Mathematics and Computing 2013 pp 1-11 | Cite as

# Propagation of Water Waves in the Presence of Thin Vertical Barrier on the Bottom Undulation

## Abstract

The problem involving diffraction of water waves by submerged thin vertical barrier over irregular bottom is examined using linearized theory of water waves. While formulating the problem mathematically, a mixed boundary value problem (BVP) occurs. The problem is solved using perturbation theory along with least-squares method and Green’s integral theorem. The first order reflection and transmission coefficients are obtained in terms of integrals involving the shape function \(c(x)\) representing the bottom undulation and the solution of the scattering problem by the submerged barrier. A special case of bottom undulation is considered to evaluate the first order reflection and transmission coefficients in detail. The numerical results of these coefficients are shown graphically.

## Keywords

Water wave scattering Bottom undulation Vertical barrier Perturbation analysis Least-squares method Green’s integral theorem Reflection and transmission coefficients## References

- 1.Davies, A.G.: The reflection of wave energy by undulations on the seabed. Dyn. Atmos. Oceans
**6**, 207–232 (1982)CrossRefGoogle Scholar - 2.Davies, A.G., Heathershaw, A.D.: Surface wave propagation over sinusoidally varying topography. J. Fluid Mech.
**144**, 419–443 (1984)Google Scholar - 3.Dean, W.R.: On the reflection of surface waves by a submerged plane barrier. Math. Proc. Camb. Phil. Soc.
**41**, 231–238 (1945)CrossRefzbMATHMathSciNetGoogle Scholar - 4.Losada, I.J., Losada, M.A., Roldan, A.J.: Propagation of oblique incident waves past rigid vertical thin barriers. Appl. Ocean Res.
**14**, 191–199 (1992)CrossRefGoogle Scholar - 5.Mandal, B.N., Basu, U.: A note on oblique water-wave diffraction by a cylindrical deformation of the bottom in the presence of surface tension. Archive of Mech.
**42**, 723–727 (1990)zbMATHGoogle Scholar - 6.Mandal, B.N., Basu, U.: Wave diffraction by a small elevation of the bottom of an ocean with an ice-cover. Archive of Appl. Mech.
**73**, 812–822 (2004)CrossRefzbMATHGoogle Scholar - 7.Mandal, B.N., Dolai, D.P.: Oblique water wave diffraction by thin vertical barriers in water of uniform finite depth. Appl. Ocean Res.
**16**, 195–203 (1994)CrossRefGoogle Scholar - 8.Mandal, B.N., Gayen, R.: Water wave scattering by bottom undulations in the presence of a thin partially immersed barrier. Appl. Ocean Res.
**28**, 113–119 (2006)CrossRefGoogle Scholar - 9.Martha, S.C., Bora, S.N.: Oblique surface wave propagation over a small undulation on the bottom of an ocean. Geophys. Astrophys. Fluid Dyn.
**101**, 65–80 (2007)CrossRefMathSciNetGoogle Scholar - 10.Martha, S.C., Bora, S.N., Chakrabarti, A.: Interaction of surface water waves with small bottom undulation in sea-bed. J. Appl. Math. Inf.
**27**, 1017–1031 (2009)Google Scholar - 11.Miles, J.W.: Oblique surface-wave diffraction by a cylindrical obstacle. Dyn. Atmos. Oceans
**6**, 121–123 (1981)CrossRefGoogle Scholar - 12.Porter, D.: The transmission of surface waves through a gap in a vertical berrier. Math. Proc. Cambridge Philos. Soc.
**71**, 411–422 (1972)CrossRefzbMATHGoogle Scholar - 13.Porter, R., Evans, D.V.: Complementary approximations to wave scattering by vertical barriers. J. Fluid Mech.
**294**, 155–180 (1995)CrossRefzbMATHMathSciNetGoogle Scholar - 14.Ursell, F.: The effect of a fixed barrier on surface waves in deep water. Math. Proc. Cambridge Philos. Soc.
**43**, 374–382 (1947)CrossRefzbMATHMathSciNetGoogle Scholar