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 1, 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. (C) 2016 Elsevier B.V. All rights reserved.