  This work presents a generalized physical interpretation of unconventional
dispersion asymmetries associated moving elastic solids. By shifting the notion
from systems with time-variant material fields to physically traveling
materials, the newly adopted paradigm provides an eloquent take on the
dispersion problem and, in the process, highlights discrepancies between both
schemes. Equations governing the motion of an elastic rod with a prescribed
moving velocity observed from a stationary reference frame are used to predict
propagation patterns and asymmetries in wave velocities induced as a result of
the induced linear momentum bias. Three distinct scenarios corresponding to a
moving rod with a constant modulus, a spatially varying one, and one that
varies in space and time are presented. These cases are utilized to extract and
interpret correlations pertaining to directional velocities, dispersion
patterns, as well as nature of band gaps between moving periodic media and
their stationary counterparts with time-traveling material properties. A linear
vertical shear transformation is derived and utilized to thoroughly neutralize
the effect of the moving velocity on the resultant dispersion characteristics.
Finally, dispersion contours associated with the transient response of a finite
moving medium are used to validate the entirety of the presented framework.
