Amplified phonon bands with tunable dispersion in active mechanical metamaterials

Parametric resonance—injecting energy into a wave by modulating the parameters of the propagation medium—is a promising mechanism for sound processing and amplification in active phononic metamaterials. However, parametric resonances typically occur in narrow frequency ranges. To amplify spectrally complex signals, the modulation must have spectral features that are dictated by the dispersion relation of the unmodulated structure. For instance, it is known that a traveling-wave modulation can amplify an entire phonon band with linear dispersion. Here, we describe strategies to create fully amplified bands when the unmodulated dispersion relation is gapped rather than linear, as commonly occurs in mechanical resonator arrays. Using Floquet theory, we design modulation waves that enable completely amplified gapped bands whose spectral bandwidth can be large or vanishingly small, enabling fine control of the propagation speed of amplified wavepackets. We also design bands whose spectra exhibit nonzero winding in the complex frequency plane, leading to a strong sensitivity to boundary conditions (non-Hermitian skin effect). The fully amplified bands we describe show promise in manipulating sound signals and as a platform to investigate non-Hermitian wave phenomena in mechanics.