Interaction of water waves with small undulations on a porous bed in a two-layer ice-covered fluid

Abstract

The scattering problem involving water waves by small undulation on the porous ocean-bed in a two-layer fluid, is investigated within the framework of the two-dimensional linear water wave theory where the upper layer is covered by a thin uniform sheet of ice modeled as a thin elastic plate. In such a two-layer fluid there exist waves with two different modes, one with a lower wave number propagate along the ice-cover whilst those with a higher wave number propagate along the interface. An incident wave of a particular wave number gets reflected and transmitted over the bottom undulation into waves of both modes. Perturbation analysis in conjunction with the Fourier transform technique is used to derive the first-order corrections of reflection and transmission coefficients for both the modes due to incident waves of two different modes. One special type of bottom topography is considered as an example to evaluate the related coefficients in detail. These coefficients are depicted in graphical forms to demonstrate the transformation of wave energy between the two modes and also to illustrate the effects of the ice sheet and the porosity of the undulating bed.