Shoaling on Steep Continental Slopes: Relating Transmission and Reflection Coefficients to Green’s Law

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.

A Path-Integral Method for Solution of the Wave Equation with Continuously Varying Coefficients

“…showing that the limiting functions w 1 (x, t), w 2 (x, t) satisfy (8) and therefore give the solution to the wave equation (2). It remains to show that w 1 , w 2 exist.…”

“…In general the pressure is given by p = w 1 + w 2 ; for the case of a pure right-going pulse (13), for which w 1 is zero, we have p(x, 0) = w 2 (x, 0) = p 0 . According to (8), along the path X(t) the value of w 2 (and hence the value of p) satisfies the ODE p (X(t)) = Z (X(t)) 2Z(X(t)) p(X(t))…”

“…Remark 1. We use C G for the quantity defined in (17) since this is the amplification factor given by Green's law in the context of shoaling, as we discuss in more detail in [8]. The linearized shallow water equations used there can be put in the form (2) by introducing p(x, t) as the depth perturbation of a small amplitude long wave on a background water depth h(x), and µ(x, t) as the momentum perturbation.…”

“…where g is the gravitational constant. This differs from the conservative form used in [8], and has the same form as (2) if we set K(x) ≡ 1 and ρ(x) = 1/(gh(x)). Then the wave speed is c(x) = gh(x), the impedance is Z(x) = 1/ gh(x), and C G = (h − /h + ) 1/4 .…”

“…In this work we propose and demonstrate a method for approximately solving the general Cauchy problem for (2) in the presence of arbitrary variation in ρ and K, by grouping characteristic paths according to the number of reflections. Our interest originated in a study of the shoaling of water waves over a continental shelf and our complementary works [7,8] contain more discussion of this application and several illustrative examples using the linearized shallow water equations, a special case that is also discussed in Remark 1 below. Code to reproduce the numerical experiments in this paper is available online.…”