Scattering of oblique surface water waves by thin vertical barrier over undulating bed topography

Abstract

The present study deals with the scattering of oblique surface water waves by small undulation on the bottom in the presence of a thin vertical barrier. Here, three different configurations of vertical barriers are investigated. Perturbation analysis is employed to determine the physical quantities, namely, the reflection and transmission coefficients. In this analysis, many different Boundary Value Problems (BVPs) are obtained out of which the first two bvps are considered. The zeroth order bvp is solved with the aid of eigenfunction expansion method. The first order reflection and transmission coefficients are derived in terms of the integrals by the method of the Green’s integral theorem. The variation of these coefficients is plotted and analyzed for different physical parameters. Furthermore, the energy balance relation, an important relation in the study of water wave scattering, is derived and checked for assuring the correctness of the numerical results for the present problem.

Keywords

oblique wave scattering / bottom undulation / vertical barrier / eigenfunction expansion / Green’s integral theorem / reflection and transmission coefficients

References

[1] Banerjea S, Kanoria M, Dolai DP, Mandal BN. Oblique wave scattering by submerged thin vertical wall with a gap in finite-depth water. Applied Ocean Research, 1996, 18(6): 319-327

[2] Chakrabarti A, Martha SC. A note on energy-balance relations in surface water wave problems involving floationg elastic plates. Journal of Advances Research in Applied Mathematics, 2009, 1(2): 27-34

[3] Chakrabarti A, Mohapatra S. Scattering of surface water waves involving semi-infinite floating elastic plates on water of finite depth. Journal of Marine Science and Application, 2013, 12(3): 325-333

[4] Davies AG. The reflection of wave energy by undulations on the seabed. Dynamics of Atmospheres and Oceans, 1982, 6(4): 207-232

[5] Davies AG, Heathershaw AD. Surface wave propagation over sinusoidally varying topography. Journal of Fluid Mechanics, 1984, 144: 419-443

[6] Dean WR. On the reflexion of the suface waves by a submerged plane barrier. Mathematical Proceedings of the Cambridge Philosophical Society, 1945, 41(3): 231-238

[7] Kirby JT. A note on Bragg scattering of surface waves by sinusoidal bars. Physics of Fluids, 1993, 5(2): 380-386

[8] Lee MM, Chwang AT. Scattering and radiation of water waves by permeable barriers. Physics of Fluids, 2000, 12(1): 54-65

[9] Losada IJ, Losada MA, Roldan AJ. Propagation of oblique incident waves past rigid vertical thin barriers. Applied Ocean Research, 1992, 14(3): 191-199

[10] Mandal BN, Basu U. A note on oblique water wave diffraction by a cylindrical deformation of the bottom in the presence of surface tension. Archives of Mechanics, 1990, 42(6): 723-727

[11] Mandal BN, Chakrabarti A. On Galerkin's method applicable to the problems of water wave scattering by barriers. Proceedings of the Indian National Science Academy: PINSA, 1999, 65(1): 61-71

[12] Mandal BN, Dolai DP. Oblique water wave diffraction by thin vertical barriers in water of uniform finite depth. Applied Ocean Research, 1994, 16(4): 195-203

[13] Mandal BN, Gayen R. Water wave scattering by bottom undulations in the presence of a thin partially immersed barrier. Applied Ocean Research, 2006, 28(2): 113-119

[14] Martha SC, Bora SN. Oblique surface wave propagation over a small undulation on the bottom of an ocean. Geophysical & Astro Fluid Dynamics, 2007, 101(2): 65-80

[15] Mei CC. Resonant reflection of surface water waves by periodic sandbars. Journal of Fluid Mechanics, 1985, 152: 315-335

[16] Miles JW. Oblique surface-wave diffraction by a cylindrical obstacle. Dynamics of Atmospheres and Oceans, 1981, 6(2): 121-123

[17] Porter D. The transmission of surface waves through a gap in a vertical barrier. Mathematical proceedings of the Cambridge Philosophical Society, 1972, 71(2): 411-422

[18] Porter R, Evans DV. Complementary approximations to wave scattering by vertical barriers. Journal of Fluid Mechanics, 1995, 294: 155-180

[19] Sahoo T, Chan AT, Chwang AT. Scattering of oblique surface waves by permeable barriers. Journal of Waterway, Port, Coastal, and Ocean Engineering, 2000, 126(4): 196-205

[20] Ursell F. The effect of a fixed barrier on surface waves in deep water. Mathematical proceedings of the Cambridge Philosophical Society, 1947, 43(3): 374-382