Wave scattering by undulating bed topography in a two-layer ocean

Abstract

The problem of wave scattering by undulating bed topography in a two-layer ocean is investigated on the basis of linear theory. In a two-layer fluid with the upper layer having a free surface, there exist two modes of waves propagating at both the free surface of the upper layer and the interface between the two layers. Due to a wave train of a particular mode incident on an obstacle which is bottom-standing on the lower layer, reflected and transmitted waves of both modes are created by the obstacle. For small undulations on the bottom of the lower layer, a perturbation method is employed to obtain first-order reflection and transmission coefficients of both modes for incident wave trains of again both modes in terms of integrals involving the bed-shape function. For sinusoidal undulations, numerical results are presented graphically to illustrate the energy transfer between the waves of different modes by the undulating bed.

Keywords

two-layer ocean / wave scattering / bottom undulations / reflection and transmission coefficients

References

[1] Miles J. W. Oblique surface-wave diffraction by a cylindrical obstacle[J]. Dynamics of Atmospheres and Oceans, 1981, 6(2): 121-123

[2] Davies A. G. The reflection of wave energy by undulations on the seabed[J]. Dynamics of Atmospheres and Oceans, 1982, 6(4): 207-232

[3] 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[J]. Arch. Mech., 1990, 42: 723-727

[4] Lamb H. Hydrodynamics[M]. 1995, Cambridge: Cambridge University Press

[5] Linton C. M., Mciver M. The interaction of waves with horizontal cylinders in two-layer fluids[J]. Journal of Fluid Mechanics, 1995, 304: 213-229

[6] Cadby J. R., Linton C. M. Three-dimensional water-wave scattering in two-layer fluids[J]. Journal of Fluid Mechanics, 2000, 423: 155-173

[7] Sherief H. H., Faltas M. S., Saad E. I. Forced gravity waves in two-layered fluids with the upper fluid having a free surface[J]. Canadian Journal of Physics, 2003, 81(4): 675-689

[8] Sherief H. H., Faltas M. S., Saad E. I. Axisymmetric gravity waves in two-layered fluids with the upper fluid having a free surface[J]. Wave Motion, 2004, 40: 143-161

[9] Shaw D. C. Perturbational results for diffraction of water waves by nearly vertical barriers[J]. IMA Journal of Applied Mathematics, 1985, 34(1): 99-117

[10] Mandal B. N., Chakrabarti A. A note on diffraction of water waves by a nearly vertical barrier[J]. IMA Journal of Applied Mathematics, 1989, 43(2): 157-165

[11] Mandal B. N., Kundu P. K. Scattering of water waves by a submerged nearly vertical plate[J]. SIAM Journal on Applied Mathematics, 1990, 50(5): 1221-1231

[12] Rhodes-Robinson P. F. On wave motion in a two-layered liquid of infinite depth in the presence of surface and interfacial tension[J]. The Journal of the Australian Mathematical Society, 1994, 35: 302-322

[13] Ten I., Kashiwagi M. Hydrodynamics of a body floating in a two-layer fluid of finite depth. Part I. Radiation problem[J]. Journal of Marine Science and Technology, 2004, 9: 127-141