Anderson localization in the time domain: Numerical studies of waves in two-dimensional disordered media

Abstract

A numerical model for the dynamics of a classical wave equation in a two dimensional Anderson disordered medium is integrated over times of the order of 27000 inverse band widths.Excitations by narrow band sources lead to wave energy densities whose ensemble averages behave diffusively at early times. The behavior at more general times and distances is, however, not diffusive. The observed transport profiles are shown to be inconsistent with predictions from the hydrodynamical continuum model of Anderson localization, and to the hypothesis of exponentially slow diffusion. The evolution of the energy density distribution in systems with varying disorders and microstructures, and over a range of length scales, is found to collapse to a single function of rescaled space and time well approximated by e(x,t) = exp{-x/1; -(x2+n/4 p i;n t)P } with n= 0.46, p= .76, where x is the distance from the source, I; is the localization length and the p is a "residual diffusivity" not equal to the bare diffusivity. Dissipation is shown to have no de-localizing effect.

Localization of seismic coda at Merapi volcano (Indonesia)

“…In a second step a strongly scattering cylinder embedded in a homogeneous half‐space is introduced. Finally we apply the time dependent Anderson localization model empirically derived by Weaver [1994].…”

Classical vibrational modes in phononic lattices: theory and experiment

“…The transition between the localized and the extended states in a two-dimensional disordered system is a question of considerable current interest [74]- [76]. It has been conjectured that there could exist the quasi-mobility edge(s) in two-dimensional disordered systems [85], which separate the strongly localized states from the weakly localized states (or power-law localized states).…”

“…This field theory (or nonlinear s-model) was later recovered and extended by diagrammatic perturbation techniques [66]. Other important theoretical [67]- [76] and experimental [77]- [86] developments related with the classical wave localization were exhaustively discussed in [3].…”

On the Bloch decomposition based spectral method for wave propagation in periodic media

“…The quantity A(t) has been introduced in [9] as a measure for the presence of Anderson's localization. In the absence of any random perturbation A(t) should grow quadratically in time whereas in the case of the Anderson localization A(t) should grow only linearly (indicating diffusive behavior) and eventually become a constant in time [9,25,27]. …”

“…It has been predicted by P. W. Anderson in the context of (quantum mechanical) electron dynamics [2] but is now regarded as a general wave phenomenon that applies to the transport of electromagnetic or acoustic waves as well, cf. [7,25,27].…”