|Table of Contents|

Citation:
 S. Mohapatra,M. R. Sarangi.A Note on the Solution of Water Wave Scattering Problem Involving Small Deformation on a Porous Channel-Bed[J].Journal of Marine Science and Application,2017,(1):10-19.[doi:10.1007/s11804-017-1392-z]
Click and Copy

A Note on the Solution of Water Wave Scattering Problem Involving Small Deformation on a Porous Channel-Bed

Info

Title:
A Note on the Solution of Water Wave Scattering Problem Involving Small Deformation on a Porous Channel-Bed
Author(s):
S. Mohapatra M. R. Sarangi
Affilations:
Author(s):
S. Mohapatra M. R. Sarangi
Department of Mathematics, Veer Surendra Sai University of Technology, Burla-768018, India
Keywords:
Porous bedbottom deformationperturbation analysisFourier TransformGreen’s functionreflection coefficienttransmission coefficientenergy identitywater wave scattering
分类号:
-
DOI:
10.1007/s11804-017-1392-z
Abstract:
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 parameterε(<<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.

References:

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.
Chwang AT, 1983. A porous-wavemaker theory. Journal of Fluid Mechanics, 132, 395-406.
DOI: https://doi.org/10.1017/S0022112083001676
Davies AG, 1982. The reflection of wave energy by undulations of the sea bed. Dynamics of Atmosphere and Oceans, 6, 207-232.
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-X
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)00068-8
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)00003-9
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.2014.03.006
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-y
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.2007.07.001
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/S0022112085000714
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/s11804-014-1241-2
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/s11804-015-1298-6
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-y
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/S0022112003004208
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/S0022112004009267
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)
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)00035-7
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.2005.09.006
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)00025-1
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)00050-8

Memo

Memo:
Received date:2016-04-09;Accepted date:2016-09-21。
Foundation item:Partially supported by a research grant from Department of Science and Technology (DST), India (No. SB/FTP/MS-003/2013).
Corresponding author:S. Mohapatra,Email:smrutiranjan.m@gmail.com
Last Update: 2017-03-25