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Scattering of oblique water waves by two thin unequal barriers with non-uniform permeability

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Abstract

The scattering of obliquely incident water waves by two unequal permeable barriers with variable permeability is investigated for two types of barriers, namely partially immersed barriers and bottom-standing barriers, under the consideration of the theory of linear water waves. The barriers are present in water of uniform finite depth. The velocity potential is expanded by using Havelock’s expansion of water wave potential and employing Havelock’s inversion formula together with the conditions on the permeable barriers; the boundary value problem is reduced to a coupled Fredholm-type vector integral equations. The integral equations are solved using the multi-term Galerkin approximation where the unknown functions are approximated in terms of Chebyshev polynomials. The numerical results for the reflection coefficient, the transmission coefficient and the energy dissipation are depicted graphically. Known results for two identical as well as two non-identical impermeable barriers and for a single and twin permeable barriers are recovered in the limiting cases.

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References

  1. Sollitt CK, Cross RH (1972) Wave transmission through permeable breakwaters. Coast Eng Proc 1:1827–1846

    Google Scholar 

  2. Chwang AT (1983) A porous-wavemaker theory. J Fluid Mech 132:395–406

    Article  MATH  Google Scholar 

  3. Yu X (1995) Diffraction of water waves by porous breakwaters. J Waterw Port Coast Ocean Eng 121:275–282

    Article  Google Scholar 

  4. Isaacson M, Premasiri S, Yang G (1998) Wave interactions with vertical slotted barrier. J Waterw Port Coast Ocean Eng 124:118–126

    Article  Google Scholar 

  5. Lee MM, Chwang AT (2000) Scattering and radiation of water waves by permeable barriers. Phys Fluids 12:54–65

    Article  MathSciNet  MATH  Google Scholar 

  6. Chen K, Chen J, Lin S, Lee Y (2004) Dual boundary element analysis of normal incident wave passing a thin submerged breakwater with rigid, absorbing and permeable boundaries. J Waterw Port Coast Ocean Eng 130:179–190

    Article  Google Scholar 

  7. McIver P (2005) Diffraction of water waves by a segmented permeable breakwater. J Waterw Port Coast Ocean Eng 131:69–76

    Article  Google Scholar 

  8. Rageh O, Koraim A (2010) Hydraulic performance of vertical walls with horizontal slots used as breakwater. Coast Eng 57:745–756

    Article  Google Scholar 

  9. Tsai CH, Young DL (2011) The method of fundamental solutions for water-wave diffraction by thin porous breakwater. J Mech 27:149–155

    Article  Google Scholar 

  10. Gayen R, Mondal A (2014) A hypersingular integral equation approach to the porous plate problem. Appl Ocean Res 46:70–78

    Article  Google Scholar 

  11. Rahman MA, Akter A (2014) The effect of porosity of submerged and emerged breakwater on wave transmission. Int J Environ Sci Dev 5:473

    Article  Google Scholar 

  12. Li AJ, Liu Y, Li HJ (2015) Accurate solutions to water wave scattering by vertical thin porous barriers. Math Probl Eng 2015:1–11

    MathSciNet  MATH  Google Scholar 

  13. Manam SR, Sivanesan M (2016) Scattering of water waves by vertical porous barriers: an analytical approach. Wave Motion 67:89–101

    Article  MathSciNet  Google Scholar 

  14. Macaskill C (1979) Reflexion of water waves by a permeable barrier. J Fluid Mech 95:141–157

    Article  MATH  Google Scholar 

  15. Tao L, Song H, Chakrabarti S (2009) Wave interaction with a perforated circular breakwater of non-uniform porosity. J Eng Math 65:257–271

    Article  MathSciNet  MATH  Google Scholar 

  16. Song H, Tao L (2010) An efficient scaled boundary FEM model for wave interaction with a nonuniform porous cylinder. Int J Numer Methods Fluids 63:96–118

    MATH  Google Scholar 

  17. Chegini V, Chegini F (2011) Experimental studies of wave transmission through single perforated sheets and upright perforated wave filters. J Persian Gulf 2:1–8

    Google Scholar 

  18. Levine H, Rodemich E (1958) Scattering of surface waves on an ideal fluid. Tech Rep, DTIC Document

  19. Jarvis RJ (1971) The scattering of surface waves by two vertical plane barriers. J Inst Math Appl 7:207–215

    MATH  Google Scholar 

  20. Das P, Dolai DP, Mandal BN (1997) Oblique wave diffraction by parallel thin vertical barriers with gaps. J Waterw Port Coast Ocean Eng 123:163–171

    Article  Google Scholar 

  21. Isaacson M, Baldwin J, Premasiri S, Yang G (1999) Wave interactions with double slotted barriers. Appl Ocean Res 21:81–91

    Article  Google Scholar 

  22. De S, Mandal BN, Chakrabarti A (2009) Water wave scattering by two submerged plane vertical barriers – Abel integral equation approach. J Eng Math 65:75–87

    Article  MathSciNet  MATH  Google Scholar 

  23. De S, Mandal BN, Chakrabarti A (2010) Use of abel integral equations in water wave scattering by two surface piercing barriers. Wave Motion 47:279–288

    Article  MathSciNet  MATH  Google Scholar 

  24. Koraim A, Heikal E, Rageh O (2011) Hydrodynamic characteristics of double permeable breakwater under regular waves. Mar Struct 24:503–527

    Article  Google Scholar 

  25. Gayen R, Mondal A (2015) Scattering of water waves by a pair of vertical porous plates. Geophys Astrophys Fluid Dyn 109:480–496

    Article  MathSciNet  Google Scholar 

  26. Li J, Fu Z, Chen W (2016) Numerical investigation on the obliquely incident water wave passing through the submerged breakwater by singular boundary method. Comput Math Appl 71:381–390

    Article  MathSciNet  Google Scholar 

  27. Morris CAN (1975) A variational approach to an unsymmetric water-wave scattering problem. J Eng Math 9:291–300

    Article  MATH  Google Scholar 

  28. Ursell F (1947) The effect of a fixed vertical barrier on surface waves in deep water. Math Proc Camb Philos Soc 43:374–382

    Article  MathSciNet  MATH  Google Scholar 

  29. Dean WR (1945) On the reflexion of surface waves by a submerged plane barrier. Math Proc Camb Philos Soc 41:231–238

    Article  MathSciNet  MATH  Google Scholar 

  30. McIver P (1985) Scattering of water waves by two surface-piercing vertical barriers. IMA J Appl Math 35:339–355

    Article  MathSciNet  MATH  Google Scholar 

  31. Abrahams I, Wickham G (1991) The scattering of water waves by two semi-infinite opposed vertical walls. Wave Motion 14:145–168

    Article  MathSciNet  MATH  Google Scholar 

  32. Evans DV, Porter R (1997) Complementary methods for scattering by thin barriers. Int Ser Adv Fluid Mech 8:1–44

    MATH  Google Scholar 

  33. Neelamani S, Vedagiri M (2002) Wave interaction with partially immersed twin vertical barriers. Ocean Eng 29:215–238

    Article  Google Scholar 

  34. Roy R, Basu U, Mandal BN (2016) Oblique water wave scattering by two unequal vertical barriers. J Eng Math 97:119–133

    Article  MathSciNet  MATH  Google Scholar 

  35. Martins-Rivas H, Mei CC (2009) Wave power extraction from an oscillating water column at the tip of a breakwater. J Fluid Mech 626:395–414

    Article  MathSciNet  MATH  Google Scholar 

  36. Chang KH, Tsaur DH, Huang LH (2012) Accurate solution to diffraction around a modified v-shaped breakwater. Coast Eng 68:56–66

    Article  Google Scholar 

  37. Porter R, Evans DV (1995) Complementary approximations to wave scattering by vertical barriers. J Fluid Mech 294:155–180

    Article  MathSciNet  MATH  Google Scholar 

  38. Newman JN (1965) Propagation of waves past long two-dimensional obstacles. J Fluid Mech 23:23–29

    Article  MATH  Google Scholar 

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Acknowledgements

The authors are thankful to the reviewers for their comments and suggestions to revise the paper into its present form. The authors are also thankful to Prof Ranadev Datta of Indian Institute of Technology Kharagpur for his comments, suggestions and stimulating discussions.

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  1. Department of Mathematics, Indian Institute of Technology Kharagpur, Kharagpur, 721302, India

    Sourav Gupta & R. Gayen

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  1. Sourav Gupta
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  2. R. Gayen
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Correspondence to R. Gayen.

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Gupta, S., Gayen, R. Scattering of oblique water waves by two thin unequal barriers with non-uniform permeability. J Eng Math 112, 37–61 (2018). https://doi.org/10.1007/s10665-018-9964-8

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