Journal of applied mathematics & informatics

The Korean Society for Computational and Applied Mathematics (한국전산응용수학회)
권호 표지

INTERACTION OF SURFACE WATER WAVES WITH SMALL BOTTOM UNDULATION ON A SEA-BED

  • Martha, S.C. (Department of Mathematics, India Institute of Science) ;
  • Bora, S.N. (Department of Mathematics, Indian Institute of Technology) ;
  • Chakrabarti, A. (Department of Mathematics, Indian Institute of Science)
  • Published : 2009.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

The problem of interaction of surface water waves by small undulation at the bottom of a laterally unbounded sea is treated on the basis of linear water wave theory for both normal and oblique incidences. Perturbation analysis is employed to obtain the first order corrections to the reflection and transmission coefficients in terms of integrals involving the shape function c(x) representing the bottom undulation. Fourier transform method and residue theorem are applied to obtain these coefficients. As an example, a patch of sinusoidal ripples is considered in both the cases as the shape function. The principal conclusion is that the reflection coefficient is oscillatory in the ratio of twice the surface wave number to the wave number of the ripples. In particular, there is a Bragg resonance between the surface waves and the ripples, which is associated with high reflection of incident wave energy. The theoretical observations are validated computationally.

Keywords

References

  1. J.C.W. Berkhoff, Computation of combined refraction-diffraction, Proceedings 13th Conference on Coastal Engineering, July 1972, Vancouver, Canada, ASCE. 2(1973), 471-490.
  2. P.G. Chamberlain and D. Porter, The modified mild-slope equations, J. Fluid Mech. 291(1995), 393-407. https://doi.org/10.1017/S0022112095002758
  3. A.G. Davies, On the interaction between surface waves and undulations on the sea bed, J. Marine Res. 40(1982), 331-368.
  4. A.G. Davies, The reflection of wave energy by undulations of the sea bed, Dynamics of Atmosphers and Oceans 6 (1982), 207-232. https://doi.org/10.1016/0377-0265(82)90029-X
  5. A.G. Davies and A.D. Heathershaw, Surface-wave propagation over sinusoidally varying topography, J. Fluid Mech. 144(1984), 419-443. https://doi.org/10.1017/S0022112084001671
  6. T. Hara and C.C. Mei, Bragg scattering of surface waves by periodic bars: theory and experiment, J. Fluid Mech. 178 (1987), 221-241. https://doi.org/10.1017/S0022112087001198
  7. A.D. Heathershaw, Seabed-wave resonance and sand bar growth, Nature 296(1982), 343- 345. https://doi.org/10.1038/296343a0
  8. J.T. Kirby, A general wave equation for waves over rippled beds, J. Fluid Mech. 162(1986), 171-186. https://doi.org/10.1017/S0022112086001994
  9. B.N. Mandal and U. Basu, A note on oblique water-wave diffraction by a cylindrical deformation of the bottom in the presence of surface tension, Archive of Mech. 42(1990), 723-727.
  10. S.C. Martha and S.N. Bora, Water wave diffraction by a small deformation of the ocean bottom for oblique incidence, Acta Mech. 185(2006), 165-177. https://doi.org/10.1007/s00707-006-0358-z
  11. S.C. Martha and S.N. Bora, Oblique surface wave propagation over a small undulation on the bottom of an ocean, Geophy. Astrophy. Fluid Dynamics, 101 (2007a), 65-80. https://doi.org/10.1080/03091920701208186
  12. S.C. Martha and S.N. Bora, Refelction and transmission coefficients for water wave scattering by a sea-bed with small undulation, Z. Angew. Math. Mech. (ZAMM). 87(2007b), 314-321. https://doi.org/10.1002/zamm.200610317
  13. C.C. Mei, Resonant reflection of surface water waves by periodic sand-bars, J. Fluid Mech. 152(1985), 315-335. https://doi.org/10.1017/S0022112085000714
  14. J.M. Miles, Oblique surface wave diffraction by a cylindrical obstacle, J. Atmos. and Oceans 6(1981), 121-123. https://doi.org/10.1016/0377-0265(81)90019-1
  15. R. Smith and T. Sprinks, Scattering of surface waves by a conical island, J. Fluid Mech. 72(1975), 373-384. https://doi.org/10.1017/S0022112075003424
  16. A.S. Warke, S.K. Das and L. Debnath, Propagation of surface waves on irregular bed topography, J. Appl. Math. and Computing 20(2006), 197-208. https://doi.org/10.1007/BF02831933