, Volume 54, Issue 1–2, pp 71–84 | Cite as

Water wave scattering by three thin vertical barriers with middle one partially immersed and outer two submerged

  • Ranita Roy
  • B. N. MandalEmail author


The problem of oblique water wave scattering by three thin vertical barriers in deep water is investigated here assuming linear theory. The geometrical configurations of the three barriers are such the inner(middle) barrier is partially immersed and the two outer barriers are completely submerged and extend infinitely downwards. A system of three simultaneous integral equations of first kind involving differences of velocity potentials across the barriers has been obtained in the mathematical analysis by employing Havelock’s expansion of water wave potential alongwith Havelock’s inversion formulae. Approximate numerical solutions of these integral equations are obtained by using single-term Galerkin approximations where the single term is chosen to be the exact solution of the integral equation obtained for a single vertical barrier in deep water for normal incidence of surface waves. Fairely accurate numerical estimates for the reflection and transmission coefficients are obtained by solving this system of linear equations. These estimates satisfy the energy identity, thereby justifying the validity of the present method. Due to energy identity, the behaviour of reflection coefficient is discussed by depicting it graphically against wavenumber. Existence of zeros of reflection coefficient is observed only when the two outer barriers are identical i.e. they are submerged from the same depth below the mean free surface. Some earlier results for two completely submerged barriers and a single barrier are recovered as special cases for a particular set of values of the parameters involved in the problem. This provides another check on the correctness of the present method.


Water wave scattering Three thin vertical barriers Havelock’s expansion of water wave potential Integral equations Single-term Galerkin approximation Reflection coefficient 

Mathematical Subject Classification




The authors thank the Reviewer for his comments and suggestions to revise the paper in the present form. This work is partially supported by CSIR through a research project 25(0253)/16/EMR-II.

Compliance with ethical standards

Conflict of interest

The authors declare that there is no conflict of interests regarding the publication of this paper.


  1. 1.
    Dean WR (1945) On the reflection of surface waves by submerged plane barriers. Proc Camb Phil Soc 41:231–238ADSCrossRefzbMATHGoogle Scholar
  2. 2.
    Ursell F (1947) The effect of a fixed vertical barrier on surface waves in deep water. Proc Camb Phil Soc 43:374–382ADSMathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Levine H, Rodemich E (1958) Scattering of surface waves on an ideal fluid. Technical Report No. 78, Standford University, USA, Mathematics and Statistics LaboratoryGoogle Scholar
  4. 4.
    Evans DV (1970) Diffraction of water waves by a submerged vertical plate. J Fluid Mech 40:433–451ADSCrossRefzbMATHGoogle Scholar
  5. 5.
    Porter D (1972) The transmission if surface waves through a gap in a vertical barrier. Proc Camb Phil Soc 71:411–421ADSCrossRefzbMATHGoogle Scholar
  6. 6.
    Williams WE (1988) Note on the scattering of water waves by a vertical barrier. Proc Camb Phil Soc 62:507–509ADSMathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Evans DV, Morris CAN (1972) The effect of a fixed vertical barrier on obliquely incident surface waves in deep water. J Inst Math Appl 9:198–204CrossRefzbMATHGoogle Scholar
  8. 8.
    Porter R, Evans DV (1995) Complementary approximations to solve wave scattering by vertical barriers. J Fluid Mech 294:155–180ADSMathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Mandal BN, Das P (1996) Oblique diffraction of surface waves by submerged vertical plate. J Eng Math 30:459–470MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Das P, Banerjea S, Mandal BN (1996) Scattering of oblique waves by a thin vertical wall with a submerged gap. Arch Mech 48:959–972MathSciNetzbMATHGoogle Scholar
  11. 11.
    Smith CM (1983) Some problems in linear water wave theory, Ph.D. Thesis, University of Bristol, BristolGoogle Scholar
  12. 12.
    Losada IJ, Losada MA, Roldan AJ (1992) Propagation of oblique incident waves past rigid vertical thin barriers. Appl Ocean Res 14:191–199CrossRefGoogle Scholar
  13. 13.
    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
  14. 14.
    Banerjea S, Kanoria M, Dolai DP, Mandal BN (1996) Oblique wave scattering by a submerged thin wall with gap in finite depth water. Appl Ocean Res 18:319–327CrossRefGoogle Scholar
  15. 15.
    Chakrabarti A, Banerjea S, Mandal BN, Sahoo T (1995) Solution of a class of mixed boundary value problem for Laplace’s equation arising in water wave scattering. J Indian Inst Sci 75:577–588MathSciNetzbMATHGoogle Scholar
  16. 16.
    Chakrabarti A, Banerjea S, Mandal BN, Sahoo T (1997) A Unified approach to problems of scattering of surface water waves by vertical barriers. J Austral Math Soc Ser B 39:93–103MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Jarvis RJ (1971) The scattering of surface waves by two vertical plane barriers. J Inst Math Appl 7:207–215ADSzbMATHGoogle Scholar
  18. 18.
    Evans DV, Morris CAN (1972) Complementary approximations to the solution of a problem in water waves. J Inst Math Appl 10:1–9CrossRefzbMATHGoogle Scholar
  19. 19.
    Morris CAN (1975) A variational approach to an unsymmetric water wave scattering problem. J Eng Math 9:291–300CrossRefzbMATHGoogle Scholar
  20. 20.
    McIver P (1985) Scattering of water waves by two surface piercing vertical barriers. IMA J Appl Math 35:339–355MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Evans DV, Porter R (1997) Complementary methods for scattering by thin barriers. In: Mandal BN (ed) Math. Tech. for water waves Comp. Chapter 1, Mech. Pub. Southampton and Boston, 1–43Google Scholar
  22. 22.
    Kanoria M, Mandal BN (1996) Oblique wave diffraction by two parallel vertical barriers with submerged gaps in water of uniform finite depth. J Tech Phys 37:187–204Google Scholar
  23. 23.
    Das P, Dolai DP, Mandal BN (1997) Oblique water wave scattering by two thin barriers with gaps. J Waterw Port Coast Ocean Eng ASCE 123:163–171CrossRefGoogle Scholar
  24. 24.
    Ranita R, Basu U, Mandal BN (2016) Oblique water wave scattering by two unequal vertical barriers. J Eng Math 97:119–133MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Roy R, Basu U, Mandal BN (2016) Water wave scattering by two submerged thin vertical unequal plates. Arch Appl Mech 86:1681–1692CrossRefGoogle Scholar
  26. 26.
    Roy R, Uma Basu BN, Mandal BN (2017) Water wave scattering by a pair of thin vertical barriers with submerged gaps. J Eng Math 105:85–97MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Kuznetsov N, McIver P, Linton CM (2001) On uniqueness and trapped modes in the water-wave problem for vertical barriers. Wave Mot 33:283–307MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Roy R, De Soumen B, Mandal N (2018) Water wave scattering by two surface-piercing and one submerged thin vertical barriers. Arch Appl Mech 88:1477–1489CrossRefGoogle Scholar
  29. 29.
    Jones DS (1986) Diffraction by three semi-infinite planes. Proc R Lond A 404:299–321ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Alkumru A (1998) Plane wave diffraction by three parallel thick impedance half-planes. J Electromag Waves Appl 12:801–819CrossRefzbMATHGoogle Scholar
  31. 31.
    Buyukaksoy A, Cinar G (2004) Diffraction by a set of three parallel impedance half-planes with the one amidst located in the opposite direction. In: Progress in electromagnetic research symposium, Pisa, Italy, pp 929–932Google Scholar
  32. 32.
    Mandal BN, Chakraborti A (2000) Water wave scattering by barriers. WIT Press, AshurstzbMATHGoogle Scholar

Copyright information

© Springer Nature B.V. 2018

Authors and Affiliations

  1. 1.Department of MathematicsSerampore CollegeSeramporeIndia
  2. 2.Physics and Applied Mathematics UnitIndian Statistical InstituteKolkataIndia

Personalised recommendations