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∗=cT2/v, where cT is the wave speed in the absence of obstacles and v is the speed of coherent wave propagation in the presence of dislocations.