  We derive exact expressions for effective elastodynamic properties of
two-phase composites in the long-wavelength (quasistatic) regime via
homogenized constitutive relations that are local in space. This is
accomplished by extending the "strong-contrast" expansion formalism that was
previously applied to the static problem. These strong-contrast expansions
explicitly incorporate complete microstructural information of the composite
via an infinite set of $n$-point correlation functions. Utilizing the
rapid-convergence properties of these series expansions (even for extreme
contrast ratios), we extract accurate approximations that depend on the
microstructure via the spectral density, which is easy to compute or measure
for any composite. We also investigate the predictive power of modifications of
such approximation formulas postulated elsewhere [J. Kim and S. Torquato, Proc.
Nat. Acad. Sci. ${\bf 117}$, 8764 (2020)] to extend their applicability ${\it
beyond ~the ~quasistatic ~regime.}$ The accuracy of these nonlocal
microstructure-dependent approximations is validated by comparison to
full-waveform simulation results for certain models of dispersions. We apply
our formulas to a variety of models of nonhyperuniform and hyperuniform
disordered composites. We demonstrate that hyperuniform systems are less lossy
than their nonhyperuniform counterparts in the quasistatic regime, and stealthy
hyperuniform media can be perfectly transparent for a wide range of wavenumber.
Finally, we discuss how to utilize our approximations for engineering
composites with prescribed elastic wave characteristics.
