Localization of elastic waves in two-dimensional (2D) and three-dimensional(3D) media with random distributions of the Lam\'e coefficients (the shear andbulk moduli) is studied, using extensive numerical simulations. We compute thefrequency-dependence of the minimum positive Lyapunov exponent $\gamma$ (theinverse of the localization length) using the transfer-matrix method, thedensity of states utilizing the force-oscillator method, and the energy-levelstatistics of the media. The results indicate that all the states may belocalized in the 2D media, up to the disorder width and the smallestfrequencies considered, although the numerical results also hint at thepossibility that there might a small range of the allowed frequencies overwhich a mobility edge might exist. In the 3D media, however, most of the statesare extended, with only a small part of the spectrum in the upper band tailthat contains localized states, even if the Lam\'e coefficients are randomlydistributed. Thus, the 3D heterogeneous media still possess a mobility edge. Ifboth Lam\'e coefficients vary spatially in the 3D medium, the localizationlength $\Lambda$ follows a power law near the mobility edge,$\Lambda\sim(\Omega-\Omega_c)^{-\nu}$, where $\Omega_c$ is the criticalfrequency. The numerical simulation yields, $\nu \simeq 1.89\pm 0.17$,significantly larger than the numerical estimate, $\nu\simeq 1.57\pm 0.01$, and$\nu=3/2$, which was recently derived by a semiclassical theory for the 3DAnderson model of electron localization...
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