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

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 27 000 inverse bandwidths. 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 a simple 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/ξ-(${\mathit{x}}^{2+\mathit{n}}$/4β${\xi }^{\mathit{n}}$t${)}^{\mathit{p}}$} with n≊0.46, p≊0.76, where x is the distance from the source, ξ is the localization length, and the β is a ‘‘residual diffusivity’’ not equal to the bare diffusivity. Dissipation is shown to have no delocalizing effect.