Journal of Marine Science and Application

, Volume 16, Issue 2, pp 190–198 | Cite as

Scattering of oblique surface water waves by thin vertical barrier over undulating bed topography

  • A. Choudhary
  • S. C. Martha


The present study deals with the scattering of oblique surface water waves by small undulation on the bottom in the presence of a thin vertical barrier. Here, three different configurations of vertical barriers are investigated. Perturbation analysis is employed to determine the physical quantities, namely, the reflection and transmission coefficients. In this analysis, many different Boundary Value Problems (BVPs) are obtained out of which the first two bvps are considered. The zeroth order bvp is solved with the aid of eigenfunction expansion method. The first order reflection and transmission coefficients are derived in terms of the integrals by the method of the Green’s integral theorem. The variation of these coefficients is plotted and analyzed for different physical parameters. Furthermore, the energy balance relation, an important relation in the study of water wave scattering, is derived and checked for assuring the correctness of the numerical results for the present problem.


oblique wave scattering bottom undulation vertical barrier eigenfunction expansion Green’s integral theorem reflection and transmission coefficients 


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A. Choudhary is grateful to the University Grants Commission (UGC), Government of India, for providing the research fellowship for pursuing Ph.D. degree at the Indian Institute of Technology Ropar, India. S. C. Martha is grateful to SERB-DST, Govt. of India for financial funding under grant number SB/FTP/MS-034/2013.


  1. Banerjea S, Kanoria M, Dolai DP, Mandal BN, 1996. Oblique wave scattering by submerged thin vertical wall with a gap in finite-depth water. Applied Ocean Research, 18(6), 319–327. DOI: 10.1016/S0141-1187(97)00002-3CrossRefGoogle Scholar
  2. Chakrabarti A, Martha SC, 2009. A note on energy-balance relations in surface water wave problems involving floationg elastic plates. Journal of Advances Research in Applied Mathematics, 1(2), 27–34.Google Scholar
  3. Chakrabarti A, Mohapatra S, 2013. Scattering of surface water waves involving semi-infinite floating elastic plates on water of finite depth. Journal of Marine Science and Application, 12(3), 325–333. DOI: 10.1007/s11804-013-1204-zCrossRefGoogle Scholar
  4. Davies AG, 1982. The reflection of wave energy by undulations on the seabed. Dynamics of Atmospheres and Oceans, 6(4), 207–232. DOI: 10.1016/0377-0265(82)90029-XCrossRefGoogle Scholar
  5. Davies AG, Heathershaw AD, 1984. Surface wave propagation over sinusoidally varying topography. Journal of Fluid Mechanics, 144, 419–443. DOI: 10.1017/S0022112084001671CrossRefGoogle Scholar
  6. Dean WR, 1945. On the reflexion of the suface waves by a submerged plane barrier. Mathematical Proceedings of the Cambridge Philosophical Society, 41(3), 231–238. DOI: 10.1017/S030500410002260XCrossRefzbMATHGoogle Scholar
  7. Kirby JT, 1993. A note on Bragg scattering of surface waves by sinusoidal bars. Physics of Fluids, 5(2), 380–386. DOI: 10.1063/1.858861CrossRefzbMATHGoogle Scholar
  8. Lee MM, Chwang AT, 2000. Scattering and radiation of water waves by permeable barriers. Physics of Fluids, 12(1), 54–65. DOI: 10.1063/1.870284MathSciNetCrossRefzbMATHGoogle Scholar
  9. Losada IJ, Losada MA, Roldan AJ, 1992. Propagation of oblique incident waves past rigid vertical thin barriers. Applied Ocean Research, 14(3), 191–199. DOI: 10.1016/0141-1187(92)90014-BCrossRefGoogle Scholar
  10. Mandal BN, Basu U, 1990. A note on oblique water wave diffraction by a cylindrical deformation of the bottom in the presence of surface tension. Archives of Mechanics, 42(6), 723–727.zbMATHGoogle Scholar
  11. Mandal BN, Chakrabarti A, 1999. On Galerkin's method applicable to the problems of water wave scattering by barriers. Proceedings of the Indian National Science Academy: PINSA, 65(1), 61–71.zbMATHGoogle Scholar
  12. Mandal BN, Dolai DP, 1994. Oblique water wave diffraction by thin vertical barriers in water of uniform finite depth. Applied Ocean Research, 16(4), 195–203. DOI: 10.1016/0141-1187(94)90020-5CrossRefGoogle Scholar
  13. 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.2006.06.002CrossRefGoogle Scholar
  14. Martha SC, Bora SN, 2007. Oblique surface wave propagation over a small undulation on the bottom of an ocean. Geophysical & Astro Fluid Dynamics, 101(2), 65–80. DOI: 10.1080/03091920701208186MathSciNetCrossRefGoogle Scholar
  15. Mei CC, 1985. Resonant reflection of surface water waves by periodic sandbars. Journal of Fluid Mechanics, 152, 315–335. DOI: 10.1017/S0022112085000714CrossRefzbMATHGoogle Scholar
  16. Miles JW, 1981. Oblique surface-wave diffraction by a cylindrical obstacle. Dynamics of Atmospheres and Oceans, 6(2), 121–123. DOI: 10.1016/0377-0265(81)90019-1CrossRefGoogle Scholar
  17. Porter D, 1972. The transmission of surface waves through a gap in a vertical barrier. Mathematical proceedings of the Cambridge Philosophical Society, 71(2), 411–422. DOI: 10.1017/S0305004100050647CrossRefzbMATHGoogle Scholar
  18. Porter R, Evans DV, 1995. Complementary approximations to wave scattering by vertical barriers. Journal of Fluid Mechanics, 294, 155–180. DOI: 10.1017/S0022112095002849MathSciNetCrossRefzbMATHGoogle Scholar
  19. Sahoo T, Chan AT, Chwang AT, 2000. Scattering of oblique surface waves by permeable barriers. Journal of Waterway, Port, Coastal, and Ocean Engineering, 126(4), 196–205. DOI: 10.1061/(ASCE)0733-950X(2000)126:4(196)CrossRefGoogle Scholar
  20. Ursell F, 1947. The effect of a fixed barrier on surface waves in deep water. Mathematical proceedings of the Cambridge Philosophical Society, 43(3), 374–382. DOI: 10.1017/S0305004100023604MathSciNetCrossRefzbMATHGoogle Scholar

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© Harbin Engineering University and Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  1. 1.Department of MathematicsIndian Institute of Technology RoparRupnagarIndia

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