  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.
