China
Haojie Li
Affiliation:
College of Ocean Science and Engineering, Shanghai Maritime University, Shanghai 201306, PR China
Hongsheng Zhang
Affiliation:
College of Ocean Science and Engineering, Shanghai Maritime University, Shanghai 201306, PR China
*
Corresponding author: Dali Xu, [email protected]
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Abstract

Steady-state Bloch wave systems at resonance with fixed frequencies and amplitudes are investigated using the homotopy analysis method. Nonlinear waves propagate over a stationary undulating bottom topography of infinite extent, modelled as a superposition of two waveforms. The wave systems are classified as type 1 if the primary transmitted and resonant wave components have equal energies, and type 2 if the energy distribution is unequal. Two subtypes of type 2 are identified, distinguished by their responses to frequency detuning and bottom topography: the wave steepness in subtype 1 shows monotonic variations with detuning, while in subtype 2 it exhibits a peak at a particular detuning value, indicating downward resonance that intensifies with greater wave steepness. A pair of peaks in wave steepness arises in each subtype at certain values of the angle $\theta$ between the waveforms of the bottom topography. In both subtypes, the peaks are slightly affected by the ratio $k_{{b}1}/k_{{b}2}$ of the two bottom wave vectors, and significantly affected by the propagation angle $\alpha$ of the primary transmitted wave, but remain stable under changes to other topographic parameters. As the topography amplitude and $\theta$ vary, significant additional contributions to the total energy of the wave system appear from components other than resonant and primary transmitted waves. The most pronounced effects occur with changes in $\theta$, with the additional components accounting for up to 12 % of the total energy. This study provides an enriched understanding of resonant Bloch wave systems and a basis for improving the effectiveness of wave energy converters.

JFM classification

Waves/Free-surface Flows: Surface gravity waves Waves/Free-surface Flows: Wave-structure interactions
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 1008 , 10 April 2025 , A35
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Copyright
© The Author(s), 2025. Published by Cambridge University Press

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