  We study the coherent propagation and incoherent diffusion of in-plane
elastic waves in a two dimensional continuum populated by many, randomly placed
and oriented, edge dislocations. Because of the Peierls-Nabarro force the
dislocations can oscillate around an equilibrium position with frequency
$\omega_0$. The coupling between waves and dislocations is given by the
Peach-Koehler force. This leads to a wave equation with an inhomogeneous term
that involves a differential operator. In the coherent case, a Dyson equation
for a mass operator is set up and solved to all orders in perturbation theory
in independent scattering approximation (ISA). As a result, a complex index of
refraction is obtained, from which an effectve wave velocity and attenuation
can be read off, for both longitudinal and transverse waves. In the incoherent
case a Bethe-Salpeter equation is set up, and solved to leading order in
perturbation theory in the limit of low frequency and wave number. A diffusion
equation is obtained and the (frequency-dependent) diffusion coefficient is
explicitly calculated. It reduces to the value obtained with energy transfer
arguments at low frequency. An important intermediate step is the obtention of
a Ward-Takahashi identity (WTI) for a wave equation that involves a
differential operator, which is shown to be compatible with the ISA.
