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

TL;DR

The work addresses how to control wave propagation in 1D lattices by combining nonreciprocal stiffness with engineered damping. It develops a theoretical framework based on complex dispersion relations, including onsite, intersite, and gyroscopic damping, revealing a decoupled mechanism where $\alpha$ sets the temporal growth while $\beta$ tunes group velocity and oscillation frequency. The key finding is that gyroscopic damping allows independent manipulation of gain and wave kinematics, enabling slower waves to accumulate more amplification and enabling boundary-induced multi-frequency interference. These insights offer design principles for active metamaterials with programmable gain, speed, and spectral content, with potential experimental routes and extensions to higher dimensions.

Abstract

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.

Figures (15)

• Figure 1: Wave propagation in a 1D lattice with nonreciprocal stiffness. (a) Schematic of the lattice with asymmetric intersite springs characterized by forward and backward stiffnesses $k_f = k(1+\alpha)$ and $k_b = k(1-\alpha)$. (b–c) Real and imaginary parts of the complex dispersion relation $\omega(q)$ for asymmetry parameter $\alpha = 0$ and $\alpha = 0.1$. (d–e) Parametric maps of group velocity and decay/growth rate signs as functions of wavenumber $q$ and asymmetry $\alpha$, highlighting directional energy transport in the absence of damping. We take intersite stiffness $k = 180$, onsite stiffness $k_g = 120$, and mass $m = 1$ in these calculations.
• Figure 2: Finite-chain simulations of wave propagation and directional amplification in a lattice with nonreciprocal stiffness. (a) Space–time evolution of the space-normalized displacement field, $u_n(t)/\|u_n(t)\|_{\infty,n}$, in a 200-particle lattice with fixed boundaries showing selective amplification in the forward direction. The wavefront travels with group velocity $v_g \approx 7.184$. (b) Logarithmic plot of the instantaneous global maximum of the lattice displacement field, with a linear fit yielding a slope of 0.71. (c–d) Velocity responses and corresponding spectral amplitudes at downstream sites ($n = 130, 160$) highlight energy amplification and a spectral peak near $\omega \approx 24.38 \, \text{rad/s}$. (e–f) Upstream sites ($n = 70, 40$) exhibit strong attenuation in both time and frequency domains, confirming unidirectional energy transport enabled by non-reciprocal stiffness.
• Figure 3: Distinct trajectories of spatial and temporal peaks in a 1D finite lattice with nonreciprocal stiffness. (a) Analytical trajectories of the Gaussian envelope maxima, comparing the instantaneous spatial peaks $n_{\mathrm{peak}}(t)$ (dots and circles) and the temporal peaks $t_{\mathrm{peak}}(n)$ (solid lines). While the forward and backward $t_{\mathrm{peak}}(n)$ branches share the same slope (indicating equal effective group velocity), they exhibit distinct temporal intercepts at the source $n=n_0$, as highlighted in the inset. (b) Spatiotemporal evolution of the displacement field normalized by the local temporal maximum, $u_n(t)/\|u_n(t)\|_{\infty,t}$. The wavefronts propagate with equal speeds but project to unequal origin times, visually confirming the positive and negative intercepts predicted in (a).
• Figure 4: Wave propagation in a 1D lattice with nonreciprocal stiffness and onsite damping. (a) Schematic of the lattice with asymmetric intersite springs and onsite viscous dampers; (b–c) Real and imaginary parts of the complex dispersion relation $\omega(q)$ for asymmetry parameters $\alpha = 0$ and $\alpha = 0.1$. (d–e) Parametric maps showing the signs of group velocity and temporal growth/decay rate as functions of $q$ and $\alpha$. We take onsite damping $c_g = 0.2$. Other parameters are the same as in Fig. \ref{['fig:FIG1']}.
• Figure 5: Finite-chain simulations of wave propagation and directional amplification in a lattice with nonreciprocal stiffness and onsite damping. (a) Space–time evolution of the space-normalized displacement field, showing amplification in the forward-propagating wave packet with measured group velocity $v_g \approx 7.184$. (b) Logarithmic plot of the instantaneous global maximum, where a linear fit yields a slope of 0.61, indicating a reduced temporal growth rate due to the presence of onsite damping. (c–d) Velocity responses and corresponding spectral amplitudes at downstream sites ($n = 130, 160$) confirm energy amplification and peak excitation near $\omega \approx 24.38$. (e–f) Upstream sites ($n = 70, 40$) show significant attenuation in both time and frequency domains, affirming unidirectional energy transport.