The solution of water wave scattering problem involving small deformation on a porous bed in a channel, where the upper surface is bounded above by an infinitely extent rigid horizontal surface, is studied here within the framework of linearized water wave theory. In such a situation, there exists only one mode of waves propagating on the porous surface. A simplified perturbation analysis, involving a small parameter ε ( ≪ 1), which measures the smallness of the deformation, is employed to reduce the governing Boundary Value Problem (BVP) to a simpler BVP for the first-order correction of the potential function. The first-order potential function and, hence, the first-order reflection and transmission coefficients are obtained by the method based on Fourier transform technique as well as Green’s integral theorem with the introduction of appropriate Green’s function. Two special examples of bottom deformation: the exponentially damped deformation and the sinusoidal ripple bed, are considered to validate the results. For the particular example of a patch of sinusoidal ripples, the resonant interaction between the bed and the upper surface of the fluid is attained in the neighborhood of a singularity, when the ripples wavenumbers of the bottom deformation become approximately twice the components of the incident field wavenumber along the positive x-direction. Also, the main advantage of the present study is that the results for the values of reflection and transmission coefficients are found to satisfy the energy-balance relation almost accurately.
Porous bed bottom deformation perturbation analysis Fourier Transform Green’s function reflection coefficient transmission coefficient energy identity water wave scattering
This is a preview of subscription content, log in to check access.
The author wishes to thank Prof. Swaroop Nandan Bora, Indian Institute of Technology Guwahati, India for his valuable discussions and suggestions to carry out the preparation of the manuscript.
Chakrabarti A, 2000. On the solution of the problem of scattering of surface-water waves by the edge of an ice cover. Proceedings of The Royal Society of London, Series A, 456, 1087–1099.MathSciNetCrossRefMATHGoogle Scholar
Chwang AT, 1983. A porous-wavemaker theory. Journal of Fluid Mechanics, 132, 395–406. DOI: https://doi.org/10.1017/S0022112083001676CrossRefMATHGoogle Scholar
Davies AG, 1982. The reflection of wave energy by undulations of the sea bed. Dynamics of Atmosphere and Oceans, 6, 207–232.CrossRefGoogle Scholar
Hur PS, Mizutani N, 2003. Numerical estimation of wave forces acting on a three-dimensional body on submerged breakwater. Coastal Engineering, 47(3), 329–345. DOI: http://dx.doi.org/10.1016/S0378-3839(02)00128-XCrossRefGoogle Scholar
Jeng DS, 2001. Wave dispersion equation in a porous seabed. Ocean Engineering, 28(12), 1585–1599. DOI: http://dx.doi.org/10.1016/S0029-8018(00)00068-8CrossRefGoogle Scholar
Linton CM, Chung H, 2003. Reflection and transmission at the ocean/sea-ice boundary. Wave Motion, 38(1), 43–52. DOI: 10.1016/S0165-2125(03)00003-9MathSciNetCrossRefMATHGoogle Scholar
Maiti P, Mandal BN, 2014. Water wave scattering by an elastic plate floating in an ocean with a porous bed. Applied Ocean Research, 47, 73–84. DOI: http://dx.doi.org/10.1016/j.apor.2014.03.006CrossRefGoogle Scholar
Mandal BN, Basu U, 2004. Wave diffraction by a small elevation of the bottom of an ocean with an ice-cover. Archive of Applied Mechanics, 73, 812–822. DOI 10.1007/s00419-004-0332-yCrossRefMATHGoogle Scholar
Martha SC, Bora SN, 2007. Oblique water-wave scattering by small undulation on a porous sea-bed. Applied Ocean Research, 29(1-2), 86–90. DOI: http://dx.doi.org/10.1016/j.apor.2007.07.001CrossRefMATHGoogle Scholar
Mei CC, 1985. Resonant reflection of surface water waves by periodic sandbars. Journal of Fluid Mechanics, 152, 315–335. DOI: https://doi.org/10.1017/S0022112085000714CrossRefMATHGoogle Scholar
Mohapatra S, 2014. 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, 13(2), 167–172. DOI: 10.1007/s11804-014-1241-2CrossRefGoogle Scholar
Mohapatra S, 2015. Scattering of oblique surface waves by the edge of a small undulation on a porous ocean bed. Journal of Marine Science and Application, 14(2), 156–162. DOI: 10.1007/s11804-015-1298-6CrossRefGoogle Scholar
Mohapatra S, 2016. The interaction of oblique flexural gravity waves with a small bottom deformation on a porous ocean-bed: Green’s function approach. Journal of Marine Science and Application, 15(2), 112–122. DOI: 10.1007/s11804-016-1353-yCrossRefGoogle Scholar
Porter R, Porter D, 2003. Scattered and free waves over periodic beds. Journal of Fluid Mechanics, 483, 129–163. DOI: https://doi.org/10.1017/S0022112003004208MathSciNetCrossRefMATHGoogle Scholar
Porter D, Porter R, 2004. Approximations to wave scattering by an ice sheet of variable thickness over undulating topography. Journal of Fluid Mechanics, 509, 145–179. DOI: https://doi.org/10.1017/S0022112004009267MathSciNetCrossRefMATHGoogle Scholar
Sahoo T, Chan AT, Chwang AT, 2000. Scattering of oblique surface waves by permeable barrierrs. Journal of Waterway, Port and Coastal Ocean Engineering, 126(4), 196–205. DOI: http://dx.doi.org/10.1061/(ASCE)0733-950X(2000)126:4(196)CrossRefGoogle Scholar
Silva R, Salles P, Palacio A, 2002. Linear wave propagating over a rapidly varying finite porous bed. Coastal Engineering, 44(3), 239–260. DOI: http://dx.doi.org/10.1016/S0378-3839(01)00035-7CrossRefGoogle Scholar
Tsai CP, Chen HB, Lee FC, 2006. Wave transfermation over submerged permeable breakwater on porous bottom. Ocean Engineering, 33(11-12), 1623–1643. DOI: http://dx.doi.org/10.1016/j.oceaneng.2005.09.006CrossRefGoogle Scholar
Wang CM, Meylan MH, 2002. The linear wave response of a floating thin plate on water of variable depth. Applied Ocean Research, 24(3), 163–174. DOI: http://dx.doi.org/10.1016/S0141-1187(02)00025-1CrossRefGoogle Scholar
Zhu S, 2001. Water waves within a porous medium on an undulating bed. Coastal Engineering, 42(1), 87–101. DOI: http://dx.doi.org/10.1016/S0378-3839(00)00050-8CrossRefGoogle Scholar