Abstract. The propagation of long waves onto a continental shelf is of great interest in tsunami modeling and other applications where understanding the amplification of waves during shoaling is important. When the linearized shallow water equations are solved with the continental shelf modeled as a sharp discontinuity, the ratio of the amplitudes is given by the transmission coefficient. On the other hand, when the slope is very broad relative to the wavelength of the incoming wave, then amplification is governed by Green's Law, which predicts a larger amplification than the transmission coefficient, and a much smaller amplitude reflection than given by the reflection coefficient of a sharp interface. We explore the relation between these results and elucidate the behavior in the intermediate case of a very steep continental shelf.
Preprint: https://arxiv.org/abs/1901.04148
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This appeared in Volume II of a special issue of Pure and Applied Geophysics on "Twenty Five Years of Modern Tsunami Science Following the 1992 Nicaragua and Flores Island Tsunamis". See also Volume I of the journal (or the resulting book) for other related papers.
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See also: The related paper A path integral method for solution of the wave equation with continuously-varying coefficients
bibtex entry:
@article{GeorgeKetchesonEtAl2020,
title = {Shoaling on Steep Continental Slopes: Relating Transmission and
Reflection Coefficients to {Green's Law}},
author = {George, Jithin and Ketcheson, David I. and LeVeque, Randall J.},
year = {2020},
volume = {177},
pages = {1659--1674},
doi = {10.1007/s00024-019-02316-y},
journal = {Pure and Applied Geophysics},
}