Diffusion of elastic waves in a two dimensional continuum with a random distribution of screw dislocations

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.