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On the Downshift of Wave Frequency for Bragg Resonance

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An Erratum to this article was published on 01 April 2022

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Abstract

For surface gravity waves propagating over a horizontal bottom that consists of a patch of sinusoidal ripples, strong wave reflection occurs under the Bragg resonance condition. The critical wave frequency, at which the peak reflection coefficient is obtained, has been observed in both physical experiments and direct numerical simulations to be downshifted from the well-known theoretical prediction. It has long been speculated that the downshift may be attributed to higher-order rippled bottom and free-surface boundary effects, but the intrinsic mechanism remains unclear. By a regular perturbation analysis, we derive the theoretical solution of frequency downshift due to third-order nonlinear effects of both bottom and free-surface boundaries. It is found that the bottom nonlinearity plays the dominant role in frequency downshift while the free-surface nonlinearity actually causes frequency upshift. The frequency downshift/upshift has a quadratic dependence in the bottom/free-surface steepness. Polychromatic bottom leads to a larger frequency downshift relative to the monochromatic bottom. In addition, direct numerical simulations based on the high-order spectral method are conducted to validate the present theory. The theoretical solution of frequency downshift compares well with the numerical simulations and available experimental data.

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References

  • Alam, M.R., Liu, Y.M. and Yue, D.K.P., 2010. Oblique sub- and super-harmonic Bragg resonance of surface waves by bottom ripples, Journal of Fluid Mechanics, 643, 437–447.

    Article  MathSciNet  Google Scholar 

  • Chamberlain, P.G. and Porter, D., 1995. The modified mild-slope equation, Journal of Fluid Mechanics, 291, 393–407.

    Article  MathSciNet  Google Scholar 

  • Chang, H.K. and Liou, J.C., 2007. Long wave reflection from submerged trapezoidal breakwaters, Ocean Engineering, 34(1), 185–191.

    Article  Google Scholar 

  • Dalrymple, R.A. and Kirby, J.T., 1986. Water waves over ripples, Journal of Waterway, Port, Coastal, and Ocean Engineering, 112(2), 309–319.

    Article  Google Scholar 

  • Davies, A.G., 1982. The reflection of wave energy by undulations on the seabed, Dynamics of Atmospheres and Oceans, 6(4), 207–232.

    Article  Google Scholar 

  • Davies, A.G. and Heathershaw, A.D., 1984. Surface-wave propagation over sinusoidally varying topography, Journal of Fluid Mechanics, 144, 419–443.

    Article  Google Scholar 

  • Dommermuth, D.G. and Yue, D.K.P., 1987. A high-order spectral method for the study of nonlinear gravity waves, Journal of Fluid Mechanics, 184, 267–288.

    Article  Google Scholar 

  • Fan, J., Zheng, J.H., Tao, A.F. and Liu, Y.M., 2021. Upstream-propagating waves induced by steady current over a rippled bottom: theory and experimental observation, Journal of Fluid Mechanics, 910, A49.

    Article  MathSciNet  Google Scholar 

  • Fan, J., Zheng, J.H., Tao, A.F., Yu, H.F. and Wang, Y., 2016. Experimental study on upstream-advancing waves induced by currents, Journal of Coastal Research, 75, 846–850.

    Article  Google Scholar 

  • Guazzelli, E., Rey, V. and Belzons, M., 1992. Higher-order Bragg reflection of gravity surface waves by periodic beds, Journal of Fluid Mechanics, 245, 301–317.

    Article  Google Scholar 

  • Guo, F.C., Liu, H.W. and Pan, J.J., 2021. Phase downshift or upshift of Bragg resonance for water wave reflection by an array of cycloidal bars or trenches, Wave Motion, 106, 102794.

    Article  MathSciNet  Google Scholar 

  • Liang, B.C., Ge, H.L., Zhang, L.B. and Liu, Y., 2020. Wave resonant scattering mechanism of sinusoidal seabed elucidated by Mathieu Instability theorem, Ocean Engineering, 218, 108238.

    Article  Google Scholar 

  • Liu, H.W., Li, X.F. and Lin, P.Z., 2019a. Analytical study of Bragg resonance by singly periodic sinusoidal ripples based on the modified mild-slope equation, Coastal Engineering, 150, 121–134.

    Article  Google Scholar 

  • Liu, H.W., Liu, Y. and Lin, P.Z., 2019b. Bloch band gap of shallow-water waves over infinite arrays of parabolic bars and rectified cosinoidal bars and Bragg resonance over finite arrays of bars, Ocean Engineering, 188, 106235.

    Article  Google Scholar 

  • Liu, H.W., Shi, Y.P. and Cao, D.Q., 2015. Optimization of parabolic bars for maximum Bragg resonant reflection of long waves, Journal of Hydrodynamics, 27(3), 373–382.

    Article  Google Scholar 

  • Liu, H.W., Zeng, H.D. and Huang, H.D., 2020. Bragg resonant reflection of surface waves from deep water to shallow water by a finite array of trapezoidal bars, Applied Ocean Research, 94, 101976.

    Article  Google Scholar 

  • Liu, W.J., Liu, Y.S. and Zhao, X.Z., 2019. Numerical study of Bragg reflection of regular water waves over fringing reefs based on a Boussinesq model, Ocean Engineering, 190, 106415.

    Article  Google Scholar 

  • Liu, Y.M. and Yue, D.K.P., 1998. On generalized Bragg scattering of surface waves by bottom ripples, Journal of Fluid Mechanics, 356, 297–326.

    Article  MathSciNet  Google Scholar 

  • Mei, C.C., 1985. Resonant reflection of surface water waves by periodic sandbars, Journal of Fluid Mechanics, 152, 315–335.

    Article  Google Scholar 

  • Mei, C.C., Hara, T. and Naciri, M., 1988. Note on Bragg scattering of water waves by parallel bars on the seabed, Journal of Fluid Mechanics, 186, 147–162.

    Article  Google Scholar 

  • Miles, J.W., 1967. Surface-wave scattering matrix for a shelf, Journal of Fluid Mechanics, 28(4), 755–767.

    Article  MathSciNet  Google Scholar 

  • Naciri, M. and Mei, C.C., 1988. Bragg scattering of water waves by a doubly periodic seabed, Journal of Fluid Mechanics, 192, 51–74.

    Article  MathSciNet  Google Scholar 

  • O’Hare, T.J. and Davies, A.G., 1993. A comparison of two models for surface-wave propagation over rapidly varying topography, Applied Ocean Research, 15(1), 1–11.

    Article  Google Scholar 

  • Peng, J., Tao, A.F., Liu, Y.M., Zheng, J.H., Zhang, J.S. and Wang, R.S., 2019. A laboratory study of class III Bragg resonance of gravity surface waves by periodic beds, Physics of Fluids, 31(6), 067110.

    Article  Google Scholar 

  • Phillips, O.M., 1960. On the dynamics of unsteady gravity waves of finite amplitude. Part 1. The elementary interactions, Journal of Fluid Mechanics, 9(2), 193–217.

    Article  MathSciNet  Google Scholar 

  • Qin, S.F., Fan, J., Zhang, H.M., Su, J.W. and Wang, Y., 2021. Flume experiments on energy conversion behavior for oscillating buoy devices interacting with different wave types, Journal of Marine Science and Engineering, 9(8), 852.

    Article  Google Scholar 

  • Short, A.D., 1975. Multiple offshore bars and standing waves, Journal of Geophysical Research, 80(27), 3838–3840.

    Article  Google Scholar 

  • Tao, A.F., Yan, J., Wang, Y., Zheng, J.H., Fan, J. and Qin, C., 2017. Wave power focusing due to the Bragg resonance, China Ocean Engineering, 31(4), 458–465.

    Article  Google Scholar 

  • Tao, A.F., **e, S.Y., Wu, D., Fan, J. and Yang, Y.N., 2021. The effects on water particle velocity of wave peaks induced by nonlinearity under different time scales, Journal of Marine Science and Engineering, 9(7), 748.

    Article  Google Scholar 

  • Tsai, L.H., Kuo, Y.S., Lan, Y.J., Hsu, T.W. and Chen, W.J., 2011. Investigation of multiply composite artificial bars for Bragg scattering of water waves, Coastal Engineering Journal, 53(4), 521–548.

    Article  Google Scholar 

  • Wang, G., Liang, Q.H., Shi, F.Y. and Zheng, J.H., 2021. Analytical and numerical investigation of trapped ocean waves along a submerged ridge, Journal of Fluid Mechanics, 915, A54.

    Article  MathSciNet  Google Scholar 

  • Wang, S.K., Hsu, T.W., Tsai, L.H. and Chen, S.H., 2006. An application of Miles’ theory to Bragg scattering of water waves by doubly composite artificial bars, Ocean Engineering, 33(3–4), 331–349.

    Article  Google Scholar 

  • Zakharov, V.E., 1968. Stability of periodic waves of finite amplitude on the surface of a deep fluid, Journal of Applied Mechanics and Technical Physics, 9(2), 190–194.

    Article  Google Scholar 

  • Zhang, H.M., Tao, A.F., Tu, J.H., Su, J.W. and **e, S.Y., 2021. The focusing waves induced by Bragg resonance with V-shaped undulating bottom, Journal of Marine Science and Engineering, 9(7), 708.

    Article  Google Scholar 

  • Zhang, Y., Guo, J., Liu, Q., Huang, W.R., Bi, C.W. and Zhao, Y.P., 2021. Storm damage risk assessment for offshore cage culture, Aquacultural Engineering, 95, 102198.

    Article  Google Scholar 

  • Zheng, J.H., Yao, Y., Chen, S.G., Chen, S.B. and Zhang, Q.M., 2020. Laboratory study on wave-induced setup and wave-driven current in a 2DH reef-lagoon-channel system, Coastal Engineering, 162, 103772.

    Article  Google Scholar 

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Authors and Affiliations

  1. Key Laboratory of Ministry of Education for Coastal Disaster and Protection, Hohai University, Nan**g, 210098, China

    Ji Peng, Ai-feng Tao, Jun Fan & **-hai Zheng

  2. Quality Supervision Department, China Renewable Energy Engineering Institute, Bei**g, 100120, China

    Ji Peng

  3. College of Harbor, Coastal and Offshore Engineering, Hohai University, Nan**g, 210098, China

    Ai-feng Tao, Jun Fan & **-hai Zheng

  4. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, 02139, USA

    Yu-ming Liu

Authors
  1. Ji Peng
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  2. Ai-feng Tao
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  3. Jun Fan
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  4. **-hai Zheng
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  5. Yu-ming Liu
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Correspondence to Ai-feng Tao.

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This research work is financially supported by the National Natural Science Foundation of China (Grant Nos. U1706230 and 51379071), the Key Project of NSFC-Shandong Joint Research Funding POW3C (Grant No. U1906230), and the National Science Fund for Distinguished Young Scholars (Grant No. 51425901).

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Peng, J., Tao, Af., Fan, J. et al. On the Downshift of Wave Frequency for Bragg Resonance. China Ocean Eng 36, 76–85 (2022). https://doi.org/10.1007/s13344-022-0006-y

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  • DOI: https://doi.org/10.1007/s13344-022-0006-y

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