Scattering mean-free path in continuous complex media: beyond the Helmholtz equation

TL;DR

This paper develops a more accurate theoretical model for wave propagation in complex media, revealing that traditional scalar potential assumptions significantly overestimate the scattering mean-free path, especially at low frequencies.

Contribution

It introduces a combined scalar and operator potential model for wave scattering, improving the accuracy of mean-free path calculations beyond the Helmholtz equation assumptions.

Findings

The traditional scalar potential approach overestimates the mean-free path by more than four times.

Including both scalar and operator terms yields more accurate scattering predictions.

Numerical experiments confirm the theoretical results.

Abstract

We present theoretical calculations of the ensemble-averaged (a.k.a. effective or coherent) wavefield propagating in a heterogeneous medium considered as one realization of a random process. In the literature, it is usually assumed that heterogeneity can be accounted for by a random scalar function of the space coordinates, termed the potential. Physically, this amounts to replacing the constant wavespeed in Helmholtz' equation by a space-dependent speed. In the case of acoustic waves, we show that this approach leads to incorrect results for the scattering mean-free path, no matter how weak fluctuations are. The detailed calculation of the coherent wavefield must take into account both a scalar and an operator part in the random potential. When both terms have identical amplitudes, the correct value for the scattering mean-free paths is shown to be more than four times smaller (13/3, precisely) in the low frequency limit, whatever the shape of the correlation function. Based on the diagrammatic approach of multiple scattering, theoretical results are obtained for the self-energy and mean-free path, within Bourret's and on-shell approximations. They are confirmed by numerical experiments.