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 effective 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.

我们研究了二维连续体中平面内弹性波的相干传播和非相干扩散，该连续体由许多随机放置和定向的边缘位错填充。由于 Peierls-Nabarro 力，位错可以以频率在平衡位置附近振荡${\omega }_0$. 波和位错之间的耦合由 Peach-Koehler 力给出。这导致波动方程具有涉及微分算子的非齐次项。在相干情况下，在独立散射近似 (ISA) 中，质量算子的戴森方程被建立并求解到微扰理论中的所有阶次。结果，获得了复折射率，从中可以读出纵波和横波的有效波速和衰减。在非相干情况下，建立 Bethe-Salpeter 方程，并在低频和波数极限下求解微扰理论中的领先阶次。获得扩散方程并明确计算（频率相关）扩散系数。它减少到在低频下通过能量转移参数获得的值。一个重要的中间步骤是获得一个包含微分算子的波动方程的 Ward-Takahashi 恒等式 (WTI)，它被证明与 ISA 兼容。