Non-reciprocity in time-periodic phononic materials with a non-zero moving velocity

Non-reciprocal wave propagation in elastic structures has received considerable attention lately. A common mechanism to break elastic wave reciprocity is the use of phononic materials with traveling-wave-like properties. Among the popular methods to study wave dispersion in periodic media are the plane wave expansion and transfer matrix method (TMM). However, owing to the time-variant nature of such non-reciprocal systems, the implementation of both methods requires the truncation of harmonic terms. In this work, we adopt the TMM to extract the dispersion patterns of a moving phononic material with a prescribed velocity. In the presence of a temporal modulation of material properties in one direction accompanied by physical motion in an opposing direction, both effects cancel out and the problem becomes effectively time-invariant. This facilitates the analysis and yields interesting results. Subsequently, we exploit the well-established relationship between the momentumenergy spaces of moving and stationary elastic media to reconstruct exact dispersion diagrams of a stationary space-time-periodic system. The proposed approach provides a platform to incorporate the TMM in the analysis of non-reciprocal time-variant materials. Finally, given the lack of harmonic truncation, the accuracy of the new method does not diminish as the modulating speed increases.