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

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.