# One-way Acoustic Guiding under Transverse Fluid Flow

Ohad Silbiger and Yakir Hadad<EMAIL_ADDRESS>School of Electrical
Engineering, Tel-Aviv University, Ramat-Aviv, Tel-Aviv, Israel, 69978

###### Abstract

In a moving acoustic medium, sound waves travel differently with and against
the fluid flow. This well-established acoustic effect is backed by the
intuition that the fluid velocity bias imparts momentum on the propagating
acoustic waves, thus violating reciprocity. Based on this conception, fluid
flow that is transverse to the wave direction of propagation will not break
reciprocity. In this letter we contrast this common wisdom and theoretically
show that the interplay between transverse mean flow and transverse structural
gliding-asymmetry can yield strong nonreciprocity and even, surprisingly, one-
way waveguiding which is a rare in acoustics.

###### pacs:

Valid PACS appear here

††preprint: APS/123-QED

Introduction.—Acoustic nonreciprocity has gained a lot of attention in recent
years due to its numerous potential applications [1, 2]. It can be achieved
using nonlinearities [3, 4, 5], and by using active elements [6, 7, 8], but
nevertheless, nonreciprocal acoustic and phononic propagation is mostly known
to occur in moving media [9, 10, 11, 12, 13, 14], or some other form of
internal motion such as rotation [15]. To illustrate that, assume that two
friends, Alfred and Beth, shown in Fig. 1(a), are located at points A and B on
the $x$ axis, the distance between them $L$, is acoustically large, and under
this high frequency limit they communicate via _local_ plane waves that travel
along the $x$ axis in a fluid with uniform mean flow velocity $U_{0}\hat{x}$,
and acoustic wave speed $c$. Using simple kinematic arguments, the time it
takes for a signal from Alfred to reach Beth is different then the time it
takes for a signal to propagate in the reciprocal direction:
$\tau_{d}^{A\leftrightarrow B}=L/(c\pm U_{0})$. Clearly, in light of this
result, nonreciprocity that is caused due to a collinear fluid flow is rather
weak at low Mach numbers, $q_{0}=U_{0}/c\ll 1$. Nevertheless, in the presence
of sharp resonances the effect may be enhanced giving rise even to isolation.
This has been demonstrated, for example, using resonant cavities [12, 16, 17,
18], waveguides connected in a sensitive Mach-Zehnder interferometer setup
[19] and near zero index metamaterial waveguides [20]. Moreover, faster
_synthetic_ motion and thereby stronger nonreciprocity, may be emulated by
space-time modulated acoustic metamaterials [21, 22, 23, 24, 25, 26].

Figure 1: (a) When the medium between Alfred and Beth flows parallel to the
plane waves that propagate between them, different behavior is expected when
the propagation is from left to right or vice versa. Since only the medium is
flowing, while Alfred and Beth are stationary, the wave _frequency_ as
observed by them does not experience a Doppler-shift. However, due to the
medium flow, the _wavelengths_ of the two counter-propagating waves are
different [11]. (b) In contrast, if the medium between Alfred and Beth flows
transversely to the plane waves that propagate between them, these plane waves
are expected to propagate reciprocally [27].

In contrast, if, as illustrated in Fig. 1(b), the fluid flows _transverse_ to
the direction of propagation, so the flow velocity is $U_{0}\hat{z}$, the time
delay for a _local plane wave_ to propagate between Alfred and Beth is
$\tau_{d}^{A\leftrightarrow B}=L/c$, independent of the direction of
propagation, as in a stationary fluid [27]. The exact results for time of
flight in both cases (longitudinal vs transverse flow) are highly intuitive
from the kinematic point of view which implies, allegedly, that in the
presence of a uniform mean flow, the communication between Alfred and Beth
will be nonreciprocal only if the wave that propagates between them has some
wavevector component that is parallel to the fluid stream.

In contrast with this common wisdom, in this letter we show that under certain
conditions it is possible to achieve strong nonreciprocity also for waves that
are propagating along a direction that is _transverse_ to the uniform flow
velocity of the ambient medium. As we show below, this is achieved by the
interplay between the transverse flow and structural the transverse gliding-
asymmetry of acoustic scatterers. Interestingly, this effect can be explained
only by considering the _complete field_ interaction of the scatterers and its
effect on the travelling mode. Remarkably, by this way we demonstrate below
strong nonreciprocity and even _one-way guiding_ that is due to _transverse
flow_ biasing. In a striking contrast with the common perception that is
illustrated in Fig. 1(b).

Description of the model.—We consider a coupled resonator waveguide that
comprises of $N$ periodic linear chains of scatterers. The waveguide is
located on the $y=0$ plane, parallel to the $x$-axis, and consists of a
periodic arrangement of scatterers that are equally spaced with acoustically-
small inter-scatterer spacing $d<\lambda/2$, where $\lambda$ is the acoustic
wavelength. See illustration in Fig. 2 with $N=5$. The $n$’th scatterer of
chain number $i$ ($i=1..N$) is located at
$\mathbf{r}_{n}^{i}=(d_{x}^{i}+nd,0,d_{z}^{i})$. Here, $d_{z}^{i}=(i-1)d_{z}$
denotes the $z$ coordinate of the $i$’th chain, where $d_{z}^{1}=0$ so that
the first ($i=1$) chain is located right on the $x$-axis, and the spacing
between the chains is $d_{z}$. And, $d_{x}^{i}$ is the _gliding_ distance of
the $i$’th chain with respect to the origin. For the first chain,
$d_{x}^{1}=0$. Thus, the relative gliding between the chains reads
$d_{x}^{ij}=d_{x}^{j}-d_{x}^{i}$. The entire structure is located inside a
duct with a _quiescent_ (stationary) medium that is sandwiched between two
half-spaces with an uniform flow with velocity
$U_{0}\hat{z}=q_{0}c_{0}\hat{z}$, _transverse_ to the coupled resonator
waveguide axis. We assume that the density, $\rho_{0}$, and the wave velocity,
$c_{0}$ are uniform in the entire space. Thus, the duct is created merely due
to the inhomogeneous medium flow velocity. This configuration may be achieved
in practice using a thin, acoustically transparent, membranes placed at the
interfaces, $y=-L/2$ and $y=L/2$, as illustrated in the left panel of Fig. 2.
We stress that by using this duct configuration we completely _avoid the
possible issue of turbulent flow_ at high Raynolds numbers which could yield a
porus-media like behaviour that may be expected by the presence of the
acoustically dense scatterers, had the flow was applied in the _entire_ space.
This implies that in the suggested configuration, the medium flow lines are
necessarily not affected by the lattice of scatterers, and hence they are
fully transverse to the waveguide axis. The boundary condition in the scenario
suggested in Fig. 2, at the interfaces $y=\pm L/2$, has been extensively
studied, for example in [28], and experimentally explored in [29].

Figure 2: A periodic waveguide with transverse asymmetry (left - side view,
right - top view). As an example, the waveguide consists of five chains of
acoustic scatterers, and is located inside a _stationary_ medium, sandwiched
between two half-spaces with mean transverse flow with velocity
$U_{0}\hat{z}$. $d$ is the inter-scatterer spacing along each one of the
chains, and $d_{z}$ is the spacing between the chains. The lattice gliding-
asymmetry is determined by the chain gliding parameters - $d_{x}^{1j}$,
$j=2,3,4,5$. The waveguide is infinite in the $x$ direction, only three unit
cells are shown.

Modelling the acoustic scatterers.—We assume that the acoustic scatterers may
exhibit either dominant monopole or dipole response. These scatterers may be
passive, for example Helmholtz resonators [11] and variations over them [30],
where the structural properties of the scatterer determine its response, or
alternatively, convenient tunable acoustic meta-atoms may be utilized [31].
The latter are essentially active scatterers that consist of a transducer that
senses the pressure wave incident on the meta-atom and an electronic feedback
circuit that manipulates the electric signal produced and drives a second
transducer that creates the desired acoustic response which may be of a
monopole scatterer, a dipole scatterer, and in principle of any multipole.
Thus, for the sake of generality, we assume that the scatterers are
characterized by their acoustic susceptibilities, $\alpha^{mm}$ and
$\alpha^{dd}_{\xi}$, which link the _local field_ , i.e., the field in the
scatterer’s location but at the absence of the scatterer itself, to the
resulting scattering response. Specifically, the monopole is characterized by
a volume $V$ (units: $m^{3}$) and is induced by a local pressure $P$, while
the dipole is characterized by a dipole moment $D_{\xi}$ (units: $m^{4}$) and
is induced by the space derivative of the local pressure $\partial
P/\partial\xi$, in the direction $\xi=x,y,z$. Thus,

$V=\alpha^{mm}P\qquad,\qquad
D_{\xi}=\alpha^{dd}_{\xi}\frac{\partial{P}}{\partial{\xi}}.$ (1)

We use a Lorentzian model for the susceptibilities, i.e.,
$1/\alpha=A(\omega^{2}-\omega_{r}^{2})-j\omega\gamma$ where $\omega_{r}$
denotes the resonance frequency, $A$ takes the role of the oscillator
strength, and $\gamma$ encapsulates the resonator damping. Here $\alpha$
stands for either $\alpha^{mm}$ or $\alpha_{\xi}^{dd}$ for monopole and dipole
scatterers, respectively. Specifically, the radiation loss for a monopole
scatterer is given by $\gamma=\gamma^{mm}=\omega^{2}\rho_{0}/4\pi c$, while
for a dipole scatterer it reads
$\gamma=\gamma^{dd}_{\xi}=\omega^{4}\rho_{0}/12\pi c^{3}$ (see [32] and Sec. I
in [33]). Additional loss mechanisms may be added to $\gamma$. Specific values
for the resonators’ resonance frequency and strength are given in the
numerical examples below. Once the _induced source_ on the scatterer is known,
the scattered field generated by the scatterer is given via the corresponding
Green’s function that takes into account the wave interaction at the
boundaries with the uniform flow domains. As an example, for a monopole
scatterer that is located at ${\bf{r}}^{\prime}=(0,0,0)$, the scattered
acoustic pressure on the $y=0$ plane is given by
$P_{scat}({\bf{r}})=G^{mm}({\bf{r}})V$, where [33] (Sec. II)

$G^{mm}({\bf{r}})\\!=\\!\iint_{-\infty}^{\infty}\\!\\!g^{mm}(k_{x},k_{z})e^{-j(k_{x}x+k_{z}z)}dk_{x}\,dk_{z}.$
(2)

The spectral green’s function in Eq. (2) reads,

$g^{mm}(k_{x},k_{z})=\frac{j\omega^{2}\rho_{0}}{8\pi^{2}k_{y}}\frac{1+Re^{-jk_{y}L}}{1-Re^{-jk_{y}L}}$
(3)

with

$R=\frac{k_{2}^{2}k_{y1}-k_{1}^{2}k_{y2}}{k_{2}^{2}k_{y1}+k_{1}^{2}k_{y2}}$
(4)

and where $k_{y1,2}=\sqrt{k_{1,2}^{2}-k_{x}^{2}-k_{z}^{2}}$, with
$k_{1}=\omega/c_{0}$ and $k_{2}=(\omega-U_{0}k_{z})/c_{0}$, in the stationary
medium and in the uniform flow domains, respectively. Additional Green’s
functions, for different fields, and due to additional sources, are listed in
Ref. [33].

Guided modes.—Given a specified spatial configuration of the waveguide, such
as in Fig. 2, our interest is to find the guided modes that may propagate
through it. To that end, we consider an infinite lattice. Due to its
structural periodicity, the _induced sources on the scatterers_ have to
respect a Bloch-form, i.e., exhibiting a _collective_ response with the
following wave behavior: $X^{i}_{n}=X^{i}_{0}e^{-j\beta nd}$, where $\beta$ is
the unknown propagation constant of the eigenmode. Henceforth we use
$X_{n}^{i}$ to denote the induced source moment on the $n$’th scatterer in
chain number $i$, which may be a monopole ($V$) or a dipole ($D_{x}$ or
$D_{z}$).

Given two scatterers, with indexes $n$ and $m$ on chains number $i$ and $j$,
i.e., $X_{n}^{i}$ and $X_{m}^{j}$, respectively, that are located at
$\mathbf{r}_{n}^{i}=(d_{x}^{i}+nd,0,d_{z}^{i})$ and
$\mathbf{r}_{m}^{j}=(d_{x}^{j}+md,0,d_{z}^{j})$ on the $y=0$ plane (See Fig.
2). We denote by $G^{ij}(\mathbf{r}_{n}^{i},\mathbf{r}_{m}^{j})$ the Green’s
function that relates the induced source $X_{n}^{i}$ to the _field that
excites_ $X_{m}^{j}$. For example, if the scatterers are both monopoles, i.e.,
$X_{n}^{i}=V_{n}^{i}$, $X_{m}^{j}=V_{m}^{j}$, then $G^{ij}$ is defined as the
Green’s function that connects a monopole source to the excited pressure wave,
that is,
$P(\mathbf{r}_{n}^{i})=G^{ij}(\mathbf{r}_{m}^{j},\mathbf{r}_{n}^{i})V_{n}^{i}$.
Note that since all the scatterers are located on $y=0$, and since the duct
structure (in the absence of the scatterers) is shift invariant on the $xz$
plane, then
$G^{ij}(\mathbf{r}_{m}^{j},\mathbf{r}_{n}^{i})=G^{ij}(\mathbf{r}_{m}^{j}-\mathbf{r}_{n}^{i})$.

Next, we close the loop, and write the equations that govern the excitation of
the scatterers on chain number $j$. In light of the structural periodicity, it
is enough to write the equation for $X_{0}^{j}$. Using its susceptibility, and
using the local field at its location, $E^{L}({\bf{r}}_{0}^{j})$,

$X_{0}^{j}=\alpha^{j}E^{L}({\bf{r}}_{0}^{j})=\alpha^{j}\sum_{i=1}^{N}S^{ij}X_{0}^{i}.$
(5)

Here, $S^{ij}X_{0}^{i}$ denotes the field at $\mathbf{r}_{0}^{j}$ due to all
the scatterers at chain number $i$. $S^{ij}$ is expressed in terms of
$G^{ij}$,

$S^{ij}(\beta)=\sum_{n=-\infty}^{\infty}e^{-j\beta
nd}\,G^{ij}(\mathbf{r}_{0}^{j}-\mathbf{r}_{n}^{i}).$ (6)

where $\mathbf{r}_{m}^{j}-\mathbf{r}_{n}^{i}=(d_{x}^{ij}-nd,0,d_{z}^{ij})$.
Note that $G^{ij}(\mathbf{r}_{0}^{j}-\mathbf{r}_{n}^{i})$ consists of _two
contributions_ , a _primary_ wave that would be excited in an infinite
homogenous medium with no flow, and a _secondary_ wave that is reflected by
the boundaries with the moving medium. Importantly, for $i=j$, and when $n=0$
in Eq. (6), $G^{ii}(\mathbf{r}_{0}^{i}-\mathbf{r}_{0}^{i})$ consists of only
the secondary wave field. This is because the local field
$E^{L}({\bf{r}}_{0}^{j})$ is by definition the field at the particle location
but in the absence of the particle _itself_. Reradiation effect of the
scatterer on itself though the boundaries are essential. The numerical
evaluation of Eq. (6) is discussed in Ref. [33] (Sec. III). By rewriting Eq.
(5) for all $j=1..N$, a self-consistent linear system that encapsulates the
modal dynamics of the waveguide with $N$ parallel chains is obtained,

$\mathbf{X}_{0}=\underline{\underline{\alpha}}(\omega)\underline{\underline{S}}(\omega,\beta)\mathbf{X}_{0}$
(7)

where $\mathbf{X}_{0}=[X_{0}^{1},..,X_{0}^{N}]^{T}$,
$\underline{\underline{\alpha}}=\mbox{diag}[\alpha^{1},..,\alpha^{N}]$, and
$\underline{\underline{S}}=\\{S^{ij}\\}$ with $i=1..N,j=1..N$. The waveguide
modes are the nontrivial solutions of Eq. (7) and as such satisfy the
dispersion equation

$\mbox{det}[\underline{\underline{\alpha}}(\omega)^{-1}-\underline{\underline{S}}(\omega,\beta)]=0$
(8)

that its solutions are $\beta(\omega)$. Real $\beta$ solutions will exist only
if $d<\lambda/2$. Therefore implying sub-wavelength mode width as expected by
a sub-diffraction waveguide (see [34, 35, 36] for akin waveguides in optics).

Nonreciprocity and one way guiding.—In the absence of flow, $q_{0}=0$,
reciprocity is expected. Mathematically, this is evident by the symmetry (or
anti-symmetry) of all Green’s functions $G^{ij}$ with respect to $x$ and $z$
which implies
$\underline{\underline{S}}(-\beta)=\underline{\underline{S}}^{T}(\beta)$.
Therefore, in light of the form of Eq. (8), for any given frequency $\omega$,
if $\beta$ solves the dispersion equation, so does $-\beta$. The same argument
is true even in the presence of transverse flow, $q_{0}\neq 0$, if the
waveguide contains _only zero or half-step gliding_ for any number of chains,
i.e., if $\forall i\neq j\quad d_{x}^{ij}=0$ or $d_{x}^{ij}=d/2$. In this case
the waveguide’s transverse spatial symmetry enforces reciprocal wave guiding.
Detailed derivation of the symmetry conditions is in [33] (Sec. IV).

Instead, as we show below, it requires to have _simultaneously transverse flow
and spatial asymmetry_ in the form of gliding in order to breach reciprocity
and get different propagation characteristics for counter propagating waves
along the waveguide axis.

Figure 3: Dispersion plots for a waveguide comprised of 5 scatterer chains -
two chains of monopoles and three chains of longitudinal dipoles. (a) With no
flow above and below the duct and no chain gliding the propagating mode is
symmetric, both solutions have equal phase and group velocities, but in
opposite directions. (b) Adding transverse flow while maintaining zero gliding
changes the dispersion plots very slightly, without affecting the reciprocal
nature of the waveguide. (c) Introducing chain gliding in a quiescent medium
changes the dispersion curves, but does not break reciprocity. (d) Combination
of chain gliding-asymmetry and transverse flow generates substantial
nonreciprocity in the waveguide.

The aforementioned discussion is summarized by the dispersion plots that are
given in Fig. 3. In this case the structure consists of total of five coupled
linear arrays, two with monopole scatterers and three with longitudinal dipole
scatterers. The linear arrays are shifted relative to each another to produce
a gliding asymmetry. The medium is uniformly characterized as air with
$c=343$m/s and $\rho_{0}=1.2\mbox{kg/m}^{3}$. The monopoles resonate at a
frequency of $f^{mm}_{r}=f_{0}=1125$Hz, while the dipoles exhibit no resonance
around this frequency, but around $f^{dd}_{z}=3377$Hz. The resonator strength
for the monopole $A^{mm}=10\mbox{kg/m}^{4}$, and for the dipoles
$A^{dd}_{x}=10^{-3}\mbox{kg/m}^{6}$. See [33] (Sec. I) for susceptibility
plots. The thickness of the layer between the flowing media was set to quarter
of a wavelength at frequency $f_{0}$, that is $L=7.5$cm in Fig. 2. The lattice
periodicity is taken to be $d=6$cm, the transverse inter-chain spacing is
$d_{z}=2$cm, and the gliding values are: $d_{x}^{12}=0.66d$,
$d_{x}^{13}=0.5d$, $d_{x}^{14}=0.33d$, and $d_{x}^{15}=0.33d$. By solving Eq.
(8) we find the dispersion relation $\omega(\beta)$ that is plotted in Fig. 3.
In the absence of medium flow ($q_{0}=0$) _or_ when the structure is
transversely symmetric (for all $i,j$ $d_{x}^{ij}=0$ or $d/2$), the dispersion
relation is symmetric (in $\beta$), as shown in Fig. 3(a)-(c), and the
waveguide is reciprocal. It is only when transverse flow is introduced
simultaneously with a transversely gliding-asymmetric structure, then
reciprocity in the longitudinal direction is broken and the dispersion
relation becomes asymmetric as shown in Fig. 3(d) with Mach number
$q_{0}=0.1$. In this case, remarkably, we observe certain frequency bands (See
the colored regions in Fig. 3(d)) at which a real $\beta$ solution exits only
for modes that are propagating to one side. At these bands the transversely
biased waveguide acts as a longitudinal one-way waveguide. We note that these
one-way guiding bands in the dispersion cannot be found by using a mere
kinematic calculation. This is demonstrated in detail in [33](Sec. V) using a
coupled mode model with non-reciprocal coupling coefficients due to the
transverse flow. Instead, a detailed calculation that takes carefully into
account the complete acoustic field form is essential.

To further demonstrate the phenomenon and validate our dispersion results, a
finite waveguide with 400 unit-cells is excited by applying a _localized_
external pressure field that oscillates at $\omega=1.6\omega_{0}$ on a
monopole scatterer in the middle of chain $i=1$. At this excitation frequency,
with flow velocity with $q_{0}=0.1$, the dispersion diagram in Fig. 8(d)
predict a single real solution with negative group velocity
($d\omega/d\beta$). Thus, we expect in the excitation problem to see
propagation to the left, and an evanescent wave to the right. The numerical
excitation results of the finite lattice are shown in Fig. 4, while the
numerical details are omitted here for brevity and are provided in [33] (Sec.
VI). These results nicely demonstrate how under transverse mean flow,
longitudinal one-way leftward propagation of the acoustic wave is obtained,
while rightward propagation is practically prohibited. Instead, rapid
exponential decay takes place in the forbidden direction, indicating on a
complex leaky mode that is excited in this direction. A detailed analysis of
this excitation problem should follow Green’s function development for the
waveguide under study here. Related work may be found in [37], and in [38].
The latter discusses the Green’s function of a one-way plasmonic sub-
diffractive waveguide under magnetic biasing. Note that additional examples
using a waveguide lattice that comprises of monopoles only, and including
additional loss terms are given in [33](SEc. VII).

Figure 4: Amplitudes of monopole/dipole strength along a chain with 400
scatterers, due to a local external pressure field $P_{inc}$ applied on a
single monopole scatterer in the middle of chain $i=1$. Color coded: blue -
$V_{n}^{1}$, orange - $D_{z_{n}}^{2}$, yellow - $D_{z_{n}}^{3}$, magenta -
$V_{n}^{4}$, green - $D_{z_{n}}^{5}$. The lattice parameters used are the same
as in Fig. 3(d), with Mach number $q_{0}=0.1$ and at frequency
$\omega=1.6\omega_{0}$. Only propagation to the left is allowed consistent
with the negative group velocity obtained from the dispersion diagram in Fig.
3(d) at this frequency.

Conclusions.—We have theoretically shown that an acoustic waveguide that is
placed in the _vicinity_ of a flow with uniform velocity _transverse_ with
respect to the waveguide axis can exhibit substantial nonreciprocity and even
one-way guiding. This counter-intuitive phenomenon stems from the interplay
between the structural transverse gliding asymmetry and the transverse
nonreciprocal interaction between the scatterers that comprise the waveguide.
In that sense this phenomenon may be regarded as the acoustic analog to one-
way optical waveguiding that is based on the Voigt magneto-optical
configuration such as in [39]. Lastly, due to the similarity between the
physical mechanisms, we expect that our proposed concept for one-way
waveguides may be extended also to systems with transverse _synthetic motion_
along the waveguide cross section, (e.g., by space-time modulation), and
thereby also to other physical realms such as electromagnetics and optics.

###### Acknowledgements.

This research was supported by the Israel Science Foundation (grant No.
1353/19). Y.H would like to thank Prof. Eldad Avital for insightful
discussions.

## References

  * [1] R. Fleury, D. Sounas, M. Haberman, A. Alú, “Nonreciprocal Acoustics,” _Acoustics Today_ 11(3), pp. 14-21 (2015).
  * [2] H. Nassar, B. Yousefzadeh, R. Fleury, M Ruzzene, A. Alú, C. Daraio, A. N. Norris, G. Huang, M. R. Haberman, “Nonreciprocity in acoustic and elastic materials,” _Nat. Rev. Mater._ 5, pp. 667–685 (2020).
  * [3] B. Liang, B. Yuan, J. C. Cheng, “Acoustic Diode: Rectiﬁcation of Acoustic Energy Flux in One-Dimensional Systems,” _Phys. Rev. Lett._ , 103, 104301 (2009).
  * [4] N. Boechler, G. Theocharis, C. Daraio, “Bifurcation-based acoustic switching and rectification,” _Nat. Mat._ , 10, pp. 665-668, (2011).
  * [5] J. Cui, T. Yang, and L. Q. Chen, “Frequency-preserved nonreciprocal acoustic propagation in a granular chain,” _Appl. Phys. Lett._ , 112 (18), 181904, (2018).
  * [6] B. I. Popa, S. A. Cummer, “nonreciprocal and highly nonlinear active acoustic metamaterials,” _Nat. Comm._ 5, 3398 (2014).
  * [7] X. Guo, H. Lissek, R. Fleury, “Improving Sound Absorption Through Nonlinear Active Electroacoustic Resonators,” _PhysRevApplied_ 13, 014018 (2020).
  * [8] Y. Zhai, H. S. Kwon, B. I. Popa, “Active Willis metamaterials for ultra-compact nonreciprocal linear acoustic devices,” _Phys. Rev. B_ , 99, 220301(R) (2019). Godin1997, Wiederhold2019, Morse, Fleury2014, Auregan2017, Auregan2015
  * [9] O. Godin, “Reciprocity and energy theorems for waves in a compressible inhomogeneous moving fluid,” _Wave Motion_ , 25(2) pp. 143-167 (1997).
  * [10] C. P. Wiederhold, D. L. Sounas, A. Alú, “Nonreciprocal acoustic propagation and leaky-wave radiation in a waveguide with flow”, _J. Acoust. Soc. Am._ 146 (2019).
  * [11] Philip M. Morse, and K. Uno Ingrad, _theoretical acoustics_ , McGraw-Hill, New York NY, (1968).
  * [12] R. Fleury, D. Sounas, C. Sieck, M. Haberman, A. Alú,“Sound Isolation and Giant Linear Nonreciprocity in a Compact Acoustic Circulator,” _Science_ , 343 (2014).
  * [13] Y. Aurégan, V. Pagneux, “$\mathcal{PT}$ Symmetric Scattering in Flow Duct Acoustics,” _Phys. Rev. Lett._ , 118, 174301 (2017).
  * [14] Y. Aurégan, P. Fromholz, F. Michel, V. Pagneux, R. Parentani, “Slow sound in a duct, effective transonic flows, and analog black holes,” _Phys. Rev. D_ , 92, 081503(R) (2015).
  * [15] P. Wang, L. Lu, K. Bertoldi, “Topological Phononic Crystals with One-Way Elastic Edge Waves,” _Phys. Rev. Lett_ 115, 104302 (2015).
  * [16] Y. Ding, Y. Peng, Y. Zhu, X. Fan, J. Yang, B. Lang, X. Zhu, X. Wan, J. Cheng, “Experimental Demonstration of Acoustic Chern Insulators,” _Phys. Rev. Lett._ 122 014302 (2019).
  * [17] Z. Yang, X. Shi, X. Lin, Z. Gao, Y. Chong, B. Zhang, “Topological Acoustics,” _Phys. Rev. Lett._ 114, 114301 (2015).
  * [18] R. Fleury, D. Sounas, A. Alú, “Subwavelength ultrasonic circulator based on spatiotemporal modulation”, _Phys. Rev. B_ , 91, 174306 (2015).
  * [19] N. Jankovic, A. Alú, “Glide-Symmetric Acoustic Waveguides for Extreme Sensing and Isolation,” _Phys. Rev. Applied_ 15, 024004 (2021).
  * [20] L. Quan, D. Sounas,A. Alú, “Nonreciprocal Willis Coupling in Zero-Index Moving Media,” _Phys. Rev. Lett._ , 123, 064301 (2019).
  * [21] Y. Wang, B. Yousefzadeh, H. Chen, H. Nassar, G. Huang, C. Daraio, “Observation of Nonreciprocal Wave Propagation in a Dynamic Phononic Lattice,” _Phys. Rev. Lett._ , 121, 194301 (2018).
  * [22] C. Croënne, J. O. Vasseur, O. Bou Matar, A.-C. Hladky-Hennion, B. Dubus, “Nonreciprocal behavior of one-dimensional piezoelectric structures with space-time modulated electrical boundary conditions,” _J. Appl. Phys._ 126, 145108 (2019).
  * [23] J. Huang, X. Zhou, “A time-varying mass metamaterial for nonreciprocal wave propagation,” _Int. J. Sol. Struc._ 164 pp. 25-36 (2019).
  * [24] H. Nassar, X. C. Xu, A. N. Norris, G. I. Huang, “Modulated phononic crystals: nonreciprocal wave propagation and Willis materials,” _J. Mech. Phy. Solids_ , 101, pp. 10-29 (2017).
  * [25] N. Swinteck, S. Matsuo, K. Runge, J. O. Vasseur, P. Lucas, P. A. Deymier, “Bulk elastic waves with unidirectional backscattering-immune topological states in a time-dependent superlattice,” _J. Appl. Phys._ 118 (2015).
  * [26] R. Fleury, A. B. Khanikaev, A. Alú, “Floquet topological insulators for sound”, _ncomms_ 7 (2016).
  * [27] To see that, we write the wave equation for the pressure field in a fluid with uniform mean flow velocity $U_{0}\hat{z}$,  $(\partial_{t}+U_{0}\partial_{z})^{2}p=c^{2}\nabla^{2}p.$ (9) Considering propagation only on the $xz$ plane, its monochromatic plane wave solution is $p(x,z,t)=p_{0}e^{-j(k_{x}x+k_{z}z)}e^{j\omega t}$, and subject to the dispersion relation  $k_{z}^{2}+k_{x}^{2}=k^{2}-2\frac{kU_{0}}{c}k_{z}+\frac{U_{0}^{2}}{c^{2}}k_{z}^{2}.$ (10) Clearly, for a plane wave that propagates parallel to the $x$-axis [$z$-axis], $k_{z}=0$ [$k_{x}=0$], and therefore $k_{x}=\pm k$ [$k_{z}=k/(q_{0}\pm 1)$], where $k=\omega/c$ is the wavenumver in the stationary medium and $q_{0}$ is the Mach number. Then, the phase velocity reads $v_{p}=\omega/k_{x,z}$, and thus $\tau_{d}^{A\leftrightarrow B}=L/v_{p}$.
  * [28] J.W. Miles, “On the Reflection of Sound at an Interface of Relative Motion,” _J. Acoust. Soc. Am._ , 29, pp. 226–228 (1957).
  * [29] R. Amiet, “Correction of open jet wind tunnel measurements for shear layer refraction,” In _2nd Aeroacoustics Conference_ , doi: 10.2514/6.1975-532, eprint: https://arc.aiaa.org/doi/pdf/10.2514/6.1975-532.
  * [30] L. Quan, Y. Ra’di, D.L. Sounas, A. Alú, “Maximum Willis Coupling in Acoustic Scatterers,” _Phys. Rev. Lett._ 120, 254301 (2018).
  * [31] B.I. Popa, L. Zigoneanu, S.A. Cummer, “Tunable active acoustic metamaterials,” _Phys. Rev. B._ 88, 024303 (2013).
  * [32] H. Esfahlani, Y. Mazor, and A. Alù, _Homogenization and design of acoustic Willis metasurfaces_ , Phys. Rev. B 103, 054306 (2021).
  * [33] Supplementary material
  * [34] M. Quinten, A. Leitner, J. R. Krenn, F. R. Aussenegg, “Electromagnetic energy transport via linear chains of silver nanoparticles,” _Opt. Lett._ 23 (17) pp. 1331-1333 (1998).
  * [35] A. Alú, N. Engheta, “Theory of linear chains of metamaterial/plasmonic particles as subdiffraction optical nanotransmission lines,” _Phys. Rev. B_ 74, 205436 (2006).
  * [36] Y. Hadad, B. Z. Steinberg, “Magnetized Spiral Chains of Plasmonic Ellipsoids for One-Way OpticalWaveguides,” _Phys. Rev. Lett._ 105, 233904 (2010).
  * [37] Y. Hadad and Ben Z. Steinberg, “Greens function theory for infinite and semi-infinite particle chains,” _Phys. Rev. B_ , 84, 125402 (2011).
  * [38] Y. Hadad, Y. Mazor, and Ben Z. Steinberg, “Greens function theory for one-way particle chains,” _Phys. Rev. B_ , 87, 035130 (2013).
  * [39] Y. Mazor, Y. Hadad, and Ben Z. Steinberg, “Planar one-way guiding in periodic particle arrays with asymmetric unit cell and general group-symmetry considerations,” _Phys. Rev. B_ 92, 125129 (2015).
  * [40] Namely, the field that is back-reflected to the location of the 0’th element by the boundaries. See [33].