Journal of Marine Science and Application

, Volume 16, Issue 3, pp 286–297 | Cite as

Wave trapping by dual porous barriers near a wall in the presence of bottom undulation



Trapping of oblique surface gravity waves by dual porous barriers near a wall is studied in the presence of step type varying bottom bed that is connected on both sides by water of uniform depths. The porous barriers are assumed to be fixed at a certain distance in front of a vertical rigid wall. Using linear water wave theory and Darcy's law for flow past porous structure, the physical problem is converted into a boundary value problem. Using eigenfunction expansion in the uniform bottom bed region and modified mild-slope equation in the varying bottom bed region, the mathematical problem is handled for solution. Moreover, certain jump conditions are used to account for mass conservation at slope discontinuities in the bottom bed profile. To understand the effect of dual porous barriers in creating tranquility zone and minimum load on the sea wall, reflection coefficient, wave forces acting on the barrier and the wall, and surface wave elevation are computed and analyzed for different values of depth ratio, porous-effect parameter, incident wave angle, gap between the barriers and wall and slope length of undulated bottom. The study reveals that with moderate porosity and suitable gap between barriers and sea wall, using dual barriers an effective wave trapping system can be developed which will exert less wave force on the barriers and the rigid wall. The proposed wave trapping system is likely to be of immense help for protecting various facilities/ infrastructures in coastal environment.


porous barriers mild-slope equation reflection coefficient wave trapping porous-effect parameter 



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The authors thank the reviewers for their valuable comments which have improved the presentation of this paper. Manisha expresses her gratitude to CTS, IIT Kharagpur for giving support under visitors program during which the revised manuscript was finalized.


  1. Behera H, Kaligatla RB, Sahoo T, 2015. Wave trapping by porous barrier in the presence of step type bottom. Wave Motion, 57, 219–230. DOI: 10.1016/j.wavemoti.2015.04.005MathSciNetCrossRefGoogle Scholar
  2. Behera H, Sahoo T, Ng Chiu-On, 2016. Wave scattering by a partial flexible porous barrier in the presence of a step-type bottom topography. Coastal Engineering Journal, 58(3), 1650008 (1-26). DOI: 10.1142/S057856341650008XCrossRefGoogle Scholar
  3. Bennetts LG, Biggs NRT, Porter D, 2009. The interaction of flexural-gravity waves with periodic geometries. Wave Motion, 46(1), 57–73. DOI: 10.1016/j.wavemoti.2008.08.002MathSciNetCrossRefMATHGoogle Scholar
  4. Berkhoff JCW, 1972. Computation of combined refractiondiffraction. Proceedings of 13th International Conference on Coastal Engineering ASCE, Vancouver, Canada, 471–490. DOI: 10.1061/9780872620490.027Google Scholar
  5. Bhattacharjee J, Guedes Soares C, 2011. Oblique wave interaction with a floating structure near a wall with stepped bottom. Ocean Engineering, 38, 1528–1544. DOI: 10.1016/j.oceaneng.2011.07.011CrossRefGoogle Scholar
  6. Billingham J, King AC, 2000. Wave Motion. Cambridge University Press, Cambridge, United Kingdom. DOI: 10.1017/CBO9780511841033MATHGoogle Scholar
  7. Cerrato A, Gonzalez JA, Rodriguez-Tembleque L, 2016. Boundary element formulation of the mild-slope equation for harmonic water waves propagating over unidirectional variable bathymetries. Engineering Analysis with Boundary Elements, 62, 22–34. DOI: 10.1016/j.enganabound.2015.09.006MathSciNetCrossRefGoogle Scholar
  8. Chamberlain PG, Porter D, 1995. The modified mild-slope equation. Journal of Fluid Mechanics, 291, 393–407. DOI: 10.1017/S0022112095002758MathSciNetCrossRefMATHGoogle Scholar
  9. Chwang AT, 1983. A porous-wavemaker theory. Journal of Fluid Mechanics, 132, 395–406. DOI: 10.1017/S0022112083001676CrossRefMATHGoogle Scholar
  10. Chwang AT, Chan AT, 1998. Interaction between porous media and wave motion. Annual Review of Fluid Mechanics, 30, 53–84. DOI: 10.1146/annurev.fluid.30.1.53MathSciNetCrossRefGoogle Scholar
  11. Das S, Bora SN, 2014. Damping of oblique ocean waves by a vertical porous structure placed on a multi-step bottom. Journal of Marine Science and Application, 13(4), 362–376. DOI: 10.1007/s11804-014-1281-7CrossRefGoogle Scholar
  12. Davies AG, Heathershaw AD, 1984. Surface-wave propagation over sinusoidally varying topography. Journal of Fluid Mechanics, 144, 419–443. DOI: 10.1017/S0022112084001671CrossRefGoogle Scholar
  13. Dhillon H, Banerjea S, Mandal BN, 2016. Water wave scattering by a finite dock over a step-type bottom topography. Ocean Engineering, 113, 1–10. DOI: 10.1016/j.oceaneng.2015.12.017CrossRefGoogle Scholar
  14. Huang Z, Li Y, Liu Y, 2011. Hydraulic performance and wave loadings of perforated/slotted coastal structures: A review. Ocean Engineering, 38(10), 1031–1053. DOI: 10.1016/j.oceaneng.2011.03.002CrossRefGoogle Scholar
  15. Kaligatla RB, Manam SR, 2016. Bragg resonance of membranecoupled gravity waves over a porous bottom. International Journal of Advances in Engineering Sciences and Applied Mathematics, 8, 222–237. DOI: 10.1007/s12572-016-0169-yMathSciNetCrossRefMATHGoogle Scholar
  16. Karmakar D, Bhattacharjee J, Guedes Soares C, 2013. Scattering of gravity waves by multiple surface-piercing floating membrane. Applied Ocean Research, 39, 40–52. DOI: 10.1016/j.apor.2012.10.001CrossRefGoogle Scholar
  17. Karmakar D, Guedes Soares C, 2014. Wave transformation due to multiple bottom-standing porous barriers. Ocean Engineering, 80, 50–63. DOI: 10.1016/j.oceaneng.2014.01.012CrossRefGoogle Scholar
  18. Koley S, Behera H, Sahoo T, 2014. Oblique wave trapping by porous structures near a wall. Journal of Engineering Mechanics, 141(3), 1–15. DOI: /10.1061/(ASCE)EM.1943-7889.0000843Google Scholar
  19. Li YC, Liu Y, Teng B, 2006. Porous effect parameter of thin permeable plates. Coastal Engineering Journal, 48(4), 309–336. DOI: 10.1142/S0578563406001441CrossRefGoogle Scholar
  20. Liu Y, Li Y, 2011. Wave interaction with a wave absorbing double curtain-wall breakwater. Ocean Engineering, 38(10), 1237–1245. DOI: 10.1016/j.oceaneng.2011.05.009CrossRefGoogle Scholar
  21. Manam SR, Kaligatla RB, 2012. A mild-slope model for membrane-coupled gravity waves. Journal of Fluids and Structures, 30, 173–187. DOI: 10.1016/j.jfluidstructs.2012.01.003CrossRefGoogle Scholar
  22. Mandal S, Behera H, Sahoo T, 2015. Oblique wave interaction with porous, flexible barriers in a two-layer fluid. Journal of Engineering Mathematics, 100(1), 1–31. DOI: 10.1007/s10665-015-9830-x.MathSciNetCrossRefGoogle Scholar
  23. Massel SR, 1993. Extended refraction-diffraction equation for surface waves. Coastal Engineering, 19(1), 97–126. DOI: 10.1016/0378-3839(93)90020-9CrossRefGoogle Scholar
  24. Michael I, Sundarlingam P, Gang Y, 1998. Wave interactions with vertical slotted barrier. Journal of Waterway, Port, Coastal, and Ocean Engineering, 124(3), 118–126. DOI: 10.1061/(ASCE)0733-950XCrossRefGoogle Scholar
  25. Mondal A, Gayen R, 2015. Wave interaction with dual circular porous plates. Journal of Marine Science and Application, 14(4), 366–375. DOI: 10.1007/s11804-015-1325-7CrossRefGoogle Scholar
  26. Porter D, 2003. The mild-slope equations. Journal of Fluid Mechanics, 494, 51–63. DOI: 10.1017/S0022112003005846MathSciNetCrossRefMATHGoogle Scholar
  27. Porter D, Porter R, 2004. Approximations to wave scattering by an ice sheet of variable thickness over undulating bed topography. Journal Fluid Mechanics, 509, 145–179. DOI: 10.1017/S0022112004009267MathSciNetCrossRefMATHGoogle Scholar
  28. Porter D, Staziker DJ, 1995. Extensions of the mild-slope equation. Journal of Fluid Mechanics, 300, 367–382. DOI: 10.1017/S0022112095003727MathSciNetCrossRefMATHGoogle Scholar
  29. Porter R, Porter D, 2000. Water wave scattering by a step of arbitrary profile. Journal of Fluid Mechanics, 411, 131–164. DOI: 10.1017/S0022112099008101MathSciNetCrossRefMATHGoogle Scholar
  30. Sahoo T, Chan AT, Chwang AT, 2000a. Scattering of oblique surface waves by permeable barriers. Journal of Waterway, Port, Coastal, and Ocean Engineering, 126(4), 196–205. DOI: 10.1061/(ASCE)0733-950XCrossRefGoogle Scholar
  31. Sahoo T, Lee MM, Chwang AT, 2000b. Trapping and generation of waves by vertical porous structures. Journal of Engineering Mechanics, 126(10), 1074–1082. DOI: 10.1061/(ASCE)0733-9399CrossRefGoogle Scholar
  32. Smith R, Sprinks T, 1975. Scattering of surface waves by a conical island. Journal of Fluid Mechanics, 72(2), 373–384. DOI: 10.1017/S0022112075003424CrossRefMATHGoogle Scholar
  33. Suh KD, Kim YW, Ji CH, 2011. An empirical formula for friction coefficient of a perforated wall with vertical slits. Coastal Engineering, 58(1), 85–93. DOI: 10.1016/j.coastaleng.2010.08.006CrossRefGoogle Scholar
  34. Suh KD, Park WS, 1995. Wave reflection from perforated-wall caisson breakwaters. Coastal Engineering, 26(3), 177–193. DOI: 10.1016/0378-3839(95)00027-5CrossRefGoogle Scholar
  35. Twu SW, Lin DT, 1990. Wave reflection by a number of thin porous plates fixed in a semi-infinitely long flume. Coastal Engineering Proceedings, 1046–1059. DOI: 10.1061/9780872627765.081Google Scholar
  36. Yu X, 1995. Diffraction of water waves by porous breakwaters. Journal of Waterway, Port, Coastal, and Ocean Engineering, 121(6), 275–282. DOI: 10.1061/(ASCE)0733-950X(1995)121:6(275)CrossRefGoogle Scholar

Copyright information

© Harbin Engineering University and Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Department of Applied MathematicsIndian Institute of Technology (ISM) DhanbadDhanbadIndia
  2. 2.Department of Ocean Engineering and Naval ArchitectureIndian Institute of Technology KharagpurKharagpurIndia

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