China Ocean Engineering

, Volume 32, Issue 5, pp 524–535 | Cite as

Scattering of Oblique Water Waves by Two Unequal Surface-Piercing Vertical Thin Plates with Stepped Bottom Topography

  • Li-xian Wang
  • Zheng-zhi Deng
  • Chen Wang
  • Peng Wang


Based on linear water-wave theory, this study investigated the scattering of oblique incident water waves by two unequal surface-piercing thin vertical rigid plates with stepped bottom topography. By using the matched eigenfunction expansion method and a least square approach, the analytical solutions are sought for the established boundary value problem. The effects of the incidence angle, location of step, depth ratio of deep to shallow waters, and column width between two plates, on the reflection coefficients, the horizontal wave forces acting on the two plates, and the mean surface elevation between the two plates, are numerically examined under a variety of wave conditions. The results show that the existence of the stepped bottom between two plates considerably impacts the hydrodynamic performances of the present system. It is found that the effect of stepped bottom on the reflection coefficient of the present two-plate structure is evident only with waves of the low dimensionless frequency. Moreover, the influence of the step location on the hydrodynamic performance of the present two-plate structure is slight if the step is placed in between the two plates.

Key words

stepped bottom topography two unequal thin vertical rigid plates oblique water waves reflection coefficient wave force 


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  1. Bhattacharjee, J. and Soares, C.G., 2010. Wave interaction with a floating rectangular box near a vertical wall with step type bottom topography, Journal of Hydrodynamics, Ser. B, 22(5 Suppl1), 91–96.CrossRefGoogle Scholar
  2. Bhattacharjee, J. and Soares, C.G., 2011. Oblique wave interaction with a floating structure near a wall with stepped bottom, Ocean Engineering, 38(13), 1528–1544.CrossRefGoogle Scholar
  3. Das, P., Dolai, D.P. and Mandal, B.N., 1997. Oblique wave diffraction by parallel thin vertical barriers with gaps, Journal of Waterway, Port, Coastal, and Ocean Engineering, 123(4), 163–171.CrossRefGoogle Scholar
  4. Dhillon, H., Banerjea, S. and Mandal, B.N., 2013. Oblique wave scattering by a semi-infinite rigid dock in the presence of bottom undulations, Indian Journal of Pure and Applied Mathematics, 44(2), 167–184.MathSciNetCrossRefzbMATHGoogle Scholar
  5. Dhillon, H., Banerjea, S. and Mandal, B.N., 2016. Water wave scattering by a finite dock over a step-type bottom topography, Ocean Engineering, 113), 1–10.CrossRefGoogle Scholar
  6. Evans, D.V. and Porter, R., 1997. Complementary methods for scattering by thin barriers, in: Mandal, B.N. (ed.), International Series on Advances in Fluid Mechanics, Computational Mechanics Publications, Southampton, pp. 1–43.Google Scholar
  7. Havelock, T.H., 1929. LIX. Forced surface-waves on water, The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science, 8(51), 569–576.CrossRefzbMATHGoogle Scholar
  8. Isaacson, M., Baldwin, J., Premasiri, S. and Yang, G., 1999. Wave interactions with double slotted barriers, Applied Ocean Research, 21(2), 81–91.CrossRefGoogle Scholar
  9. Karmakar, D., Bhattacharjee, J. and Sahoo, T., 2010. Oblique flexural gravity-wave scattering due to changes in bottom topography, Journal of Engineering Mathematics, 66(4), 325–341.MathSciNetCrossRefzbMATHGoogle Scholar
  10. Karmakar, D. and Sahoo, T., 2008. Gravity wave interaction with floating membrane due to abrupt change in water depth, Ocean Engineering, 35(7), 598–615.CrossRefGoogle Scholar
  11. Karmakar, D. and Soares, C.G., 2012. Oblique scattering of gravity waves by moored floating membrane with changes in bottom topography, Ocean Engineering, 54), 87–100.CrossRefGoogle Scholar
  12. Liu, P.L.F. and Abbaspour, M., 1982. Wave scattering by a rigid thin barrier, Journal of the Waterway, Port, Coastal and Ocean Division, 108(4), 479–491.Google Scholar
  13. Liu, Y. and Li, Y.C., 2011. Wave interaction with a wave absorbing double curtain-wall breakwater, Ocean Engineering, 38(10), 1237–1245.CrossRefGoogle Scholar
  14. Losada, I.J., Losada, M.A. and Roldán, A.J., 1992. Propagation of oblique incident waves past rigid vertical thin barriers, Applied Ocean Research, 14(3), 191–199.CrossRefGoogle Scholar
  15. Mandal, B.N. and Das, P., 1996. Oblique diffraction of surface waves by a submerged vertical plate, Journal of Engineering Mathematics, 30(4), 459–470.MathSciNetCrossRefzbMATHGoogle Scholar
  16. Mandal, B.N. and 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.CrossRefGoogle Scholar
  17. Martha, S.C. and Bora, S.N., 2007. Reflection and transmission coefficients for water wave scattering by a sea-bed with small undulation, Journal of Applied Mathematics and Mechanics, 87(4), 314–321.MathSciNetzbMATHGoogle Scholar
  18. McIver, P., 1985. Scattering of water waves by two surface-piercing vertical barriers, IMA Journal of Applied Mathematics, 35(3), 339–355.MathSciNetCrossRefzbMATHGoogle Scholar
  19. Morris, C.A.N., 1975. A variational approach to an unsymmetric water- wave scattering problem, Journal of Engineering Mathematics, 9(4), 291–300.CrossRefzbMATHGoogle Scholar
  20. Neelamani, S. and Vedagiri, M., 2002. Wave interaction with partially immersed twin vertical barriers, Ocean Engineering, 29(2), 215–238.CrossRefGoogle Scholar
  21. Newman, J.N., 1965. Propagation of water waves over an infinite step, Journal of Fluid Mechanics, 23(2), 399–415.MathSciNetCrossRefGoogle Scholar
  22. Newman, J.N., 1974. Interaction of water waves with two closely spaced vertical obstacles, Journal of Fluid Mechanics, 66(1), 97–106.CrossRefzbMATHGoogle Scholar
  23. Porter, R. and Evans, D.V., 1995. Complementary approximations to wave scattering by vertical barriers, Journal of Fluid Mechanics, 294), 155–180.MathSciNetCrossRefzbMATHGoogle Scholar
  24. Reddy, M.S. and Neelamani, S., 1992. Wave transmission and reflection characteristics of a partially immersed rigid vertical barrier, Ocean Engineering, 19(3), 313–325.CrossRefGoogle Scholar
  25. Rezanejad, K., Bhattacharjee, J. and Soares, C.G., 2015. Analytical and numerical study of dual-chamber oscillating water columns on stepped bottom, Renewable Energy, 75), 272–282.CrossRefGoogle Scholar
  26. Rhee, J.P., 1997. On the transmission of water waves over a shelf, Applied Ocean Research, 19(3–4), 161–169.CrossRefGoogle Scholar
  27. Roy, R., Basu, U. and Mandal, B.N., 2016. Oblique water wave scattering by two unequal vertical barriers, Journal of Engineering Mathematics, 97(1), 119–133.MathSciNetCrossRefzbMATHGoogle Scholar
  28. Shin, D.M. and Cho, Y., 2016. Diffraction of waves past two vertical thin plates on the free surface: a comparison of theory and experiment, Ocean Engineering, 124), 274–286.CrossRefGoogle Scholar
  29. Ursell, F. and Dean, W.R., 1947. The effect of a fixed vertical barrier on surface waves in deep water, Mathematical Proceedings of the Cambridge Philosophical Society, 43(3), 374–382.MathSciNetCrossRefzbMATHGoogle Scholar
  30. Wiegel, R.L., 1960. Transmission of waves past a rigid vertical thin barrier, Journal of the Waterways and Harbors Division, 86(1), 1–12.Google Scholar

Copyright information

© Chinese Ocean Engineering Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  • Li-xian Wang
    • 1
    • 2
  • Zheng-zhi Deng
    • 3
  • Chen Wang
    • 3
  • Peng Wang
    • 3
  1. 1.Key Laboratory of High Performance Ship Technology (Wuhan University of Technology)Ministry of EducationWuhanChina
  2. 2.Departments of Naval Architecture, Ocean and Structural Engineering, School of TransportationWuhan University of TechnologyWuhanChina
  3. 3.Institute of Port, Coastal and Offshore Engineering, Ocean CollegeZhejiang UniversityZhoushanChina

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