Kinetic modeling of multiple scattering of acoustic waves in randomly heterogeneous flows

TL;DR

This paper develops a kinetic model for acoustic wave propagation in randomly fluctuating flows, capturing scattering, refraction, and phase effects using a radiative transfer equation derived from the linearized Euler equations.

Contribution

It introduces a novel derivation of a radiative transfer equation for acoustic waves in random flows, accounting for multiple scattering and inhomogeneities at wavelengths comparable to correlation lengths.

Findings

Derivation of a comprehensive radiative transfer equation for acoustic waves in random flows.

Inclusion of scattering, refraction, and phase effects in the model.

The model encompasses phenomena like spectral broadening and multiple scattering.

Abstract

We study the propagation of sound waves in a three-dimensional, infinite ambient flow with weak random fluctuations of the mean particle velocity and speed of sound. We more particularly address the regime where the acoustic wavelengths are comparable to the correlation lengths of the weak inhomogeneities--the so-called weak coupling limit. The analysis is carried on starting from the linearized Euler equations and the convected wave equation with variable density and speed of sound, which can be derived from the nonlinear Euler equations. We use a multi-scale expansion of the Wigner distribution of a velocity potential associated to the waves to derive a radiative transfer equation describing the evolution of the angularly resolved wave action in space/time phase space. The latter experiences convection, refraction and scattering when it propagates through the heterogeneous ambient flow, although the overall wave action is conserved. The convection and refraction phenomena are accounted for by the convective part of the transport equation and depend on the smooth variations of the ambient quantities. The scattering phenomenon is accounted for by the collisional part of the transport equation and depends on the cross-power spectral densities of the fluctuations of the ambient quantities at the wavelength scales. The refraction, phase shift, spectral broadening, and multiple scattering effects of the high-frequency regimes described in various previous publications are thus encompassed by the proposed model. The overall derivation is based on the interpretation of spatial-temporal Wigner transforms in terms of semiclassical operators in their standard quantization.