Propagation and localization of acoustic and elastic waves in heterogeneous materials: renormalization group analysis and numerical simulations

摘要

We describe and discuss the recent progress in the study of propagation and localization of acoustic and elastic waves in heterogeneous media. The heterogeneity is represented by a spatial distribution of the local elastic moduli. Both randomly distributed elastic moduli as well as those with long-range correlations with a nondecaying power-law correlation function, are considered. The motivation for the study is twofold. One is that recent analysis of experimental data for the spatial distribution of the elastic moduli of rock indicated that the distribution is characterized by the type of long-range correlations that we consider in this study. The second motivation for the problem is to understand whether localization of electrons (which, in quantum mechanics, are described by wave functions) has any analogy in the propagation of classical waves in disordered media. The problem is studied by two approaches. One of them is based on developing a dynamic renormalization group (RG) approach to analytical analysis of the governing equations for wave propagation. The RG analysis indicates that, depending on the type of the disorder (correlated vs. uncorrelated), one may have a transition between localized and extended regimes in any spatial dimension. The second approach utilizes numerical simulations of the governing equations in two- and three-dimensional media. The results obtained by the two approaches are in agreement with each other. Using numerical simulations, we also describe how the characteristics of a propagating wave may be used for probing the differences between heterogeneous media with short- and long-range correlations. To do so, we study the evolution of several distinct characteristics of the waves, such as the amplitude of the coherent wave front, its width, the spectral densities, the scalogram (wavelet transformation of the waves' amplitudes at different scales and times), and the dispersion relation. It is demonstrated that such properties have completely different characteristics in uncorrelated and correlated media. Finally, it is shown how wave propagation may be used for establishing a link between the static and dynamical properties of heterogeneous media.