Transport in a stochastic Goupillaud medium

TL;DR

This paper models wave transport in a one-dimensional stochastic Goupillaud medium, showing how solutions converge to Levy process-based limits, providing a foundation for more complex random media models.

Contribution

It introduces explicit stochastic models of irregular transport speeds in Goupillaud media, linking layered media solutions to Levy processes and their limits.

Findings

Solutions for layered media converge to Levy process limits

Fourier integral operator representations also converge

Provides a basis for modeling wave propagation in complex random media

Abstract

This paper is part of a project that aims at modelling wave propagation in random media by means of Fourier integral operators. A partial aspect is addressed here, namely explicit models of stochastic, highly irregular transport speeds in one-dimensional transport, which will form the basis for more complex models. Starting from the concept of a Goupillaud medium (a layered medium in which the layer thickness is proportional to the propagation speed), a class of stochastic assumptions and limiting procedures leads to characteristic curves that are L\'evy processes. Solutions corresponding to discretely layered media are shown to converge to limits as the time step goes to zero (almost surely pointwise almost everywhere). This translates into limits in the Fourier integral operator representations.