Resonant scattering of edge waves by longshore periodic topography: finite beach slope

Abstract

The resonant scattering of low-mode progressive edge waves by small-amplitude longshore periodic depth perturbations superposed on a plane beach has recently been investigated using the shallow water equations (Chen & Guza 1998). Coupled evolution equations describing the variations of edge wave amplitudes over a finite-size patch of undulating bathymetry were developed. Here similar evolution equations are derived using the full linear equations, removing the shallow water restriction of small (2N + 1)θ, where N is the maximum mode number considered and θ is the unperturbed planar beach slope angle. The present results confirm the shallow water solutions for vanishingly small (2N + 1)θ and allow simple corrections to the shallow water results for small but finite (2N + 1)θ. Additionally, multi-wave scattering cases occurring only when (2N + 1)θ = O(1) are identified, and detailed descriptions are given for the case involving modes 0, 1, and 2 that occurs only on a steep beach with θ = π/12.