Transport in Stochastic Goupillaud Media

TL;DR

This paper studies wave transport in a random layered medium, showing that as layers become infinitesimally thin, the characteristic curves converge to Levy processes with explicitly computed distributions, especially for inverse Gaussian cases.

Contribution

It introduces a stochastic model for wave transport in Goupillaud media and derives explicit distributions for the limiting Levy processes.

Findings

Characteristic curves converge to Levy processes as layer thickness approaches zero.

Explicit probability distributions are derived for the limiting processes.

Special case analysis for inverse Gaussian Levy processes.

Abstract

The paper addresses one-dimensional transport in a Goupillaud medium (a layered medium in which the layer thickness is proportional to the propagation speed), as a prototypical case of wave propagation in random media. Suitable stochastic assumptions and limiting procedures lead to characteristic curves that are L\'evy processes. Solutions corresponding to the discretely layered medium are shown to converge to limits as the thickness of the layers goes to zero. The probability distribution of the limiting characteristic curves is explicitly computed and exemplified when the underlying L\'evy process is an inverse Gaussian process.