  The propagation of incoherent elastic energy in a three-dimensional solid due
to the scattering by many, randomly placed and oriented, pinned dislocation
segments, is considered in a continuum mechanics framework. The scattering
mechanism is that of an elastic string of length L that re-radiates as a
response to an incoming wave. The scatterers are thus not static but have their
own dynamics. A Bethe-Salpeter (BS) equation is established, and a
Ward-Takahashi Identity (WTI) is demonstrated. The BS equation is written as a
spectral problem that, using the WTI, is solved in the diffusive limit. To
leading order a diffusion behavior indeed results, and an explicit formula for
the diffusion coeffcient is obtained. It can be evaluated in an Independent
Scattering Approximation (ISA) in the absence of intrinsic damping. It depends
not only on the bare longitudinal and transverse wave velocities but also on
the renormalized velocities, as well as attenuation coeffcients, of the
coherent waves. The influence of the length scale given by L, and of the
resonant behavior for frequencies near the resonance frequency of the strings,
can be explicitly identified. A Kubo representation for the diffusion constant
can be identified. Previous generic results, obtained with an energy transfer
formalism, are recovered when the number of dislocations per unit volume is
small. This includes the equipartition of diffusive energy density which,
however, does not hold in general. The formalism bears a number of similarities
with the behavior of electromagnetic waves in a medium with a random
distribution of dielectric scatterers; the elastic interaction, however, is
momentum dependent.
