Nonreciprocal parametric amplification of elastic waves in supersonic spatiotemporally modulated media

Spatiotemporally modulated elastic media, whose material properties vary in both space and time, have attracted significant attention as a promising route to nonreciprocal propagation of elastic waves. To date, most studies have focused on temporally bounded yet spatially unbounded media, particularly in the subsonic regime. In contrast, wave propagation in spatially bounded media under supersonic modulation remains largely unexplored, and the associated nonreciprocal phenomena present an open challenge. In this work, we investigate the dynamic response of elastic longitudinal waves propagating through a spatially bounded medium with supersonically modulated elastic properties. We show that supersonic modulation gives rise to directional wavenumber bandgaps in the κ ( ω ) dispersion representation, manifested as separations between adjacent wavenumber bands, with no imaginary wavenumber components involved. Using a theoretical framework based on mode-coupling theory, we further demonstrate that such spatially bounded modulated media can support both parametric amplification and frequency conversion. Remarkably, the observed parametric amplification is not driven by temporally growing modes, but instead arises from the interference between counter-propagating Floquet–Bloch modes at the spatial interfaces. Numerical simulations corroborate the theoretical predictions and illustrate nonreciprocal amplification and frequency conversion effects. Our findings pave the way for the design of nonreciprocal elastic wave devices with advanced functionalities such as signal processing, energy amplification, and frequency conversion.