Observation of Nonreciprocal Wave Propagation in a Dynamic Phononic Lattice

Acoustic waves in a linear time-invariant medium are generally reciprocal; however, reciprocity can break down in a time-variant system. In this Letter, we report on an experimental demonstration of nonreciprocity in a dynamic one-dimensional phononic crystal, where the local elastic properties are dependent on time. The system consists of an array of repelling magnets, and the on-site elastic potentials of the constitutive elements are modulated by an array of electromagnets. The modulation in time breaks time-reversal symmetry and opens a directional band gap in the dispersion relation. As shown by experimental and numerical results, nonreciprocal mechanical systems like the one presented here offer opportunities to create phononic diodes that can serve for rectification applications.

Figure 1

Experimental setup for the nonreciprocal dynamic phononic lattice. (a) Top: Schematic of the experimental setup. Middle: Discrete mechanical representation of the system with masses and springs. Bottom: Schematic illustration of the modulation concept by changing the gounding spring stiffness ($k_g$) in a wavelike fashion. (b) Scattering analysis: The red solid curve describes the original dispersion relation of the unmodulated monatomic lattice. The black dashed and grey dash-dotted curves correspond to Floquet-Bloch replicas of the original curves obtained by translation along the solid blue arrows $\pm ({\omega }_{\text{mod\hspace{0.17em}}},q_{\text{mod\hspace{0.17em}}})=\pm (15\mathrm{Hz},\pi /2)$. Parity-breaking crossings (circled) are where Bragg’s condition is satisfied and nonreciprocal wave scattering is anticipated. (c) Force-displacement curve for neighboring magnetic masses, measurement (solid) and fitted curve (dashed). (d) Measured force-dispacement curves between the ring magnet and its surrounding coil at different currents. The red shaded regions in both (c) and (d) correspond to the dynamic operating regime of our experiments.

Figure 2

Nonreciprocal wave propagation for ${\mathit{f}}_{\mathit{mod}}=15\mathrm{Hz}$. (a) Dispersion diagram of the modulated lattice calculated by Fourier analysis of simulated velocity fields (color map) and analytically by coupled mode theory (solid black line). (b) Measured velocity response function. The amplitude ratio at 19.6 Hz is $r=2.9$. (c) Measured velocity time series at $f_{dr}=19.6\mathrm{Hz}$. The time series for $q_{\text{mod}}=-\pi /2$ is shown along the negative time axis for better illustration. (d) and (e) are the simulation results corresponding to (b) and (c), respectively. The simulated amplitude ratio at 19.6 Hz is $r=1.9$ in panel (d).

Figure 3

Nonreciprocal wave propagation for ${\mathit{f}}_{\mathit{mod}}=40\mathrm{Hz}$. (a) Dispersion diagram of the modulated lattice calculated by Fourier analysis of simulated velocity fields (color map) and analytically by coupled mode theory (solid black line). (b) Measured velocity response function. The amplitude ratios are $r=1.8$ at 9.8 Hz and $r=0.4$ at 31.6 Hz. (c) Measured velocity time series at $f_{dr}=31.6\mathrm{Hz}$. The time series for $q_{\text{mod}}=-\pi /2$ is shown along the negative time axis for better illustration. (d) and (e) Simulation results corresponding to (b) and (c), respectively. The simulated bias ratios are $r=1.6$ at 9.8 Hz and $r=0.7$ at 31.6 Hz in panel (c).