Coherent propagation and incoherent diffusion of elastic waves in a two dimensional continuum with a random distribution of edge dislocations

TL;DR

This paper investigates how elastic waves propagate coherently and diffuse incoherently in a 2D medium with randomly distributed edge dislocations, deriving wave velocities, attenuation, and diffusion coefficients through advanced theoretical models.

Contribution

It develops a comprehensive theoretical framework for elastic wave behavior in dislocation-rich media, including Dyson and Bethe-Salpeter equations, and introduces a Ward-Takahashi identity for differential operator wave equations.

Findings

Derived complex index of refraction for elastic waves.

Calculated frequency-dependent diffusion coefficient.

Established compatibility of Ward-Takahashi identity with ISA.

Abstract

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.