Numerical analysis of dispersion and elastic wave propagation in spatiotemporally modulated spring–mass metamaterials

This study presents a numerical framework for analyzing dispersion and wave propagation in spatiotemporally modulated metamaterials. A one-dimensional periodic spring–mass system with space- and time-dependent stiffness is modeled, and dispersion relations are extracted using a spatial perturbation method combined with two-dimensional Fourier transformation. Random initial velocity perturbations—analogous to thermal excitation in molecular dynamics—are applied to excite all possible wave modes, thereby eliminating the need for complex analytical derivations. The computed dispersion curves agree closely with analytical solutions based on the Bloch wave assumption for spatial, temporal, and spatiotemporal modulations. The proposed method is computationally adaptable and applicable to systems with arbitrary unit-cell configurations and various forms of stiffness modulation in both space and time, accurately capturing both primary and modulation-induced secondary dispersion branches. Transient simulations further reveal symmetric bidirectional propagation in spatially periodic systems, frequency conversion in temporally modulated systems, and direction-dependent propagation in spatiotemporal systems. However, complete non-reciprocity—i.e., one-way transmission—is not observed because directional bandgaps are absent. Overall, the spatial perturbation–Fourier framework provides a robust and generalizable tool for investigating dynamic metamaterials, enabling rapid design and optimization of structures with tunable and asymmetric wave-manipulation capabilities.