  Localization of elastic waves in two-dimensional (2D) and three-dimensional
(3D) media with random distributions of the Lam\'e coefficients (the shear and
bulk moduli) is studied, using extensive numerical simulations. We compute the
frequency-dependence of the minimum positive Lyapunov exponent $\gamma$ (the
inverse of the localization length) using the transfer-matrix method, the
density of states utilizing the force-oscillator method, and the energy-level
statistics of the media. The results indicate that all the states may be
localized in the 2D media, up to the disorder width and the smallest
frequencies considered, although the numerical results also hint at the
possibility that there might a small range of the allowed frequencies over
which a mobility edge might exist. In the 3D media, however, most of the states
are extended, with only a small part of the spectrum in the upper band tail
that contains localized states, even if the Lam\'e coefficients are randomly
distributed. Thus, the 3D heterogeneous media still possess a mobility edge. If
both Lam\'e coefficients vary spatially in the 3D medium, the localization
length $\Lambda$ follows a power law near the mobility edge,
$\Lambda\sim(\Omega-\Omega_c)^{-\nu}$, where $\Omega_c$ is the critical
frequency. 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 3D
Anderson model of electron localization...
