Exponential asymptotics in wave propagation problems

Abstract

We use the methods of exponential asymptotics to study the solutions of a one dimensional wave equation with a non-constant wave speed c(x,t) modelling, for example, a slowly varying spatio-temporal topography. The equation reads htt(x,t) = (c2(x,t)hx(x,t))x' (1) where the subscripts denote differentiation w.r.t. the parameters x and t respectively. We focus on the exponentially small reflected wave that appears as a result of a Stokes phenomenon associated with the complex singularities of the speed. This part of the solution is not captured by the standard WKBJ (geometric optics) approach. We first revisit the time-independent propagation problem using resurgent analysis. Our results recover those obtained using Meyers integral-equation approach or the Kruskal-Segur (K-S) method. We then consider the time-dependent propagation of a wavepacket, assuming increasingly general models for the wave speed: time independent, c(x), and separable, c1(x)c2(t). We also discuss the situation when the wave speed is an arbitrary function, c(x,t), with the caveat that the analysis of this setup has yet to be completed. We propose several methods for the computation of the reflected wavepacket. An integral transform method, using the Dunford integral, provides the solution in the time independent case. A second method exploits resurgence: we calculate the Stokes multiplier by inspecting the late terms of the dominant asymptotic expansion. In addition, we explore the benefits of an integral transform that relates the coefficients of the dominant solution in the time-dependent problem to the coefficients of the dominant solution in the time-independent problem. A third method is a partial differential equation extension of the K-S complex matching approach, containing details of resurgent analysis. We confirm our asymptotic predictions against results obtained from numerical integration.