Boundary effects in Radiative Transfer of acoustic waves in a randomly fluctuating half-space

TL;DR

This paper derives radiative transfer equations for acoustic waves in a randomly fluctuating half-space, analyzing boundary effects through asymptotic analysis of the wave's Wigner transform, revealing how boundary conditions influence wave intensity.

Contribution

It introduces a novel asymptotic analysis of boundary effects in radiative transfer for acoustic waves in random media, using the method of images and Wigner transform techniques.

Findings

Boundary effects cause intensity doubling with Neumann conditions.

Boundary effects cancel out with Dirichlet conditions.

Two energy density contributions identified: one propagating in full-space, one near boundary.

Abstract

This paper concerns the derivation of radiative transfer equations for acoustic waves propagating in a randomly fluctuating half-space in the weak-scattering regime, and the study of boundary effects through an asymptotic analysis of the Wigner transform of the wave solution. These radiative transfer equations allow to model the transport of wave energy density, taking into account the scattering by random heterogeneities. The approach builds on the method of images, where the half-space problem is extended to a full-space, with two symmetric sources and an even map of mechanical properties. Two contributions to the total energy density are then identified: one similar to the energy density propagation in a full-space, for which the resulting lack of statistical stationarity of the medium properties has no leading-order effect; and one supported within one wavelength of the boundary, which describes interference effects between the waves produced by the two symmetric sources. In the case of a homogeneous Neumann boundary conditions, this boundary effect yields a doubling of the intensity, and in the case of homogeneous Dirichlet boundary conditions, a canceling of that intensity.