Dynamic renormalization group analysis of propagation of elastic waves in two-dimensional heterogeneous media

TL;DR

This paper applies dynamic renormalization group analysis to study how elastic waves propagate and localize in two-dimensional heterogeneous media with different correlation structures, revealing phase-dependent behaviors and localization conditions.

Contribution

It introduces a one-loop RG framework for elastic wave localization in 2D heterogeneous media with long-range correlations, analyzing phase behavior based on correlation exponent.

Findings

Localization depends on the correlation exponent $ ho$

Gaussian fixed point stability varies with $ ho$

Delocalization occurs for $ ho<1$ in certain regions

Abstract

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. The Matin-Siggia-Rose method is used, and the one-loop renormalization group (RG) equations for the the coupling constants are derived in the limit of long wavelengths. The various phases of the coupling constants space, which depend on the 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, implying that the disorder is irrelevant and the waves are delocalized. In the rest of the disorder 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.