Wave propagation in an elastic lattice with non-reciprocal stiffness and engineered damping

Nonreciprocal wave propagation allows for directional energy transport. In this work, we systematically investigate wave dynamics in an elastic lattice that combines nonreciprocal stiffness with viscous damping. After establishing how conventional damping counteracts the system's gain, we introduce a non-dissipative form of nonreciprocal damping in the form of gyroscopic damping. We find that the coexistence of nonreciprocal stiffness and nonreciprocal damping results in a decoupled control mechanism. The nonreciprocal stiffness is shown to govern the temporal amplification rate, while the nonreciprocal damper independently tunes the wave's group velocity and oscillation frequency. This decoupling gives rise to phenomena such as the enhancement of net amplification for slower-propagating waves, and also boundary-induced wave interference arising from divergent and convergent reflected wave trajectories with varying growth rates. These findings provide a theoretical framework for designing active metamaterials with more versatile control over their wave propagation characteristics.