Resonant scattering of edge waves by longshore periodic topography: finitebeach slope

The resonant scattering of low-mode progressive edge waves by small-amplitu de longshore periodic depth perturbations superposed on a plane beach has r ecently been investigated using the shallow water equations (Chen & Guza 19 98). Coupled evolution equations describing the variations of edge wave amp litudes over a finite-size patch of undulating bathymetry were developed. H ere similar evolution equations are derived using the full linear equations , removing the shallow water restriction of small (2N + 1)theta, where N is the maximum mode number considered and theta is the unperturbed planar bea ch slope angle. The present results confirm the shallow water solutions for vanishingly small (2N + 1)theta and allow simple corrections to the shallo w water results for small but finite (2N + 1)theta. Additionally, multi-wav e scattering cases occurring only when (2N + 1)theta = 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 theta = pi/12.