Abstract This study presents a novel numerical framework for analyzing wave propagation in spatiotemporally modulated metamaterials. A one-dimensional periodic spring-mass system is considered, wherein the spring stiffness varies in both space and time. To extract the dispersion relations of such systems, we propose a spatial perturbation method combined with two-dimensional Fourier transformation. The method leverages random initial velocity perturbations—analogous to thermal excitations in molecular dynamics—to excite all possible wave modes without requiring complex derivations. The computed dispersion curves exhibit excellent agreement with analytical solutions derived under the Bloch wave assumption for spatial, temporal, and spatiotemporal metamaterials. In addition to confirming the method’s accuracy, we demonstrate its general applicability to systems with arbitrary unit cell configurations. Transient wave propagation simulations reveal distinct behaviors across different metamaterial types. Spatial systems exhibit symmetric bidirectional wave propagation, while temporally modulated systems generate secondary wave components at frequencies \ (\: \: ₑ\: \: \: n\: \) due to modulation-induced frequency conversion. In spatiotemporal systems, asymmetric dispersion relations result in directionally dependent energy propagation, though strictly non-reciprocal behavior is not observed due to the absence of directional bandgaps. The proposed method provides a robust and efficient tool for analyzing complex periodic structures, enabling the accelerated design and optimization of spatiotemporal metamaterials for advanced wave manipulation.
Liao et al. (Tue,) studied this question.