Derivation of a Schrödinger-like equation for elastic waves in granular media

Abstract

A theoretical model for the propagation of acoustic waves in dry granular media is presented within the framework of the nonlinear elastic theory recently proposed by Jiang and Liu. Paralleling the analysis of Tregoures and van Tiggelen for multiple scattering of acoustic waves in heterogeneous elastic media, we derive a quantum field formulation of the dynamics of deformation in static granular ensembles in which the total wave energy satisfies a Schrodinger-like equation. An additional term appears in the corresponding time-evolution matrix, which together with the strain-dependent elastic coefficients, leads to a description of intrinsic features of granular dynamics such as volume dilatancy, mechanical yield, and anisotropies in the stress distribution. Starting from the Laplace transform of the Schrodinger-like equation and introducing the disorder perturbation as a small fluctuation of the time-evolution operator, we derive the radiative transport equation and the related diffusion equation for the propagation of elastic waves in a granular medium. The present approach provides an accurate description of acoustic wave propagation in granular packings and represents a powerful tool to interpret the results of current experiments.