Scattering of surface waves over an uneven sea-bed

This work is concerned with the scattering of a train of progressive waves by a small undulation of the bottom of a laterally unbounded sea, using two-dimensional linear water wave theory. Assuming irrotational motion, a simplified perturbation analysis is employed to obtain the first-order corrections to the velocity potential by using the Green’s integral theorem in a suitable manner, and the reflection and transmission coefficients in terms of integrals involving the shape of the function c(x)c(x) representing the bottom undulation. Particular forms of the shape function are considered and the integrals for reflection and transmission coefficients are evaluated for these shape functions. For a varying sinusoidal sea bottom when Bragg resonance occurs, the reflection coefficient becomes a multiple of the number of ripples, and high reflection of the incident wave energy occurs if this number is large. We present the result in graphical form for one, three and five ripples in the patch. The results find absolute agreement with available results.