  The propagation of acoustic and elastic waves in time-varying, spatially
homogeneous media can exhibit different phenomena when compared to traditional
spatially-varying, temporally-homogeneous media. In the present work, the
response of a one-dimensional phononic lattice with time-periodic elastic
properties is studied with experimental, numerical and theoretical approaches.
The system consists of repelling magnetic masses with grounding stiffness
controlled by electrical coils driven with electrical signals that vary
periodically in time. For small amplitude excitation, in agreement with
theoretical predictions, wavenumber bandgaps emerge. The underlying
instabilities associated to the wavenumber bandgaps are investigated with
Floquet theory and the resulting parametric amplification is observed in both
theory and experiments. In contrast to genuinely linear systems, large
amplitude responses are stabilized via the nonlinear nature of the magnetic
interactions of the system. In particular, the parametric amplification induced
by the wavenumber bandgap can lead to bounded and stable responses that are
temporally quasi-periodic. Controlling the propagation of acoustic and elastic
waves by balancing nonlinearity and external modulation offers a new dimension
in the realization of advanced signal processing and telecommunication devices.
For example, it could enable time-varying, cross-frequency operation, mode- and
frequency-conversion, and signal-to-noise ratio enhancements.
