The interaction of oblique flexural gravity waves with a small bottom deformation on a porous ocean-bed: Green’s function approach

Abstract

The interaction of oblique incident water waves with a small bottom deformation on a porous ocean-bed is examined analytically here within the framework of linear water wave theory. The upper surface of the ocean is assumed to be covered by an infinitely extended thin uniform elastic plate, while the lower surface is bounded by a porous bottom surface having a small deformation. By employing a simplified perturbation analysis, involving a small parameter δ(=1), which measures the smallness of the deformation, the governing Boundary Value Problem (BVP) is reduced to a simpler BVP for the first-order correction of the potential function. This BVP is solved using a method based on Green’s integral theorem with the introduction of suitable Green’s function to obtain the first-order potential, and this potential function is then utilized to calculate the first-order reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom deformation. Consideration of a patch of sinusoidal ripples shows that when the quotient of twice the component of the incident field wave number propagating just below the elastic plate and the ripple wave number approaches one, the theory predicts a resonant interaction between the bed and the surface below the elastic plate. Again, for small angles of incidence, the reflected wave energy is more as compared to the other angles of incidence. It is also observed that the reflected wave energy is somewhat sensitive to the changes in the flexural rigidity of the elastic plate, the porosity of the bed and the ripple wave numbers. The main advantage of the present study is that the results for the values of reflection and transmission coefficients obtained are found to satisfy the energy-balance relation almost accurately.

Keywords

oblique waves / bottom deformation / porous bed / elastic plate / Green’s function / reflection coefficient / transmission coefficient / energy identity

References

[1] Bennetts LG, Biggs NRT, Porter D. A multi-mode approximation to wave scattering by ice sheets of varying thickness. Journal of Fluid Mechanics, 2007, 579: 413-443

[2] Chamberlain PG, Porter D. The modified mild-slope equations. Journal of Fluid Mechanics, 1995, 291: 393-407

[3] Chwang AT. A porous-wavemaker theory. Journal of Fluid Mechanics, 1983, 132: 395-406

[4] Davies AG. The reflection of wave energy by undulations of the sea bed. Dynamics of Atmosphere and Oceans, 1982, 6: 207-232

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

[6] Fox C, Squire VA. On the oblique reflection and transmission of ocean waves at shore fast sea ice. Philosophical Transactions: Royal Society of London, Series A, 1994, 347: 185-218

[7] Gu Z, Wang H. Gravity waves over porous bottoms. Coastal Engineering, 1991, 15(5-6): 497-524

[8] Jeng DS. Wave dispersion equation in a porous seabed. Coastal Engineering, 2001, 28(12): 1585-1599

[9] Kirby JT. A general wave equation for waves over rippled beds. Journal of Fluid Mechanics, 1986, 162: 171-186

[10] Linton CM, Chung H. Reflection and transmission at the ocean/sea-ice boundary. Wave Motion, 2003, 38(1): 43-52

[11] Mandal B D S. Surface wave propagation over small undulations at the bottom of an ocean with surface discontinuity. Geophysical and Astrophysical Fluid Dynamics, 2009, 103(1): 19-30

[12] 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

[13] Martha SC, Bora SN. Oblique water-wave scattering by small undulation on a porous sea-bed. Applied Ocean Research, 2007, 29(1-2): 86-90

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

[15] Mohapatra S. Scattering of surface waves by the edge of a small undulation on a porous bed in an ocean with ice-cover. Journal of Marine Science and Application, 2014, 13(2): 167-172

[16] Mohapatra S. Scattering of oblique surface waves by the edge of a small undulation on a porous ocean bed. Journal of Marine Science and Application, 2015, 14(2): 156-162

[17] Porter D, Porter R. Approximations to wave scattering by an ice sheet of variable thickness over undulating topography. Journal of Fluid Mechanics, 2004, 509: 145-179

[18] Porter R, Porter D. Scattered and free waves over periodic beds. Journal of Fluid Mechanics, 2003, 483: 129-163

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

[20] Silva R, Salles P, Palacio A. Linear wave propagating over a rapidly varying finite porous bed. Coastal Engineering, 2002, 44(3): 239-260

[21] Staziker DJ, Porter D, Stirling DSG. The scattering of surface waves by local bed elevations. Applied Ocean Research, 1996, 18(5): 283-291

[22] Wang CM, Meylan MH. The linear wave response of a floating thin plate on water of variable depth. Applied Ocean Research, 2002, 24(3): 163-174

[23] Zhu S. Water waves within a porous medium on an undulating bed. Coastal Engineering, 2001, 42(1): 87-101