Self-consistent theory for sound propagation in a simple model of a disordered, harmonic solid

We present a self-consistent theory for sound propagation in a simple model of a disordered solid. The solid is modeled as a collection of randomly distributed particles connected by harmonic springs with strengths that depend on the interparticle distances, i.e., the Euclidean random matrix model of Mézard et al. [Nucl. Phys. B 559, 689 (1999)0550-321310.1016/S0550-3213(99)00428-9]. The derivation of the theory combines two exact projection operator steps and a factorization approximation. Within our approach, the square of the speed of sound is non-negative. We expect that an unjamming transition would manifest itself through the vanishing of the speed of sound.