Abstract
Using the Martin-Siggia-Rose method, we study localization of elastic waves in two-dimensional heterogeneous solids with randomly distributed Lam\'e coefficients, as well as those with long-range correlations with a power-law correlation function. We derive the one-loop renormalization group (RG) equations of the coupling constants in the limit of long wavelengths. The various phases of the coupling constant space, which depend onthe value $\rho$, the exponent that characterizes the power-law correlation function, are determined and described. Qualitatively different behaviors emerge for $\rho<1$ and $\rho>1$. The Gaussian fixed point (FP) is stable (unstable) for $\rho<1$ ($\rho>1$). For $\rho<1$ there is a region of the coupling constants space in which the RG flows are toward the Gaussian FP, i.e., the disorder is irrelevant and the waves are delocalized. In the rest of the space the elastic waves are localized. We compare the results with those obtained previously for acoustic wave propagation in the same type of heterogeneous media, and describe the similarities and differences between the two phenomena.