  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.
