Water wave scattering by bottom undulations in the presence of a thin partially immersed barrier

The problem of water wave scattering by bottom undulations in the presence of a partially immersed thin vertical barrier is investigated here by employing a simplified perturbation analysis. The first-order reflection and transmission coefficients are obtained in terms of integrals involving the shape function describing the bottom undulations and the solution of the scattering problem involving the barrier present in uniform finite depth water. For sinusoidal undulations of bottom symmetric about the plane of the barrier, the first-order transmission coefficient vanishes identically. The corresponding first-order reflection coefficient is computed numerically and the numerical results are depicted graphically against the wave number. Resonant interaction of the incident wave with the bottom undulations is seen to occur for a particular value of the ratio of the surface wavelength to sinusoidal wavelength. As the depth of the lower edge of the barrier tends to zero, the curve for this reflection coefficient almost coincides with the same for the reflection coefficient for the scattering problem involving sinusoidal bottom undulations only.