Abstract:
The problem involving scattering of oblique waves by small undulation on the porous ocean bed in
a two-layer fluid is investigated within the framework of linearised theory of water waves where the
upper layer is free to the atmosphere. In such a two-layer fluid, there exist waves with two different
wave numbers (modes): wave with lower wave number propagates along the free surface whilst that
with higher wave number propagates along the interface. When an oblique incident wave of a particular
mode encounters the undulating bottom, it gets reflected and transmitted into waves of both modes
so that some of the wave energy transferred from one mode to another mode. Perturbation analysis in
conjunction with Fourier transform technique is used to derive the first-order corrections of velocity
potentials, reflection and transmission coefficients at both modes due to oblique incident waves of both
modes. One special type of undulating bottom topography is considered as an example to evaluate
the related coefficients in detail. These coefficients are shown in graphical forms to demonstrate the
transformation of water wave energy between the two modes. Comparisons between the present results
with those in the literature are made for particular cases and the agreements are found to be satisfactory.
In addition, energy identity, an important relation in the study of water wave theory, is derived with
the help of the Green’s integral theorem.
