×

Equilibrium states with finite amplitudes at exactly and nearly class-I Bragg resonances. (English) Zbl 1477.76020

Summary: The exactly and nearly class-I Bragg resonances of strongly nonlinear waves are studied analytically by the homotopy analysis method. Two types of equilibrium states with time-independent wave spectra and different energy distributions are obtained. Effects of the incident wave height, the seabed height, and the frequency detuning on resonant waves are investigated. Bifurcation points of the equilibrium states are found and tend to greater value of relatively incident wave height for a steeper wave. The wave steepness of the whole wave system grows linearly with the seabed height. Meanwhile, the resonant peak can shift to up or down side when the near resonance is considered. This work provides us a deeper understanding on class-I Bragg resonance and enlightens further studies of higher-order wave-bottom interactions.

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Liu, Y.; Yue, D. K. P., On generalized bragg scattering of surface waves by bottom ripples, Journal of Fluid Mechanics, 356, 297-326 (1998) · Zbl 0908.76014 · doi:10.1017/s0022112097007969
[2] Heathershaw, A. D., Seabed-wave resonance and sand bar growth, Nature, 296, 5855, 343-345 (1982) · doi:10.1038/296343a0
[3] Davies, A. G., The reflection of wave energy by undulations on the seabed, Dynamics of Atmospheres and Oceans, 6, 4, 207-232 (1982) · doi:10.1016/0377-0265(82)90029-x
[4] Mei, C. C., Resonant reflection of surface water waves by periodic sandbars, Journal of Fluid Mechanics, 152, 315-335 (1985) · Zbl 0588.76022 · doi:10.1017/s0022112085000714
[5] Mei, C. C.; Hara, T.; Naciri, M., Note on bragg scattering of water waves by parallel bars on the seabed, Journal of Fluid Mechanics, 186, 147-162 (1988) · Zbl 0643.76009 · doi:10.1017/s0022112088000084
[6] Liu, H.-W.; Li, X.-F.; Lin, P., Analytical study of bragg resonance by singly periodic sinusoidal ripples based on the modified mild-slope equation, Coastal Engineering, 150, 121-134 (2019) · doi:10.1016/j.coastaleng.2019.04.015
[7] Liu, H.-W.; Zeng, H.-D.; Huang, H.-D., Bragg resonant reflection of surface waves from deep water to shallow water by a finite array of trapezoidal bars, Applied Ocean Research, 94, 101976 (2020) · doi:10.1016/j.apor.2019.101976
[8] Couston, L.-A.; Jalali, M. A.; Alam, M.-R., Shore protection by oblique seabed bars, Journal of Fluid Mechanics, 815, 481-510 (2017) · Zbl 1387.86010 · doi:10.1017/jfm.2017.61
[9] Xu, D.; Lin, Z.; Liao, S., Equilibrium states of class-i bragg resonant wave system, European Journal of Mechanics - B/Fluids, 50, 38-51 (2015) · Zbl 1408.76093 · doi:10.1016/j.euromechflu.2014.10.006
[10] Kirby, J. T., A general wave equation for waves over rippled beds, Journal of Fluid Mechanics, 162, -1, 171-186 (1986) · Zbl 0596.76017 · doi:10.1017/s0022112086001994
[11] Xu, D.-l.; Liu, Z., A study on nonlinear steady-state waves at resonance in water of finite depth by the amplitude-based homotopy analysis method, Journal of Hydrodynamics, 32, 5, 888-900 (2020) · doi:10.1007/s42241-020-0013-5
[12] Stiassnie, M., On the strength of the weakly nonlinear theory for surface gravity waves, Journal of Fluid Mechanics, 810, 1-4 (2017) · Zbl 1383.76059 · doi:10.1017/jfm.2016.632
[13] Liu, Z.; Xu, D. L.; Liao, S. J., Finite amplitude steady-state wave groups with multiple near resonances in deep water, Journal of Fluid Mechanics, 835, 624-653 (2018) · Zbl 1421.76026 · doi:10.1017/jfm.2017.787
[14] Liao, S.; Xu, D.; Stiassnie, M., On the steady-state nearly resonant waves, Journal of Fluid Mechanics, 794, 175-199 (2016) · Zbl 1445.76023 · doi:10.1017/jfm.2016.162
[15] Liu, Z.; Xie, D., Finite-amplitude steady-state wave groups with multiple near-resonances in finite water depth, Journal of Fluid Mechanics, 867, 348-373 (2019) · Zbl 1415.76075 · doi:10.1017/jfm.2019.150
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.