Localization of Elastic Waves in Heterogeneous Media with Off-Diagonal Disorder and Long-Range Correlations

Using the Martin-Siggia-Rose method, we study propagation of acoustic waves in strongly heterogeneous media which are characterized by a broad distribution of the elastic constants. Gaussian-white distributed elastic constants, as well as those with long-range correlations with nondecaying power-law correlation functions, are considered. The study is motivated in part by a recent discovery that the elastic moduli of rock at large length scales may be characterized by long-range power-law correlation functions. Depending on the disorder, the renormalization group (RG) flows exhibit a transition to localized regime in any dimension. We have numerically checked the RG results using the transfer-matrix method and direct numerical simulations for one- and two-dimensional systems, respectively.

Figure 1

RG flows in the coupling constants space for $0<\rho <d/2$.

Figure 2

Same as in Fig. 1, but for $\rho >d/2$.

Figure 3

Localization length $\xi$ for a disordered chain with 50 000 site and its dependence on the frequency $\omega$.

Figure 4

Amplitude of the waves in 2D systems, for both randomly and uniformly distributed and power-law correlated elastic constants.