Journal of Engineering Mathematics

, Volume 65, Issue 1, pp 75–87 | Cite as

Water-wave scattering by two submerged plane vertical barriers—Abel integral-equation approach

Article

Abstract

The classical problem of surface water-wave scattering by two identical thin vertical barriers submerged in deep water and extending infinitely downwards from the same depth below the mean free surface, is reinvestigated here by an approach leading to the problem of solving a system of Abel integral equations. The reflection and transmission coefficients are obtained in terms of computable integrals. Known results for a single barrier are recovered as a limiting case as the separation distance between the two barriers tends to zero. The coefficients are depicted graphically in a number of figures which are identical with the corresponding figures given by Jarvis (J Inst Math Appl 7:207–215, 1971) who employed a completely different approach involving a Schwarz–Christoffel transformation of complex-variable theory to solve the problem.

Keywords

Abel integral equations Reflection and transmission coefficients Two barriers Wave scattering 

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References

  1. 1.
    Ursell F (1947) The effect of a fixed vertical barrier on surface waves in deep water. Proc Camb Phil Soc 43: 374–382MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Evans DV (1970) Diffraction of water waves by a submerged vertical plate. J Fluid Mech 40: 433–451MATHCrossRefADSGoogle Scholar
  3. 3.
    Porter D (1972) The transmission of surface waves through a gap in a vertical barrier. Proc Camb Phil Soc 71: 411–421MATHCrossRefGoogle Scholar
  4. 4.
    Kuznetsov N, McIver P, Linton CM (2001) On uniqueness and trapped modes in the water wave problem for vertical barriers. Wave Motion 33: 283–307MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Williams WE (1966) Note on the scattering of water waves by a vertical barrier. Proc Camb Phil Soc 62: 507–509MATHCrossRefGoogle Scholar
  6. 6.
    Levine H, Rodemich F (1958) Scattering of surface waves on an ideal fluid. Stanford University Technical Report Number 78, Math Stat LabGoogle Scholar
  7. 7.
    Jarvis RJ (1971) The scattering of surface waves by two vertical plane barriers. J Inst Maths Appl 7: 207–215MATHADSGoogle Scholar
  8. 8.
    Evans DV, Morris CAN (1972) Complementary approximations to the solution of a problem in water waves. J Inst Maths Appl 10: 1–9MATHCrossRefGoogle Scholar
  9. 9.
    Newman JN (1974) Interaction of water waves with two closely spaced vertical obstacles. J Fluid Mech 66: 97–106MATHCrossRefADSGoogle Scholar
  10. 10.
    McIver P (1985) Scattering of water waves by two surface-piercing vertical barriers. IMA J Appl Math 35: 339–355MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Mandal BN, Dolai DP (1994) Oblique water wave diffraction by thin vertical barriers in water of uniform finite depth. Appl Ocean Res 16: 195–203CrossRefGoogle Scholar
  12. 12.
    Das P, Dolai DP, Mandal BN (1997) Oblique water wave diffraction by two parallel thin barriers with gaps. J Waterw Port Coast Ocean Eng 123: 163–171CrossRefGoogle Scholar
  13. 13.
    Porter R, Evans DV (1995) Complementary approximations to waves scattering by vertical barriers. J Fluid Mech 294: 155–180MATHCrossRefADSMathSciNetGoogle Scholar
  14. 14.
    Dean WR (1945) On the reflection of surface waves by submerged plane barrier. Proc Camb Phil Soc 4: 231–238CrossRefMathSciNetGoogle Scholar
  15. 15.
    Faulkner TR (1966) The diffraction of any obliquely incident surface wave by a vertical barrier of finite depth. Proc Camb Phil Soc 62: 829–838MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2009

Authors and Affiliations

  1. 1.Physics and Applied Mathematics UnitIndian Statistical InstituteKolkataIndia
  2. 2.Department of MathematicsIndian Institute of ScienceBangaloreIndia

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