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Journal of Marine Science and Application

, Volume 16, Issue 1, pp 10–19 | Cite as

A note on the solution of water wave scattering problem involving small deformation on a porous channel-bed

  • S. Mohapatra
  • M. R. Sarangi
Article

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 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.

Keywords

Porous bed bottom deformation perturbation analysis Fourier Transform Green’s function reflection coefficient transmission coefficient energy identity water wave scattering 

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Notes

Acknowledgement

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.

References

  1. 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.MathSciNetCrossRefzbMATHGoogle Scholar
  2. Chwang AT, 1983. A porous-wavemaker theory. Journal of Fluid Mechanics, 132, 395–406. DOI: https://doi.org/10.1017/S0022112083001676CrossRefzbMATHGoogle Scholar
  3. Davies AG, 1982. The reflection of wave energy by undulations of the sea bed. Dynamics of Atmosphere and Oceans, 6, 207–232.CrossRefGoogle Scholar
  4. 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
  5. 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
  6. 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)00003-9MathSciNetCrossRefzbMATHGoogle Scholar
  7. 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
  8. 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-yCrossRefzbMATHGoogle Scholar
  9. 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.2007.07.001CrossRefzbMATHGoogle Scholar
  10. 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/S0022112085000714CrossRefzbMATHGoogle Scholar
  11. 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
  12. 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
  13. 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
  14. 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/S0022112003004208MathSciNetCrossRefzbMATHGoogle Scholar
  15. 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/S0022112004009267MathSciNetCrossRefzbMATHGoogle Scholar
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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

Copyright information

© Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsVeer Surendra Sai University of TechnologyBurlaIndia

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