Effective dynamics in lattices with random mass perturbations

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 lattice dynamics in two frameworks: a transient dynamics problem and a time-harmonic transmission problem. By a stochastic, multiscale analysis we provide expressions for the transient displacement field, that propagates through the random perturbations, in the first case and for the time-harmonic transmission coefficients in the second case. These theoretical predictions are supported by illustrations of their agreements with numerical simulations.