  We study the diffusion of anti-plane elastic waves in a two dimensional
continuum by many, randomly placed, screw dislocations. Building on a
previously developed theory for coherent propagation of such waves, the
incoherent behavior is characterized by way of a Bethe Salpeter (BS) equation.
A Ward-Takahashi identity (WTI) is demonstrated and the BS equation is solved,
as an eigenvalue problem, for long wavelengths and low frequencies. A diffusion
equation results and the diffusion coefficient $D$ is calculated. The result
has the expected form $D = v^* l /2$, where $l$, the mean free path, is equal
to the attenuation length of the coherent waves propagating in the medium and
the transport velocity is given by $v^*= c_T^2/v$, where $c_T$ is the wave
speed in the absence of obstacles and $v$ is the speed of coherent wave
propagation in the presence of dislocations.
