Sound Waveguiding by Spinning: An Avenue toward Unidirectional Acoustic Spinning Fibers
Item Type Article
Authors Farhat, Mohamed;Chen, Pai-Yen;Wu, Ying
Citation Farhat, M., Chen, P.-Y., & Wu, Y. (2023). Sound Waveguiding by Spinning: An Avenue toward Unidirectional Acoustic Spinning Fibers. Physical Review Applied, 19(4). https://doi.org/10.1103/
physrevapplied.19.l041002 Eprint version Publisher's Version/PDF
DOI 10.1103/physrevapplied.19.l041002 Publisher American Physical Society (APS) Journal Physical Review Applied
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Supplementary Material for ”Sound waveguiding by spinning: An avenue toward unidirectional acoustic spinning fibers”
Mohamed Farhat,1 Pai-Yen Chen,2 and Ying Wu1,∗
1Computer, Electrical, and Mathematical Science and Engineering (CEMSE) Division, King Abdullah University of Science and Technology
(KAUST), Thuwal 23955-6900, Saudi Arabia
2Department of Electrical and Computer Engineering, University of Illinois at Chicago, Chicago, Illinois 60607, USA
(Dated: April 3, 2023)
Abstract
This Supplementary Material details the derivation of the governing equations for pressure acous- tics in spinning media as well as some results with finite-element method (FEM, via COMSOL Multiphysics). It also describes the Hamiltonian formulation that shows the Zeeman acoustic analogue, which derives from the symmetry breaking induced by spinning.
CONTENTS
I. Background and problem setup 2
II. Origin of the nonreciprocity: Acoustic Zeeman effect 6
A. Hamiltonian formulation 6
B. Eigensolutions for the unbiased system 7
C. Eigensolutions for the biased system 8
III. Dispersion of the ASF 9
IV. COMSOL modeling and potential experimental realization 13
A. FEM governing equations 13
B. FEM-validations of the ASF waveguiding 15
C. Potential experimental realization 16
References 17
I. BACKGROUND AND PROBLEM SETUP
The basic conservation equations of acoustics at rest are momentum conservation and mass conservation, i.e.,
ρ∂u
∂t =−∇P , β˜∂P
∂t =−∇ ·u, (1)
with ρ denoting the density and ˜β the compressibility of the background ( ˜β = 1/˜κ, with ˜κ denoting the bulk modulus of the medium). These equations are supplied with boundary
conditions, i.e., continuity of the normal component of the velocity ˆn·u, with ˆn being the normal to the boundary and of the pressure p.
Consider a medium that is uniformly rotating [with rotation axis coinciding with ˆez, as schematized in Fig. 1 of the manuscript (MS)] at angular velocity Ω. As shown in Refs.
[1–6], the first part of Eq. (1) has to be re-written as ρDu
Dt′ =−∇′P , (2)
with ∇′ the del operator and D/Dt′ the total time-derivative in the co-spinning frame R′ with unit vectors ˆeϕ′ and ˆer′. A realistic approximation is to consider that the density ρ is static (assuming reasonable rates of rotation and a low compressibility, in addition to low-amplitude acoustic perturbations [1]). The parameterP denotes the total pressure, i.e., the sum of the pressure due to acoustic waves and the one due to rotations of the object, whereasudenotes the total velocity. It can be shown that in the static (non-spinning) frame of reference R, Eq. (2) is transformed into [1, 3, 7]
∂
∂t + (u0 · ∇)
v+ (v· ∇)u0 =−ρ−1∇p , (3) whereu=u0+v. Herevdenotes the acoustic velocity andu0 = Ωreϕ for uniform spinning with angular velocity Ω. We assume in this derivation that P =p0+p, with p0 the pressure induced by the rotation of the frame and pthe acoustic wave pressure. A similar derivation can lead to the modified mass conservation equation, i.e.,
∂
∂t+ (u0· ∇)
p+c2ρ∇ ·v= 0, (4)
taking advantage of the relation ∇ ·u0 = 0 by defining the velocity c= 1/
q
ρβ˜[1, 7].
In these equations, as mentioned, we neglect the effect of viscosity. Adding viscosity would change the momentum conservation equation by adding a term of the formµ∆u, i.e., Eq. (3) may be re-written as
∂
∂t+ (u0· ∇)
v+ (v· ∇)u0 =−ρ−1∇p+µ ρ∆v,
with µ the dynamic viscosity of air. To compare orders of magnitude of each term we can replace time derivatives (harmonic regime) with multiplication byiωand spatial derivatives by multiplication byik. By doing so, it can be shown that the viscosity term is seven orders of
magnitude smaller than the remaining ones. Basically, this means that with the frequencies we are using and the remaining physical parameters, we can safely assume inviscid fluid approximation. In fact, several recent studies that considered spinning flows assumed non- viscous fluids, e.g., Refs. [31], [34], [35] (cited in the main manuscript), to name a few.
Moreover, we consider that the compressibility is low, and hence for reasonable rates of spinning and small-amplitude pressure waves, the density is assumed constant.
As we assume linear regime of pressure waves, we expand fields using Fourier series and thus replace time-dependence with e−iωt factors, where ω represents the angular frequency of the monochromatic wave. In order to derive analytical expressions, we consider as shown in Fig. 1 (of the main manuscript) cylindrical (polar) structures (the spherical case is more tricky and thus more complex to consider, at least analytically, when rotation is added, as the modified boundary condition would be different for different polar angles). The fields are thus proportional to eimϕ with ϕ the polar angle, and it is clear that azimuthal derivatives (∂ϕ) are of the form ∝im, withm the azimuthal order and i2 =−1. By combining the two modified conservation equations, i.e., Eqs. (3)-(4), and by linearizing them (we only keep the first order acoustic perturbations; for example the term (v· ∇)vis neglected in the final equation) in cylindrical coordinates, we derive the following coupled differential system
vr
r + ∂vr
∂r + 1
rimvϕ+ikvz+ ˜βγmp= 0, γmvr−2Ωvϕ+1
ρ
∂p
∂r = 0, (5)
2Ωvr+γmvϕ+ 1
ρrimp= 0, γmvz+ik
ρp= 0,
where k denotes here the wavenumber in the z-direction (or propagation constant), and where γm =i(mΩ−ω) is the modified spinning (Doppler) angular frequency. This system can be recast to the following equation of the pressure field
∂2p
∂r2 +1 r
∂p
∂r +
kt2− m2 r2
p= 0, (6)
where we can define the following dispersion relation
kt2+ηm2k2 =k2m = −(4Ω2+γm2)
c2 , (7)
with the the transverse and propagating wavenumber kt and k, respectively, and where η2m = 1 + 4Ω2/γm2. Equation (6) is actually a Helmholtz-like equation, expressed in polar
coordinates, with the transverse wavenumberkt. When there is no spinning (i.e., Ω = 0), we can see from Eq. (7) that we recover ηm = 1 and k2t +k2 =k02 =ω2/c2. The behavior ofkm, i.e., the spinning wavenumber, can be found in Ref. [7]. As the parameter γm is complex, km has both propagating (real part) and evanescent (imaginary part) components. One more important feature revealed by Eq. (7) is the anisotropy that appears in the spinning medium between the transverse (planer−φ) and longitudinal (z) directions, via the factor ηm. In fact, the presence of this term is the consequence of the inherent anisotropy that is induced by spinning and thus is anticipated. Moreover, the regime of propagation, i.e., km2 >0 coincides with the regime whereη2m >0.
Similar to the case at rest, the governing equation has to be complemented by appropriate continuity conditions at the physical interfaces of the problem [1]. For spinning media, the continuity conditions must take into account the relative movement. It can be shown that pshould remain continuous as before; However, the continuity of 1/ρ∂rpshould be replaced by the continuity of the normal displacement
ζrm= γmvr+ Ωvϕ
γm2 + Ω2 = (2Ω2−γm2)∂rp−(3iγmΩm/r)p
ρ(4Ω2+γm2) (Ω2+γm2) . (8) By inspection of Eq. (8), again by letting Ω = 0, we get the usual continuity as acoustics at rest.
Regarding the modified wavenumber with azimuthal order m, we could identify several regimes of propagating/evanescent regions, depending on m. For instance
k2m=−4Ω2+γm2
c2 =−1
c2 [(2−m) Ω +ω] [(2 +m) Ω +ω] , (9) where we have used γm =i(mΩ−ω).
If −2< m <2, then
[(2−m) Ω +ω]>0,but [(2 +m) Ω +ω]>0,when ω <(2 +m) Ω. (10) Hence, for small ω, k2m < 0 (evanescent regime). But, if ω > (2 +m)Ω, then km2 > 0 (propagating regime).
Now, if m= 2, km2 =−ω/c2×(4Ω−ω). Thus, if ω <4Ω we have the evanescent regime, while if ω >4Ω we have propagating regime.
If m >2, then when |2−m|Ω< ω <(2 +m)Ω, km is imaginary and we have evanescent regime. If ω <|2−m|Ω or ω >(2 +m)Ω, then km is real-valued.
Finally, if m≤ −2,km is always positive and we have the propagating regime.
The analysis of the modified spinning wavenumber shows thus the existence of frequen- cies of forbidden propagation, even for the ”fundamental mode”. This zone (highlighted by gray color in Fig. 2 and Fig. 3 of the main text) is different from classical cutoff frequencies of guided modes propagating in a duct. We denote the propagating and trans- verse wavenumbers in the fiber by k and kt, respectively, the wave dispersion becomes kt2+ηm2k2 =k2m= (ω2−2mωΩ +m2Ω2−4Ω2)/c2. k2m can become negative. For example, if m= 0, k2m becomes (ω2−4Ω2)/c2; thus if the frequency is below 2Ω, unless the transversekt is imaginary, there will be no propagating mode. For the ”fundamental mode”, propagation starts at 2Ω. Regular cutoff frequencies is illustrated in Fig. 1(c) as an example, if one continues the dark blue lines corresponding to the guided modes m = 0.
II. ORIGIN OF THE NONRECIPROCITY: ACOUSTIC ZEEMAN EFFECT
A. Hamiltonian formulation
The system of Eq. (5) may be re-written as
−i β˜
1 r + ∂
∂r
vr+ 1
βr˜ mvϕ+ k
β˜vz+mΩp=ωp , mΩvr+ 2iΩvϕ− i
ρ
∂p
∂r =ωvr,
−2iΩvr+mΩvϕ+ 1
ρrmp=ωvϕ, mΩvz +k
ρp=ωvz, (11)
In the absence of bias, i.e., of spinning Ω, a convenient way to solve the problem is to work with the state vectors|ψ⟩= (p,v)T. These states are endorsed with the scalar product defined as
⟨ξ|ψ⟩= Z
V
dV {p∗1p2+v1∗·v2} , (12) where the more classical dot product is used inside the integral. V denotes the volume of the acoustic fiber. In this way, Eq. (11) can be simply written in the more elegant way
H(0)+δH
|Ψ⟩=ω|Ψ⟩ , (13)
with the state |Ψ⟩ given by
|Ψ⟩=
p vr vϕ vz
. (14)
H(0) is the time-evolution operator of the system in the absence of bias (spinning), i.e.,
H(0) =
0 −i˜
β
1
r + ∂r∂ m˜
βr k β˜
−i ρ
∂
∂r 0 0 0
m
ρr 0 0 0
k
ρ 0 0 0
. (15)
The perturbation operator δH depends on the spinning angular frequency Ω
δH =
mΩ 0 0 0
0 mΩ 2iΩ 0
0 −2iΩ mΩ 0
0 0 0 mΩ
. (16)
The eigenmodes of the ASF can be obtained by diagonalizing the Hamiltonian of Eq. (15) for the unbiased case or of Eq. (16) for the biased scenario.
B. Eigensolutions for the unbiased system
When Ω = 0, i.e., no bias, the problem of Eq. (11) reduces to the well-known Helmholtz equation
∆p+ω2
c20p= 0, (17)
with its boundary conditions. The derivations and solutions of this system are described in Section III, in the framework of an optical fiber. These lead to the dispersion relation, giving the eigenfrequencies of the system, as
κ1 ρ1
Km(κ2r1)×Jm′ (κ1r1)−κ2 ρ2
Jm(κ1r1)×Km′ (κ2r1) = 0, (18)
whereκ1,2 satisfy the dispersion relations. We focus on the modesm=±1 which correspond to left and right propagation (LHP and RHP, respectively). Owing to the properties of the Bessel and modified Bessel functions, it is easy to show that m= 1 andm =−1 give rise to the same dispersion relation and hence eigenfrequencies and eigenmodes, which shows the degeneracy of them =±1 modes in the absence of bias.
The eigenpressure field (waveguided modes in opposite directions) corresponding to the m=±1 modes can be expressed as
p± ∝J±1(κ1r)ei(kz±ϕ). (19)
We may thus define the normalized eigenpressure fields |±⟩and use the state vector as
|±⟩=α±ζ±
iωρJ±1(κr) κ1J±1′ (κr)
±i
r J±1(κr) ikJ±1(κr)
, (20)
where the constants ζ± are the renormalization factors, computed by using the scalar product of Eq. (12) and α± =ei(kz±ϕ)/iωρ.
C. Eigensolutions for the biased system
When we turn spinning on (bias), we make the assumption that the new eigenvectors lie in the subspace described by the unbiased waveguide. We can thus write the new eigenvectors as a linear combination of |±⟩, with some complex unknown coefficientsµ+ and µ−, i.e.,
H(0)+δH
|ψ⟩=ω|ψ⟩ , (21)
with |ψ⟩=µ+|+⟩+µ−|−⟩, where we have
H(0)|±⟩=ω0|±⟩. (22)
Taking into account that {|+⟩,|−⟩} is an orthonormal basis (with respect to the scalar product of Eq. (12)),
⟨+|δH|+⟩ ⟨+|δH|−⟩
⟨−|δH|+⟩ ⟨−|δH|−⟩
µ+ µ−
= (ω−ω0)
µ+ µ−
. (23)
We could show that ⟨+|δH|−⟩ = ⟨−|δH|+⟩ = 0 and that ⟨+|δH|+⟩ = − ⟨−|δH|−⟩.
Hence, the eigenvalues of the biased system are
ω± =ω0± ⟨+|δH|+⟩ , (24)
where ⟨+|δH|+⟩= Ω for the modes that we consider in this demonstration, i.e., m=±1.
From this, we can clearly see that the spinning background mean flow acts here as the counterpart of the Zeeman effect [8]. Its role is to break the reciprocity and to lift the degeneracy of the right and left propagating waves.
III. DISPERSION OF THE ASF
The m = 0 mode does not result in neff > 1 and therefore is not a propagating mode, as seen in Fig. 2(a) (in the MS). Yet, it is interesting to start with this fundamental mode to understand the basic effects. We suppose that a column of a fluid with n1 =√
5 and of radiusr1 is spinning and we apply the boundary conditions at r=r1, assumingm= 0, i.e., continuity of the pressure field p and of ζrm=0 [See Eq. (8)]. Hence, the linear system that we obtain is
J0(κr1) −K0(βr1)
˜ α
ρ1ω2κJ0′(κr1) −ρ1
2ω2βK0′(βr1)
≡F0(ω, k) = 0, (25)
with ˜α = (2α2+ 1)/{(4α2−1)(α2−1)}. The transverse wavenumbers κ and β are related via
κ2
η02 +β2 = ω2 η02
1−4α2 c21 − 1
c22
, (26)
withη02 = 1−4Ω2/ω2 >0 in the propagation regime. Equation (25) represents the dispersion of the fundamental mode (propagation constant k versus ω) and can be solved graphically, as shown in Fig. 1(a) [9]. The graphical solution consists in separating terms depending on κ and those depending on β in Eq. (25), that is the first and second columns, respectively.
Hence, we plot the function fκ = [ ˜αxJ0′(x)]/[ρ1J0(x)] [J0′(x) = J1(x)] as function of x2 (x= κr1). On top of the same graph, we superpose the function fβ(y) = −[yK0′(y)]/[ρ2K0(y)], with x2/η02 + y2 = ω2r21[(1 −4α2)/c21 − 1/c22]/η20. The intersections of these two curves correspond to solutions of Eqs. Eq. (25)-(26). The function fβ(y) is always positive and decays rapidly withy, with an asymptote aty= 0. The functionfκ(x) possesses asymptotes at the zeros of J0, i.e., at the values x = x0l (i.e., J0(x0l) = 0). The modified origin of
FIG. 1. (a) Graphical solution to the dispersion relation of the first mode m = 0. The blue line shows the function fκ while the dashed lines show the function fβ for different angular frequen- cies (1500, 1250, 1000, 500, and 250 rad/s, from right to left, respectively) versus the transverse wavenumber κ2 (for fκ) and β2 = ωη22
0
1−4α2 c21 − 1
c22
−κ2
η02 (for fβ). (b) Behavior of the cutoff an- gular frequencyω01r1/x01 vs ω/Ω. (c) Contourplot of the dispersion relation of the mode m= 0, i.e, 10×log10(|F0(ω, k)|) vs ω/Ω and the propagation constant k, represented by Eq. (25), for a spinning angular frequency Ω/2π = 16 Hz. The red curve plots the dispersion in air, i.e., ω/c2
while the black curve gives the dispersion in the spinning medium, i.e., (ω/η0c1)√
1−4α2. (d) Snapshot of the amplitude of the pressure field|p|/|p0|inside the ASF and in the outside, showing the confinement.
the function fβ(y) depends on ω and Ω. For small values of ω, no intersection between the two functions is possible as seen from Fig. 1(a). The first intersection occurs when (ωr1)2
1−4α2 c21 −c12
2
= x201, meaning that ω01/x01 = {r1q
1−4α2 c21 −c12
2}−1, which represents the cutoff angular frequency. At this frequency, we have β = 0, thus ω01 = c2k: The wave propagates inside the surrounding as if there is no waveguiding. The behavior of this cutoff angular frequency with the spinning ratio is depicted in Fig. 1(b). At each time when x crosses a zero of J0 (i.e., x0l) a new (higher-order) mode appears with a cutoff angular frequency ω0l/x0l={r1q
1−4α2 c21 − c12
2}−1.
Although for the fundamental mode waveguiding is not induced by spinning but rather by n1 > 1 inside the fiber, spinning still has some intriguing effect on the dispersion and the induced waveguiding overall. In fact, Fig. 1(c) gives the contourplot of Eq. (25), i.e., of F0(ω, k) vs the propagation constant k and the normalized frequency ω/Ω for Ω/2π = 16 Hz (for other different spinning angular frequencies, see Fig. 2 and Fig. 3). The red solid line indicates the limit of the sound cone, i.e., ω/c2. The modes appearing below this line correspond to leaking (radiative regime), i.e., acoustic waves that propagate into the surrounding space. The second limit plotted in black line correspond to the forbidden zone (neither modes or leaking are possible above the black curve, i.e., the white region of the 2D plot). This curve does not start from zero but rather has a finite frequency due to the imaginary part of km. This frequency depends on the modes, indicating tunability of acoustic waveguiding (negative and positive spinning lead to the same effect). Zeros of Eq. (25) correspond to waveguiding, which is delineated by the dark blue in Fig. 1(c). It is further noted from Fig. 1(c) that the fundamental mode is symmetric by nature, i.e., same dispersion for both directions of propagation inside the ASF. The snapshot of the pressure
FIG. 2. Contourplot of the dispersion relation of the mode m = 1 vs the spinning ratio 1/α = ω/Ω and the wavenumber, represented by Eq. (8) of the MS, for Ω =
−100,−200,−400,−600,−800,−1000 rad/s, from left to right and from top to bottom, respec- tively.
FIG. 3. Contourplot of the dispersion relation of the modes vs the spinning ratio 1/α and the wavenumber, represented by Eq. (8) of the MS, for the modes m = 1,3,5,2,4,6 for a spinning frequency of Ω/2π= 80 Hz, from left to right and from top to bottom, respectively.
FIG. 4. Contourplot of the variable α[(m2 −4)α−2m] versus the spinning ratio α and m. The white regions depicts location where waveguiding is not possible, i.e., α[(m2−4)α−2m]<0.
field is also given in Fig. 1(d) and it shows complete confinement of the wave inside the ASF (r < r1) as anticipated.
Figure 2 depicts the dispersion relation of the first order multipole m = 1 and shows the effect of spinning on the cutoff frequency and the waveguiding mechanism.
Figure 3 depicts the dispersion relation of the first five modes of the ASF at the spinning frequency Ω = 100 rad/s.
Figure 4 plots the variation of α[(m2 −4)α −2m] as two-dimensional function of the spinning ratio α and the multipole order m. The regions of α[(m2 −4)α−2m] < 0 are depicted in white, which correspond to radiative regime. The colored regions correspond to α[(m2 −4)α−2m] > 0 where waveguiding takes place. For each m, a specific range of α (and hence of Ω) is enforced. It can be clearly seen that for m = 0, no waveguiding is possible, for ∀α. For example, for m = 1, waveguiding is possible for α ∈ [−2/3,0[ (See Fig. 4).
IV. COMSOL MODELING AND POTENTIAL EXPERIMENTAL REALIZATION
A. FEM governing equations
We make use of the model ”Linearized Navier-Stokes” (LNS), frequency domain of COM- SOL Multiphysics 5.6. This model belongs to the Aeroacousics branch of the acoustic in- terface. This model permits to compute variations in pressure, temperature, and velocity in the presence of stationary background mean flow. The governing equations of this model, based on the continuity of momentum and energy equations are
∂ρt
∂t +∇ ·(ρ0ut+ρtu0) =M , (27)
ρ0 ∂ut
∂t + (ut· ∇)u0+ (u0· ∇)ut
+ρt(u0· ∇)u0 =∇ ·σ+F−u0M , (28) ρ0CP
∂Tt
∂t + (ut· ∇)T0+ (u0· ∇)Tt
+ρCP(u0· ∇)T0
−αPT0 ∂pt
∂t + (ut· ∇)p0+ (u0· ∇)pt
−αPTt(u0· ∇)p0 =∇ ·(k∇Tt) + Φ +Q , (29) where pt, ut, and Tt are the acoustic perturbations. The subscript ”t” refers to the total fields. The stress tensor isσ and Φ is the viscous dissipation function.
The constitutive equations are σ =−ptI+µ
∇ut+ (∇uT)T
+
µB−2 3µ
(∇ ·ut)I, (30)
ρt =ρ0(βTpt−αPTt) . (31)
A material is defined by the properties: density, dynamic viscosity, bulk viscosity, thermal conductivity, and heat capacity at constant pressure, denoted by ρ0, µ, µB, k, and CP,
respectively.
A background acoustic field may be added to the domain, defined by pressure, velocity, and temperature pb, ub, and Tb, respectively.
Generally slip boundary conditions for the velocity are defined as
n·ut= 0, (32)
σn·(σn·n)n = 0, σn=σn, (33)
where σ is the stress tensor.
FIG. 5. Scattering cross-section from a fixed cylinder of radius 0.35 m and relative density of 10 (to test the model in an extreme scenario), for a spinning frequency of (a) 0 Hz (b) 10 Hz, (c) 20 Hz, and (d) 30 Hz. The black solid curves correspond to the analytical (Mie) model while the red dashed curves correspond to the linearized Navier-Stokes full-wave simulations, using COMSOL Multiphysics [10].
B. FEM-validations of the ASF waveguiding
The equations of this model treat a linearization of the full compressible, nonisothermal, and viscous flow. These equations, shown in the previous section, may be found in [10]
for more details on this model. We employ this model in the frequency-domain. As these equations are general, they can model with high fidelity complex interactions such as the one that we are treating. To validate the LNS, we consider the same fiber, but now we analyze its scattering response: A spinning column of air in the trajectory of a plane wave, and we compute the scattering cross-section (SCS). We compute both SCS obtained from the model of Ref. [3] and the LNS.
In Fig. 5, we present a numerical study of nonspinning/spinning cylinders with various spinning velocities. In Fig. 5(a) where there is no spinning, the SCS is dominated by the Mie resonant modes. In particular the ±m scattering modes are identical as showcased in Fig. 6(a). Yet, in Figs. 5(b)-(d) where spinning is induced (2π×10,20,30 rad/s respectively) we can see that both Mie model and the FEM-based results predict the emergence of a spinning-induced resonance, around similar frequencies. In this case, and contrary to the spinning scenario, the ±m modes are shifted due to the Zeeman effect as can be seen from Fig. 5(b) for a spinning frequency of 20 Hz. Despite some discrepancies, the main observation is that spinning-induced resonances do exist, including in the low frequency regime. This demonstrates that the Mie model (modified Helmholtz equation and modified boundary
FIG. 6. Scattering coefficients from a fixed cylinder of radius 0.35 m and relative density of 10 (to test the model in an extreme scenario), for (a) non-spinning and (b) spinning frequency of 20 Hz.
FIG. 7. FEM validation using the linearized Navier-Stokes interface of COMSOL Multiphysics.
(a) and (b) depict the pressureptandT, respectively. (b) shows the phase of the pressureφ(pt)/π, while (d) and (e) show the radial and azimuthal velocity ur and uϕ, respectively.
conditions) used in the works is adequate in predicting these modes and proves the physical origin of these resonances, as demonstrated by the full-wave simulations. These results further confirm and demonstrate that the Mie-based model used in the main manuscript is a convenient model that describes spinning acoustic objects, in its domain of validity.
Figure 7 depicts the asymmetric waveguided eigenmode. In Fig. 7(a) we plot the pressure amplitude which fits well with that of Fig. 4(a) in the main manuscript. Figure 7(b) shows the phase of pin units ofπand it also fits very well with the phase given in Fig. 4(b) of the main manuscript. In addition to the pressure field that we could access from the Mie model, the LNS gives access to the temperature T shown in Fig. 7(c). The lowest temperatures coincide with the lowest pressures in the center of the spinning fiber. Figures 7(d) and 7(e) plot the radial and azimuthal velocities, respectively.
C. Potential experimental realization
This Letter endeavors to show for the first time a new mechanism for airborne sound waveguiding (without the use of metamaterials). This concept is showcased theoretically
through analytical modeling that uses a set of ”reasonable” approximations and also via FEM, by using COMSOL Multiphysics [10]. We believe that these results show at least in principle the possibility of the acoustic spinning fiber. Yet, an experimental realization of our ideas may be within reach. To do so, we first need to separate the spinning air (inside the fiber) from the surrounding air at rest. This may be done via a very thin interface that is impermeable to air particles but permeable to sound (i.e., impedance-matched). The permeability to sound is a very important condition, as we wish the guiding mechanism to be realized via our new concept and not by a kind of a barrier to sound (e.g., hard walls).
This membrane can be made for example from Latex of polyurethane [11]. The second experimental challenge consists in enforcing the background homogeneous flow of air inside the fiber, Ω= Ωez (or angular velocity Ωreϕ). To do so, we can use a somehow analogous setup to the one used in Refs. [8, 12]. In these setups, the propagation of sound is in the same plane of the spinning. But in our work, as we are interested in waveguiding, sound propagates perpendicular to the air flow direction. Yet, by arranging some fans (thin enough to not disturb the sound propagation) along thez-direction, we can get a flow of the required form. Some imperfections are expected due to some viscosity or nonuniform spinning, but we believe that the main phenomena predicted in this Letter, i.e., the unidirectional sound airborne guiding, can still be observed. Moreover, in Refs. [37] and [38] of the main text, spinning metasurfaces were leveraged to obtain nonreciprocal behavior of acoustic vortex beams. Yet, the main difference with our work is that in these papers, only the metasurface is spinning, while our air medium is spinning. In the absence of spinning of the metasurface in these works, the vortex beam still exists. Spinning can only isolate or enhance the energy.
In our case, the waveguiding mechanism itself is fully induced by spinning.
[1] P. M. Morse, Ku ingard, theoretical acoustics, Princeton University Press, 949p4, 150 (1968).
[2] E. Graham and B. Graham, Effect of a shear layer on plane waves of sound in a fluid, The Journal of the Acoustical Society of America 46, 169 (1969).
[3] D. Censor and J. Aboudi, Scattering of sound waves by rotating cylinders and spheres, Journal of Sound and Vibration 19, 437 (1971).
[4] M. Schoenberg and D. Censor, Elastic waves in rotating media, Quarterly of Applied Mathe- matics 31, 115 (1973).
[5] D. Censor and M. Schoenberg, Two dimensional wave problems in rotating elastic media, Applied Scientific Research 27, 401 (1973).
[6] S. Farhadi, Acoustic radiation of rotating and non-rotating finite length cylinders, Journal of Sound and Vibration 428, 59 (2018).
[7] M. Farhat, S. Guenneau, A. Al`u, and Y. Wu, Scattering cancellation technique for acoustic spinning objects, Physical Review B101, 174111 (2020).
[8] R. Fleury, D. L. Sounas, C. F. Sieck, M. R. Haberman, and A. Al`u, Sound isolation and giant linear nonreciprocity in a compact acoustic circulator, Science 343, 516 (2014).
[9] J. D. Jackson, Classical electrodynamics (American Association of Physics Teachers, 1999).
[10] COMSOL Multiphysics, V5.6 (build: 280).
[11] S. Varghese, K. Gatos, A. Apostolov, and J. Karger-Kocsis, Morphology and mechanical properties of layered silicate reinforced natural and polyurethane rubber blends produced by latex compounding, Journal of applied polymer science 92, 543 (2004).
[12] L. Quan, S. Yves, Y. Peng, H. Esfahlani, and A. Al`u, Odd willis coupling induced by broken time-reversal symmetry, Nature communications 12, 1 (2021).
