Elastic wave propagation in moving phononic crystals and correlations with stationary spatiotemporally modulated systems

This work presents a generalized physical interpretation of unconventional dispersion asymmetries associated with moving phononic crystals (PCs). By shifting the notion from systems with time-variant material fields to physically traveling materials, the newly adopted paradigm provides a novel approach to the elastic dispersion problem and, in the process, highlights discrepancies between moving PCs and stationary ones with dynamic material fields. 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 obtained as a result of the induced linear momentum bias. Three distinct scenarios are presented corresponding to a moving rod with a constant modulus, a spatially varying one, and one that varies in space and time. 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 then derived and utilized to neutralize the effect of the moving velocity on the resultant band diagrams. Finally, dispersion contours associated with the transient response of a finite moving medium are used to validate the entirety of the presented framework.