  We present a new computational method for the accurate identification of the
propagation modes and polarizations of elastic waves propagating in periodic
solid structures and metamaterials. The method uses the eigenvectors calculated
at each propagating wave solution eigenfrequency to identify the contribution
of each translational and rotational component of the total mass in motion. We
use this information to identify the dominant wave propagation mode by defining
a relative effective modal mass vector. Further, we associate each wave
solution with its correct polarization by defining a new polarization factor
that quantifies the relative orientation between the wave propagation and
lattice motion directions and provides a positive numerical value between 0
(pure S-wave) and 1 (pure P-wave). Further, we suggest a graphical
representational scheme for an easier visualization of the wave polarization
within traditional dispersion plots. The method is validated by comparing our
predictions against previously published results for various elastodynamic
problems. Finally, we use the proposed method to analyze the effect of various
lattice and structural parameter perturbations on the elastic wave propagation
and bandgap behavior of a square planar beam lattice and show the emergence of
previously unobserved dynamic characteristics, including various polarized
bandgaps, fluid-like behavior, and ultralow-frequency SH- and SV-bandgaps that
extend to 0 Hz. The proposed method provides an alternative computational
approach to the typically employed visual mode inspection technique and
provides a robust technique for analyzing the elastic wave response of periodic
solid media.
