|Table of Contents|

Citation:
　Smrutiranjan Mohapatra.The Interaction of Oblique Flexural Gravity Waves With a Small Bottom Deformation on a Porous Ocean-Bed: Green’s Function Approach[J].Journal of Marine Science and Application,2016,(2):112-122.[doi:10.1007/s11804-016-1353-y]
Click and Copy

The Interaction of Oblique Flexural Gravity Waves With a Small Bottom Deformation on a Porous Ocean-Bed: Green’s Function Approach

Info

Title:
The Interaction of Oblique Flexural Gravity Waves With a Small Bottom Deformation on a Porous Ocean-Bed: Green’s Function Approach
Author(s):
Smrutiranjan Mohapatra
Affilations:
Author(s):
Smrutiranjan Mohapatra
Department of Mathematics, Veer Surendra Sai University of Technology, Burla 768018, India
Keywords:
oblique incident waves; bottom deformation; porous bed; elastic plate; Green’s function; reflection coefficient; transmission coefficient; energy identity
分类号:
-
DOI:
10.1007/s11804-016-1353-y
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.

References:

Bennetts, LG, Biggs NRT, Porter D, 2007. A multi-mode approximation to wave scattering by ice sheets of varying thickness. Journal of Fluid Mechanics, 579, 413-443.
DOI: 10.1017/S002211200700537X
Chamberlain PG, Porter D, 1995. The modified mild-slope equations. Journal of Fluid Mechanics, 291, 393-407.
DOI: 10.1017/S0022112095002758
Chwang AT, 1983. A porous-wavemaker theory. Journal of Fluid Mechanics, 132, 395-406.
Das D, Mandal BN, Chakrabarti A, 2008. Energy identities in water wave theory for free-surface boundary condition with higher-order derivatives. Fluid Dynamics Research, 40(4), 253-272.
DOI: 10.1016/j.fluiddyn.2007.10.002
Davies AG, 1982. The reflection of wave energy by undulations of the sea bed. Dynamics of Atmosphere and Oceans, 6, 207-232.
Davies AG, Heathershaw AD, 1984. Surface wave propagation over sinusoidally varying topography. Journal of Fluid Mechanics, 144, 419-443.
Fox C, Squire VA, 1994. On the oblique reflection and transmission of ocean waves at shore fast sea ice. Philosophical Transactions: Royal Society of London, Series A, 347, 185-218.

DOI: 10.1098/rsta.1994.0044
Gu Z, Wang H, 1991. Gravity waves over porous bottoms. Coastal Engineering, 15(5-6), 497-524.
DOI: 10.1016/0378-3839(91)90025-C
Jeng DS, 2001. Wave dispersion equation in a porous seabed. Coastal Engineering, 28(12), 1585-1599.
DOI: 10.1016/S0029-8018(00)00068-8
Kirby JT, 1986. A general wave equation for waves over rippled beds. Journal of Fluid Mechanics, 162, 171-186.
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
Mandal BN, De Soumen, 2009. Surface wave propagation over small undulations at the bottom of an ocean with surface discontinuity. Geophysical and Astrophysical Fluid Dynamics, 103(1), 19-30.
Mandal BN, Gayen R, 2006. Water wave scattering by bottom undulations in the presence of a thin partially immersed barrier. Applied Ocean Research, 28(2), 113-119.
DOI: 10.1016/j.apor.2006.06.002
Martha SC, Bora SN, Chakrabarti A, 2007. Oblique water-wave scattering by small undulation on a porous sea-bed. Applied Ocean Research, 29(1-2), 86-90.
DOI: 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: 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
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: 10.1017/S0022112004009267
Porter R, Porter D, 2003. Scattered and free waves over periodic beds. Journal of Fluid Mechanics, 483, 129-163.
DOI: 10.1017/S0022112003004208
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.
Silva R, Salles P, Palacio A (2002). Linear wave propagating over a rapidly varying finite porous bed. Coastal Engineering, 44(3), 239-260.
DOI: 10.1016/S0378-3839(01)00035-7
Staziker DJ, Porter D, Stirling DSG, 1996. The scattering of surface waves by local bed elevations. Applied Ocean Research, 18(5), 283-291.
DOI: 10.1016/S0141-1187(96)00021-1
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: 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: 10.1016/S0378-3839(00)00050-8

Memo

Memo:
Received date: 2015-07-28;Accepted date: 2015-12-23。
Foundation item: Partially Supported by a Research from Department of Science and Technology (DST), India under Grant No. SB/FTP/MS-003/2013
Corresponding author: Smrutiranjan Mohapatra,E-mail:smrutiranjan.m@gmail.com
Last Update: 2016-07-06