Green's Law and the Riemann Problem in Layered Media
George, Jithin Donny
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The propagation of long waves onto a continental shelf is of great interest in tsunami modeling, where understanding the amplification of waves during shoaling is of significant importance. 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 that can be obtained from the solution of a Riemann problem. 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 reflection than given by the reflection coefficient of a sharp interface. Exploring the relation between these results offers a perspective that allows us to view the solution to the shallow water equations as combinations of infinitely many transmitted and reflected waves. The thesis focuses on an asymptotic approximation to this solution, general enough for continuous non-homogeneous media, and presents some interesting results and scenarios along the way. The same phenomena and similar results exist for other physical settings described by wave equations, including those of linear acoustic waves and electromagnetic waves.
- Applied mathematics