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Resonant scattering of edge waves by longshore periodic topography: Finite beach slope. (English) Zbl 0958.76011

Summary: 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 [the authors, ibid. 369, 91-123 (1998)], where coupled evolution equations describing the variations of edge wave amplitudes over a finite-size patch of undulating bathymetry were developed. Here, we derive similar evolution equations using the full linear equations and removing the shallow water restriction of small \((2N+1)\theta\), where \(N\) is the maximum mode number and \(\theta\) is the unperturbed planar beach slope angle. The present results confirm the shallow water solutions for vanishingly small \((2N+1)\theta\) and allow simple corrections to the shallow water results for small but finite \((2N+1)\theta\). Additionally, we identify the multi-wave scattering cases occurring only when \((2N+1)\theta=O(1)\), and give detailed descriptions for the case involving modes 0, 1, and 2 that occurs only on a steep beach with \(\theta=\pi/12\).

MSC:

76B15 Water waves, gravity waves; dispersion and scattering, nonlinear interaction
86A05 Hydrology, hydrography, oceanography
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