Effective dynamics in lattices with random mass perturbations

TL;DR

This paper analyzes how random mass perturbations affect wave propagation in a one-dimensional lattice, providing asymptotic formulas for displacements and transmission, supported by numerical validation.

Contribution

It introduces a stochastic multiscale analysis to derive asymptotic expressions for dynamics in lattices with random mass perturbations, including transmission coefficients.

Findings

Asymptotic expressions for displacement fields

Theoretical predictions match numerical simulations

Insights into wave transmission in disordered lattices

Abstract

We consider a one-dimensional mono-atomic lattice with random perturbations of masses spread over a finite number of particles. Assuming Newtonian dynamics and linear nearest-neighbour interactions and allowing for a provision of pinning due to substrate interaction, we discuss a transient dynamics problem and a time-harmonic transmission problem. By a stochastic, multiscale analysis we provide asymptotic expressions for the displacement field that propagates through the random perturbations and for the time-harmonic transmission coefficients. These theoretical predictions are supported by illustrations of their agreements with numerical simulations.