Temporal effective medium for programmable acoustic metamaterials with multiple resonances

Abstract We extend effective medium theory (EMT) to time-modulated, frequency-dispersive acoustic metamaterials with multiple resonances. While previous studies focused on frequency non-dispersive or single-resonance systems, advances in programmable materials now enable precise control of time-varying resonances. We derive explicit averaging rules that account for the interplay between resonant and modulation frequencies. When resonant frequencies are much lower than the modulation frequency, modulating the resonant strength yields the temporal average of monopolar susceptibility χ, while modulating the resonant frequency results in the average of 1/χ, applied per resonance mode. In hybrid cases with both high and low-frequency resonances (relative to the modulation) are present, the high-frequency resonances behave effectively as a frequency-nondispersive background and can be averaged to yield their monopolar susceptibility contribution, while the low-frequency resonance is averaged after renormalizing the wave equations by this background. This generalized temporal EMT offers a unified framework for designing compact, topological, or non-Hermitian acoustic devices, leveraging the possible programmability of time-dependent material parameters in future.