Brillouinzone (BZ) definition in a class of nonreciprocal Willis monatomic lattices (WMLs) is analytically quantified. It is shown that BZ boundaries only shift in response to nonreciprocity in onedimensional WMLs, implying a constant BZ width, with asymmetric dispersion diagrams exhibiting unequal wavenumber ranges for forward and backward going waves. An extension to square WMLs is briefly discussed, analogously demonstrating the emergence of shifted and irregularly shaped BZs, which maintain constant areas regardless of nonreciprocity strength.
1. Introduction
Exploration of new wave phenomena has been a driving motivation in wave propagation research^{1} with a growing interest in elastodynamic nonreciprocity.^{2} Within linear, timeinvariant elastic media, a wave emanating from a source towards a receiver must not behave differently if the locations of the source and receiver are interchanged, thanks to elastodynamic reciprocity. Designs of artificial materials have enabled new avenues to manipulate waves beyond what is possible in natural materials, and breakage of wave reciprocity is no exception. Given its potential in enabling novel applications (e.g., oneway vibrational isolators^{3}) researchers have introduced a variety of structural designs tailored for nonreciprocity, such as exploiting nonlinear instability with supratransmission phenomenon.^{4} Comparably, granular crystals, with their nonlinear and bifurcatingchaotic behavior, have been exploited to asymmetrically transmit energy, with potential application in sensing and energy harvesting.^{5} Another choice to achieve nonreciprocity is by spatiotemporally modulating mechanical properties, with artificial momentum bias forcing waves to propagate differently in the forward and backward directions.^{3,6–10}
When elastic media defy elastodynamic reciprocity, their dispersion diagram becomes skewed and asymmetric about a zero wavenumber, and conventional Brillouinzone (BZ) definitions are rendered obsolete. Cassedy and Oliner are amongst the first contributors to study the skewed (nonreciprocal) nature of BZ in an attempt to generalize its definition for wave propagation in spacetime modulated electrical circuits.^{11} Unlike spatialonly modulation with dispersion diagrams perfectly repeating along the wavenumber axis, the presence of time modulation creates dispersiondiagram periodicity along a line with a slope that is a function of such modulation (under small modulation amplitude assumption). However, the adequacy of traditional BZs to fully capture wave nonreciprocity remains an ongoing concern, especially for twodimensional systems.^{8}
2. Onedimensional Willis monatomic lattices
2.1 Mathematical model
Examining Eq. (4) and Fig. 1, it is first deduced that the motion of individual masses m is physically coupled through springs k, as in typical MLs.^{14,15} Having the momentum bias introduced by the rod's constant speed, the physics of the lattice's motion is influenced in two ways:

Besides the spring k, the ith unitcell displacement is coupled with its next neighbors through negative springs of coupling coefficient $ \u2212 m v 0 2 / a 2$, which remains negative regardless of the sign of v_{0} (or β). Consequently, the effective stiffness ( $ k e = k \u2212 m v 0 2 / a 2$) of the WML reduces as the rod's speed v_{0} increases, jeopardizing dynamical stability when $ m v 0 2 / a 2 > k$ (equivalent to $  \beta  > 1$) as a consequence of its effective stiffness k_{e} becoming negative.

Nonlocal coupling terms emerge, which are proportional to the velocity of nextneighboring masses of the ith unit cell. Such nonlocal interactions (related to Willis coupling^{16}) arise from discretizing the mixed derivative in Eq. (1) and dictate the strength of lattice's nonreciprocity. While the physical motion of the elastic medium enables such nonlocal coupling, it may be alternatively achieved via feedback control.^{17–19} Lastly, it is noteworthy that such nonlocal (positivenegative) couplings are akin to skewsymmetric gyroscopic coupling studied in literature.^{20–25}
2.2 Dispersion relation
2.3 Brillouinzone definition
Equations (10) and (13) uncover that the BZ in its entirety is shifted and it is within the range $ q \u2208 [ \u2212 \pi + \varphi , \pi + \varphi ]$, with its width remaining constant at $ 2 \pi $ in analogy to a reciprocal ML. Depending on the sign of β (and subsequently $\varphi $), the BZ moves forward or backward, as seen in the middle row of Fig. 2. Not only that nonreciprocity shifts the BZ, but it also forces the range of the wavenumber corresponding to the forwardgoing and backwardgoing waves to be unequal. To maintain a constant BZ width of $ 2 \pi $, the amount of shrinkage (or extension) in the forwardgoing wave wavenumber range (i.e., $ q \u2208 [ 0 , \pi + \varphi ]$) is compensated by a larger (smaller) wavenumber range of the backwardgoing waves (i.e., $ q \u2208 [ \u2212 \pi + \varphi , 0 ]$), depending on the sign of $\varphi $. This shift $\varphi $ disappears when the relative speed β is zeroed out as expected, which is verifiable from Eq. (13), and the traditional BZ range ( $ q \u2208 [ \u2212 \pi , \pi ]$) is recovered.
A few additional observations from the drivenwave dispersion in Fig. 2. (i) The attenuation at a frequency higher than the cutoff frequency $ \Omega max = 2$ is smaller in WMLs compared to a reciprocal ML. (ii) The real component of the wavenumber with $ \Omega max > 2$ in WMLs does not have a constant value and varies as the frequency increases, unlike its reciprocal counterpart with a constant real wavenumber of $ \xb1 \pi $ within the attenuation zone ( $ \Omega max > 2$). (iii) The BZ defined in MLs with β = 0 is symmetric about its center, allowing for an irreducible BZ to be defined within the range $ q \u2208 [ 0 , \pi ]$. As WMLs have asymmetric dispersion branches, reducing the first BZ may be elusive. A zero wavenumber, however, remains the point that divides the forward and backwardgoing waves for both lattices. Finally, the inequivalence of the forward and backward wavenumber ranges is numerically verified via a spatiotemporal fastFourier transform (FFT) of the time response of a finite WML (bottom row of Fig. 2), exhibiting excellent agreement with the analytical results of the newly defined BZ definition in the middle row (see supplementary notes 2 for more details).
3. Extension to twodimensional lattices
In a reciprocal square ML, the vertical (horizontal) boundaries of its perfectly square BZ correspond to a vanishing group velocity in the xdirection (ydirection) only. In extension, the same condition shall be imposed on the nonreciprocal WML case to find the horizontal and vertical boundaries of the shifted BZ. By doing so, it is found that the BZ for WML is no longer a square and has irregularly shaped boundaries. Yet, its area remains a constant value of $ 4 \pi 2$, identical to its reciprocal case, regardless of the magnitude of β_{x} and β_{y} [see Figs. 3(a), 3(b) and supplementary Fig. S3(b)].
Another way to depict the twodimensional dispersion relation is by using directivity plots and reducing the two variables q_{x} and q_{y} into a single (direction) angle equal to $ \psi = tan \u2212 1 ( q y / q x )$. In the reciprocal case, all four quadrants of the angle ψ show identical profiles of the dispersion surface, and the same cannot be said when $ \beta x \u2260 0$ and/or $ \beta y \u2260 0$ [Fig. 3(c)]. Complete dispersion surfaces in the directivity plot are generated if the wavenumber values within the nonreciprocal BZ boundaries [shown in Fig. 3(b)] are substituted to the dispersion relation, which is not the case if the traditional BZ values of the wavenumbers q_{x} and q_{y} are used instead [see Fig. 3(c) and supplementary Fig. S4]. Finally, a spatiotemporal FFT of the time response of a square WML of finite size is shown in Fig. 3(d) for $ \beta x = \beta y = 0$ (reciprocal case) and $ \beta x = 0.5$ and $ \beta y = 0.75$ (nonreciprocal case). The FFT contours exhibit a close agreement with their analytical counterparts in Fig. 3(c), substantiating the approach's validity and the completeness of the newly defined BZ (see supplementary note 2 for more details).
4. Concluding remarks
In summary, this study demonstrates that the Brillouin zone (BZ) remains constant in width/area with nonreciprocity in a class of Willis monatomic lattices (WMLs), yet its boundaries are shifted in a precisely quantifiable amount. Synthesized from a modified wave equation of longitudinal waves in moving elastic rod,^{12} the onedimensional WMLs is studied analytically and its drivenwave dispersion relation is proven vital for the quantification of the BZ shift $\varphi $. It is shown that the proposed theory perfectly agrees with the numerical simulation, validating BZ shifting, its constant width, and inequivalence of wavenumber ranges occupying the forwardgoing and backward going waves. It is also established that an increase (decrease) in the forwardgoing wavenumber range is compensated by a shrinkage (enlargement) in the backwardgoing one. An extension to a square WML is also established, demonstrating the shifted nature of the BZ and its area being constant and unaffected by the degree of nonreciprocity, dictated by the velocity modulation in the x and y direction. Additionally, it is shown that the twodimensional BZ is no longer a square, and its deformed vertical (horizontal) boundaries are found by setting the group velocity in the x (y) propagation direction to zero. The established clarification on the definition of BZs is envisioned to be a stepping stone for further investigations in BZ quantification for different types of modulations for wave nonreciprocity, as well as for periodic media with multimode dispersion relations.
Supplementary Material
See supplementary material at https://doi.org/10.1121/10.0022535 for a discussion on group velocity and BZ phase shift, numerical validation procedure, and derivations related to square WMLs (notes 1, 2, and 3, respectively). In addition, supplementary video animations are provided for numerical simulations of onedimensional WMLs, as well as BZ definition of square WMLs with various values of parameters β_{x} and β_{y}.
Author Declarations
Conflicts of interest
The author has no conflicts to disclose.
Data Availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.