# The mixing of polarizations in the acoustic excitations of disordered media with local isotropy

## TL;DR

This paper develops an approximate solution to the Dyson equation for acoustic waves in disordered isotropic media, capturing the mixing of longitudinal and transverse modes at wavelengths comparable to heterogeneity sizes.

## Contribution

It introduces a novel approximate analytical method that preserves wavevector dependence of the self-energy, enabling detailed analysis of mode mixing in disordered elastic media.

## Key findings

- Quantitative description of mode mixing at relevant wavelengths
- Mathematical validation and validity region of the solution
- Application to media with exponential covariance decay

## Abstract

An approximate solution of the Dyson equation related to a stochastic Helmholtz equation, which describes the acoustic dynamics of a three-dimensional isotropic random medium with elastic tensor fluctuating in space, is obtained in the framework of the Random Media Theory. The wavevector-dependence of the self-energy is preserved, thus allowing a description of the acoustic dynamics at wavelengths comparable with the size of heterogeneity domains. This in turn permits to quantitatively describe the mixing of longitudinal and transverse dynamics induced by the medium's elastic heterogeneity and occurring at such wavelengths. A functional analysis aimed to attest the mathematical coherence and to define the region of validity in the frequency-wavevector plane of the proposed approximate solution is presented, with particular emphasis dedicated to the case of disorder characterized by an exponential decay of the covariance function.