Reflection and transmission coefficients for water wave scattering by a sea‐bed with small undulation

Abstract

AbstractThe scattering of a train of progressive waves by a small undulation of the bottom of a laterally unbounded sea is considered with 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 also to the reflection and transmission coefficients in terms of integrals involving the shape of the function $c(x)$ representing the bottom undulation. Two particular forms of the shape function are considered and the integrals for reflection and transmission coefficients are evaluated for these shape functions. Out of these two cases, we focus mainly on a patch of sinusoidal ripples at the bottom and find that the reflection coefficient up to the first order is an oscillatory function of β which is twice the ratio of the wave numbers of the wave and the ripple bed. When this ratio becomes 0.5, i.e., when β = 1 (the situation 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, two, and six ripple(s) in the patch. The results find absolute agreement with available results. We also show the graphical comparison of the reflection coefficient against the wave number for one, three, and five ripple(s).