### The effect of free-surface tension on scattering of water waves by small bottom undulation

#### Abstract

We examine the influence of surface tension at the free-surface on the problem of scattering of normal incident waves by small bottom undulation in a two-layer fluid within the framework of linearised wave theory. In such a two-layer fluid, there exist two different modes of time-harmonic waves for the given frequency, one of which corresponds to a free-surface disturbance and the other to an interfacial wave motion. Using perturbation technique, the problem is reduced up to first-order to a coupled boundary value problem which is solved by a method based on Green's integral theorem with the introduction of appropriate Green's functions. The first-order approximations to the reflection and transmission coefficients evaluated in terms of computable integrals for both the modes. For a sinusoidal bottom topography, these coefficients are determined and the numerical results depicted graphically. When the wave is incident due to the free-surface disturbance or the interfacial wave motion in a two-layer fluid, we always find energy transfer from the incident wave mode to the other wave mode. We conclude that realistic changes in surface tension on the free-surface have a significant effect on the problem of water wave scattering by bottom undulation in a two-layer fluid.

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#### Keywords

Two-layer fluid; surface tension; bottom undulation; Green's function; perturbation technique; reflection coefficient; transmission coefficient.

DOI: http://dx.doi.org/10.21914/anziamj.v58i0.11232

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