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Bragg scattering of nonlinear surface waves by sinusoidal sandbars

Published online by Cambridge University Press:  11 January 2024

Haiqi Fang Open the ORCID record for Haiqi Fang [Opens in a new window] ,
Lian Tang Open the ORCID record for Lian Tang [Opens in a new window]  and
Pengzhi Lin
Haiqi Fang
Affiliation:
State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, PR China
Lian Tang*
Affiliation:
State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, PR China
Pengzhi Lin*
Affiliation:
State Key Laboratory of Hydraulics and Mountain River Engineering, Sichuan University, Chengdu 610065, PR China
*
†Email addresses for correspondence: tanglian@scu.edu.cn, cvelinpz@scu.edu.cn
†Email addresses for correspondence: tanglian@scu.edu.cn, cvelinpz@scu.edu.cn
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Abstract

Based on the multiple-scale expansion technique, a new set of extended nonlinear Schrödinger (ENLS) equations up to the third order is derived to account for the additional high-order bottom and dispersion effects as well as the nonlinear wave interaction on wave transformation over periodic sandbars of sinusoidal geometry. By employing the small-amplitude wave assumption, a closed-form analytical solution for Bragg scattering is obtained from the linearised ENLS equations, which demonstrates that a downshift of wave frequency of the maximum reflection is mainly due to the inclusion of the high-order bottom effect. The factors that affect the downshift of the resonant frequency are identified and a theoretical expression in parabolic form is derived to quantify the downshift magnitude. The fully ENLS equations are further analysed to reveal the additional wave nonlinear effects on Bragg scattering characteristics. Under the condition of infinitesimal sandbar amplitude, the ENLS equations render a theoretical expression of the critical value of $kh$ when the nonlinear wave self-modulation effect and the nonlinear wave cross-modulation effect are equal, whereas the former effect is responsible for wavenumber upshifting and the latter downshifting. When $kh$ is larger than the critical value, the increase of wave nonlinearity will enhance the downshift magnitude of the Bragg resonance, and vice versa. For finite amplitude of the bottom sandbar, the ENLS equations are solved numerically to examine the influence of both wave nonlinearity and sandbar amplitude on the characteristics of Bragg resonance. The results reveal that as the increase of sandbar amplitude, the critical $kh$ increases monotonically.

JFM classification

Waves/Free-surface Flows: Wave scattering Waves/Free-surface Flows: Surface gravity waves Geophysical and Geological Flows: Coastal engineering
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 979 , 25 January 2024 , A13
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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