Surface waves in randomly perturbed discrete models

TL;DR

This paper investigates how surface waves propagate across structured surfaces with random inhomogeneities using a discrete model, deriving statistical measures of wave reflectance, transmittance, and radiative loss, and validating findings with numerical simulations.

Contribution

It introduces a discrete analogue of the Gurtin-Murdoch model to analyze surface wave scattering and provides new statistical descriptions of radiative loss in randomly perturbed surfaces.

Findings

Mean radiative loss is proportional to patch size, variance, and an effective spectral parameter in weak scattering.

In strong scattering, radiative loss depends on a different spectral parameter, independent of variance.

Numerical simulations agree with theoretical predictions across various surface parameters.

Abstract

We study the propagation of surface waves across structured surfaces with random, localized inhomogeneities. A discrete analogue of Gurtin-Murdoch model is employed and surface elasticity, in contrast to bulk elasticity, is captured by distinct point masses and elastic constants for nearest-neighbour interactions parallel to the surface. Expressions for the surface wave reflectance and transmittance, as well as the radiative loss, are provided for every localized patch of point mass perturbation on the surface. As the main result in the article, we provide the statistics of surface wave reflectance and transmittance and the radiative loss for an ensemble of random mass perturbations, independent and identically distributed with mean zero, on the surface. In the weakly scattering regime, the mean radiative loss is found to be proportional to the size of the perturbed patch, to the variance of the mass perturbations, and to an effective parameter that depends on the continuous spectrum of the unperturbed system. In the strongly scattering regime, the mean radiative loss is found to depend on another effective parameter that depends on the continuous spectrum, but not on the variance of the mass perturbations. Numerical simulations are found in quantitative agreement with the theoretical predictions for several illustrative values of the surface structure parameters.