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Slowsoundinaduct,effectivetransonicflows,andanalogblackholes

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Slow sound in a duct, effective transonic flows, and analog black holes

Yves Aurégan,1,* Pierre Fromholz,2,† Florent Michel,3,‡ Vincent Pagneux,1,§ and Renaud Parentani3,∥1Laboratoire d’Acoustique, Université du Maine, UMR CNRS 6613, Avenue O Messiaen,

72085 Le Mans Cedex 9, France2Département de Physique, ENS, 24 rue Lhomond, 75005 Paris, France

3Laboratoire de Physique Théorique, CNRS, Université Paris-Sud, Université Paris-Saclay,91405 Orsay, France

(Received 16 March 2015; published 21 October 2015)

We propose a new system suitable for studying analog gravity effects, consisting of a gas flowing in aduct with a compliant wall. Effective transonic flows are obtained from uniform, low-Mach-number flowsthrough the reduction of the one-dimensional speed of sound induced by the wall compliance. We showthat the modified equation for linear perturbations can be written in a Hamiltonian form. We perform aone-dimensional reduction consistent with the canonical formulation, and deduce the analog metric alongwith the first dispersive term. In a weak dispersive regime, the spectrum emitted from a sonic horizon isnumerically shown to be Planckian, and with a temperature fixed by the analog surface gravity.

DOI: 10.1103/PhysRevD.92.081503 PACS numbers: 04.62.+v, 04.70.Dy, 43.20.+g, 43.20.Wd

I. INTRODUCTION

Engineering flows that are transonic and regular offersthe possibility to test well-known predictions concerningastrophysical black holes [1]. Of particular interest isHawking’s discovery that black holes should spontane-ously emit a steady thermal flux [2]. Although this effectwas originally phrased in the context of quantum relativ-istic fields, it rests on the anomalous mode mixingoccurring near the black-hole horizon [3]. This mixing,which is stationary and conserves the wave energy, is calledanomalous because it leads to a mode amplification andinvolves negative energy waves. Because of the preciseanalogy between the equation governing sound propaga-tion and that used by Hawking, these key elements arerecovered in a stationary transonic flow. Indeed, in theacoustic approximation, for long wavelengths, the modemixing possesses the main properties of the one responsiblefor the Hawking effect [4,5].To complete the comparison, one should take into

account the dispersive properties of sound waves, whichhave no counterpart in general relativity. (Note, however,that dispersive terms appear in certain theories of modifiedgravity where Lorentz invariance is broken [6–8].)Analytical and numerical studies have established thatthe correspondence is quantitatively preserved providedthe two relevant scales are well separated [4,5,9–12],namely, when the dispersive length is sufficiently smallerthan the typical length scale associated with the inhomo-geneity of the flow (which then plays the role of theinverse surface gravity of the black hole). Therefore, thereis no conceptual obstacle preventing the testing of the

Hawking prediction by observing the mode mixing acrossa sonic horizon. In practice, the difficulty is findingappropriate setups. Many have been proposed, involvingfor instance ultracold atomic clouds [13], surface waves influmes [14], and light in nonlinear media [15]. Recently,the first experiments have been carried out [16–19].In this paper, we propose a new framework which is a

variant of the original one [1,4]. Its main interest resides inobtaining a large reduction of the low-frequency one-dimensional sound speed in a duct so that a stationaryflow with a uniform low Mach number M possesses ahorizon. In realistic settings, M could be close to 0.3.The reduction of the effective sound speed is achievedby means of a compliant wall composed of thin tubeswhich modify the upper boundary condition, see Fig. 1.

FIG. 1 (color online). Schematic drawing of the configuration.At y ¼ 0, the wall is rigid. At y ¼ 1, the compliant wall is madeof a succession of tiny tubes of height bðxÞ. As explained in thetext, the effective sound speed is a decreasing function of b. In thepresent profile bðxÞ and for a given uniform flow with Machnumber M < 1, an effective supersonic region can be created inthe right region.

*yves.auregan@univ‑lemans.fr†pierre.fromholz@ens.fr‡florent.michel@th.u‑psud.fr§vincent.pagneux@univ‑lemans.fr∥renaud.parentani@th.u‑psud.fr

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Interestingly, for near-critical flows, the wave equationfollows from a well-defined action principle. From this,we derive the acoustic black-hole metric in the long-wavelength regime, and a conserved norm which estab-lishes the anomalous character of the mode mixing. At theend of this paper we briefly present the practical advantagesof this proposal.

II. THE MODEL

We consider the propagation of sound waves in atwo-dimensional horizontal channel of uniform height H.x� denotes theCartesian horizontal coordinate, y� the verticalone, and t� the time. We assume the flow of air is uniform,with a horizontal velocity U0. Denoting by c0 the soundspeed, ρ0 the air density, v� the velocity perturbation, and p�the pressure perturbation, the time evolution is given by

c−10 D�t p� ¼ −ρ0∇� · v�; ρ0D�

t v� ¼ −∇�p�; ð1Þ

where D�t ≡ ∂t� þ U0∂x� is the convective derivative. We

define dimensionless quantities as x ¼ x�=H, y ¼ y�=H,t ¼ t�c0=H, v ¼ v�=c0, p ¼ p�=ðρ0c20Þ, and M ¼ U0=c0.The potential ϕ gives the velocity by v ¼ ∇ϕ, and thepressure by p ¼ −Dtϕ. It obeys

D2tϕ − ð∂2

x þ ∂2yÞϕ ¼ 0: ð2Þ

At the lower wall, the impenetrability condition is simply∂yϕðt; x; y ¼ 0Þ ¼ 0, see Fig. 1. At the upper wall y ¼ 1,the continuity of the displacement and pressure gives rise toa nonlocal expression in time, see [20–22] for details.However, for near-critical flows and small frequencies, itcan be written as

∂yϕþDtðbðxÞDtϕÞ ¼ 0 at y ¼ 1; ð3Þ

which is second order in ∂t.For a homogeneous stationary flow, we can look for

solutions of the form φk ∝ coshðαkyÞeiðkx−ωktÞ. Equation (2)and the two boundary conditions respectively give

α2k ¼ k2 − Ω2k and αk tanhðαkÞ ¼ bΩ2

k; ð4Þ

where Ωk ≡ ωk −Mk is the frequency in the comovingframe. At low wave number, the dispersion relation reads

Ω2k ¼ c2SðbÞk2 − k4=Λ2

b þOðk6Þ; ð5Þ

where

c2SðbÞ ¼1

1þ b; Λ2

b ¼3ð1þ bÞ3

b2: ð6Þ

One sees the important effect of the boundary condition ofEq. (3): the low-frequency group velocity with respect to the

fluid (¼ ∂kΩ) is reduced by the compliant wall. One alsosees that the dispersive length 1=Λb given by the quartic termvanishes in the limit b → 0.To obtain flows crossing the effective sound speed,

we make b vary with x, see Fig. 1. We call b1 > b2 itsasymptotic values, and db its typical variation length. Wechoose the following form for bðxÞ:

bðxÞ ¼ b1 þ b22

þ b2 − b12

tanh

�xdb

�: ð7Þ

We then adjust the flow speed M to obtain

cSðb1Þ ¼1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ b1p < jMj < 1ffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ b2p ¼ cSðb2Þ: ð8Þ

Since the background flow is stationary, we shall work with(complex) stationary waves,

~ϕωðx; yÞ ¼Z þ∞

−∞

dt2π

eiωtϕðt; x; yÞ: ð9Þ

III. WAVE EQUATION

Since the height H of the duct is much smaller thantypical longitudinal wavelengths, we expect that stationarywaves obey an effective one-dimensional equation in x, asis the case in elongated atomic Bose condensates [23] andin flumes [14,24]. To obtain such a reduction is nontrivial,as the x dependence of bðxÞ prevents us from factorizingout a y-dependent factor. To proceed, and to make contactwith the works mentioned above, it is useful to exploit thefact that Eq. (2) and Eq. (3) can be derived from thefollowing action:

S ¼ 1

2

ZdtZ

dxZ

1

0

dyL

L ¼ ðDtϕÞ2 − ð∇ϕÞ2 þ δðy− 1ÞbðxÞðDtϕÞ2: ð10Þ

We notice that the nontrivial condition of Eq. (3) isincorporated by the above boundary term. Introducingthe conjugate momentum πðt; x; yÞ ¼ ∂L=∂ð∂tϕÞ oneobtains the Hamiltonian H by the usual Legendre trans-formation. In addition, as in [4,24], the conserved innerproduct ð·j·Þ is not positive definite, and has the Klein-Gordon form

ðϕ1;ϕ2Þ ¼ iZ þ∞

−∞dx

Z1

0

dyðπ�1ϕ2 − ϕ�1π2Þ; ð11Þ

where ϕ1;ϕ2 are two complex solutions of Eqs. (2), (3), andπ1; π2 their associated momenta. The inner product isconserved by virtue of Hamilton’s equations. As for soundwaves in other media, it identically vanishes for all realsolutions. However it provides key information whenstudying stationary modes, namely, the sign of their normðϕω;ϕωÞ. Indeed, as we shall see, for a fixed ω > 0, there

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will be both positive and negative norm modes. When theflow is stationary, for any complex solution ϕ, the waveenergy is conserved and related to Eq. (11) by

H½2ReðϕÞ� ¼ ðϕ; i∂tϕÞ: ð12Þ

Moreover, when the flow is also asymptotically homo-geneous, for every asymptotic plane wave φk ∝coshðαkyÞeiðkx−ωktÞ, the sign of H is that of ωkΩk. Thisrelation will allow us to identify the negative energy waveswithout ambiguity.We can now proceed following the hydrodynamic treat-

ment of [24]. As a first step, it is useful to derive a (1þ 1)-dimensional equation from which an effective space-timemetric can be read out. When the (adimensional) wave-length in the x direction is much larger than 1, we canassume that ∂2

yϕ is independent of y. As ∂yϕ ¼ 0 at y ¼ 0,we write the field as

ϕðx; y; tÞ ≈ Φðx; tÞ þ y2Ψðx; tÞ: ð13Þ

Plugging this into the action Eq. (10) and varying it withrespect to Φ and Ψ, we get two coupled equations.Combining them, we obtain ðO2 þ O4ÞΦ ¼ 0, where Onis an nth-order operator in ∂t and ∂x. The quadratic term isO2 ¼ ∂μFμνðxÞ∂ν, where

FμνðxÞ ¼�c2SðbðxÞÞ −M2 M

M 1

�; ð14Þ

and c2SðbðxÞÞ is given in Eq. (6). Up to a conformal factor,we obtain the d’Alembert equation in a two-dimensionalspace-time with metric gμν ∝ Fμν. This metric has a Killinghorizon where c2SðxÞ −M2 vanishes [25]. This correspon-dence with gravity relates the anomalous scatteringdescribed below to the Hawking effect.Contrary to what happens for sound waves in atomic

Bose-Einstein condensate, or water waves in the incom-pressible limit, O4 also contains third and fourth derivativesin time. This prevents us from applying the standard treat-ment on the sole field Φ. However, the set of two coupledequations on ðΦ;ΨÞ is Hamiltonian and can be used to studythe scattering. Alternatively, one can work with the originalmodel in 2þ 1 dimensions based on Eq. (10). We performednumerical simulations with both models and found similarresults.

IV. ANOMALOUS MODE MIXING

Since stationary waves with different frequencies ω donot mix, the scattering only concerns the discrete set ofmodes with the same ω. To characterize it, we identify itsdimensionality and the norms of the various asymptoticmodes for x → �∞.Figure 2 shows the dispersion relation and the roots at

fixed ω in a subsonic flow for M > 0. For this sign of M,

the flow associated with Eq. (7) passes from supersonic tosubsonic along the direction of the stream. It thus corre-sponds to a white-hole flow, like those studied in[17,24,26].1 For definiteness, we discuss only the caseω > 0. The same results are directly applicable to ω < 0after complex conjugation. In the subsonic region, on theright of the horizon, there exists a critical frequency ωmax(close to 0.42 in Fig. 2) at which two roots merge. Forω < ωmax, the dispersion relation has four real roots.2

Following [24], we call their wave vectors koutω , kinω ,kco;outω , and −kout−ω. The corresponding asymptotic modesare, respectively, φout

ω , φinω , φ

co;outω , and ðφout

−ωÞ�. They arecharacterized by three important properties:(1) In or out character: φin

ω is incoming (it movestowards the horizon) while the three other modesare outgoing (they move away from the horizon).

(2) Energy sign: ðφout−ωÞ� carries a negative energy and a

negative norm, see Eq. (12). (We are considering thecomplex conjugated so that φout

−ω is a positive normmode). The three other modes have positive energyand norm.

(3) Co- or counterpropagating nature: φco;outω is copro-

pagating (its group velocity in the frame of the fluidis positive) while the three others are counterpropa-gating. This separation is useful because only thelatter are significantly mixed in a transonic flow[9,10]. In effect, φco;out

ω acts essentially as a spectator.

FIG. 2 (color online). Dispersion relation ω versus k in ahomogeneous subsonic flow. The blue solid curves show theroots with positive comoving frequency Ω, and the red dashedones show those with Ω < 0. The dotted black line representsω ¼ 0.4, and the dot-dashed one shows the value ωmax of ω atwhich the two roots with k < 0merge. The parameters are b ¼ 1,M ¼ 0.4, and the effective sound velocity cS is equal to1=

ffiffiffi2

p≈ 0.71.

1For M < 0, one would describe a black-hole flow. Theforthcoming analysis applies by reversing the sign of velocitiesand the “in” or “out” character of the modes, see [9].

2In the supersonic region, only two real wave vectors remain:kco;inω and −kin−ω. They both describe incoming modes.

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To get the S matrix, we need to identify the basis ofincoming (outgoing) modes ϕin

ω (ϕoutω ) which contain only

one incoming (respectively outgoing) asymptotic planewave. For ω < ωmax, there are three modes, so the scatter-ing matrix has a size 3 × 3, and is an element of the Liegroup Uð2; 1Þ since ðφ−ωÞ� has a negative norm [9]. In thispaper we focus on the incident mode ϕin

ω for a white-holeflow. It is a good candidate to probe the analog Hawkingeffect, and was studied in hydrodynamic flows[16,17,26,27]. For x → ∞, it is a sum of four asymptoticmodes

ϕinω ≈ φin

ω þ αωφoutω þ βωðφout

−ωÞ� þ Aωφco;outω : ð15Þ

In transonic flows, there is no transmitted wave [26]. ϕinω

thus vanishes for x → −∞. When working with asymptoticmodes of unit norm, the norm of ϕin

ω evaluated at late time(in the sense of a broad wave packet),

ðNoutω Þ2 ¼ jαωj2 − jβωj2 þ jAωj2; ð16Þ

must be exactly 1 because of the conservation of Eq. (11).[In stationary flows, ðNout

ω Þ2 ¼ 1 also expresses the con-servation of the wave energy, see Eq. (12), and that of theenergy flux of Möhring [28].] The minus sign in front ofjβωj2 is the signature of an anomalous scattering. It stemsfrom the negative norm carried by ðϕout

−ωÞ�, see the abovepoint (2). The coefficient βω thus mixes modes of oppositenorms and energies. In quantum settings, jβωj2 would givethe mean number of spontaneously produced particles fromamplifying vacuum fluctuations, that is, the Hawkingradiation [2]. The gravitational analogy [1,4] indicates thatjβωj2 should follow a Planck law when dispersion effects(and grey body factors [29,30]) are negligible. Moreover, itpredicts that the effective temperature T should be given byTH ¼ κ=2π, where κ is the surface gravity obtained fromthe analog metric of Eq. (14). (We choose the units so thatkB=ℏ ¼ 1. TH and κ are thus both frequencies.) UsingEq. (6), one gets

κ ¼ ∂xcSjcS¼M ¼ −M3

2∂xbjcS¼M: ð17Þ

V. SPECTRAL ANALYSIS

We numerically solved the set of coupled equations onthe ð1þ 1Þ-dimensional fields Φ and Ψ, using the methodof [26] adapted to the present case. The results concerningthe incoming mode of Eq. (15) propagating in a transonicflow described by Eq. (7) are shown in Fig. 3.3 We stoppedthe integration for ω slightly below the critical frequencyωmax, where βω and Aω both vanish. We tuned the various

parameters (given in the caption of Fig. 3), so that the flowis near critical: M=cSðb1Þ ≈ 1.054 and M=cSðb2Þ ≈ 0.943.Using these parameters, one has ωmax ≈ 0.0053, and κ ≈0.019 of the same order as the dispersive frequency scaleΛbc2S evaluated at the horizon. This means that we workedjust outside the weak dispersive regime [11]. Yet, forfrequencies up to ωmax, jβωj2 follows rather well theHawking prediction jβHω j2 ¼ 1=ðeω=TH − 1Þ, that is, aPlanck law with TH given by κ=ð2πÞ, see Eq. (17). Atlow frequency, the relative difference ðjβωj2=jβHω j2Þ − 1 isof order 20% (when we used db ¼ 3, the difference reducedto about 0.3%, as expected since we were then in a weaklydispersive regime). Moreover, we see that the coefficientAω involving the copropagating mode can be safelyneglected as jAωj2 remains smaller that 0.1%. In order toestimate the numerical errors, we also show the quantityðNout

ω Þ2 − 1. As explained below Eq. (16), this quantitywould identically vanish in the absence of numerical errors.In brief, the properties we obtain are in close agreementwith those found in other media [9,12,23,31].

VI. EXPERIMENTAL ASPECTS

The incident waves will be sent by a loudspeaker in arectangular channel, and the scattered waves will beobserved using two arrays of microphones, see Fig. 1.The frequency range will be chosen to fulfill the low-frequency hypothesis used in Eq. (3). A stationary flowwith a mean Mach numberM ∼ 0.3 will be provided in thechannel. The compliant wall will be realized with ahoneycomb structure. Its height HbðxÞ will vary in sucha way that the flow is transcritical, i.e., that M=cS crossesunity at some x, something which has not yet been reachedin water tank experiments [16,17] aimed at detecting theanalog Hawking radiation. Another important advantage isthe possibility of sending the three types of incident waves,

FIG. 3 (color online). Plot of ωTH

jβωj2 (blue, solid) andjAωj2 (orange, dashed) of Eq. (15) as functions of the frequency.The black dot-dashed curve shows ω

THjβωj2 for a Planck law at

the Hawking temperature TH ¼ κ=ð2πÞ. The parameters areM ¼ 1=3, b1 ¼ 9, b2 ¼ 7, and db ¼ 1. The green dotted linerepresents ðNout

ω Þ2 − 1 where ðNoutω Þ2 is given in Eq. (16). Its

nonvanishing value quantifies the numerical errors.

3When sending a localized wave packet on a white-holehorizon, we observed at late times the formation of an undulation,see [22].

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and therefore to measure the nine scattering coefficients, inboth black-hole and white-hole flows. We suspect thatworking in these conditions can produce turbulence effectsand whistling. To reduce these effects, the compliant wallwill be covered by a wire gauze with a very low flowresistance. Special attention should also be devoted todissipation. In spite of these experimental difficulties, thesystem seems to be a good candidate to probe the variousaspects of the analog Hawking radiation.

VII. CONCLUSIONS

We showed that a low-Mach-number uniform flow of airin a tube with a compliant wall can produce a sonic horizonby reducing the local effective one-dimensional speedof sound. Despite the unusual boundary condition at thecompliant wall, the problem was phrased in a Hamiltonianformalism. For near-critical flows, a (1þ 1)-dimensional

reduction was performed, exhibiting an analog metric andthe first dispersion terms while retaining the Hamiltonianstructure. The Hawking spectrum was numerically recov-ered for sufficiently slowly varying profiles of the com-pliant wall. We thus hope to be able to verify that the normof the anomalous coefficient jβωj2 grows as κ=ð2πωÞ forlow frequency.

ACKNOWLEDGMENTS

We thank Scott Robertson for his remarks on a prelimi-nary version of this paper. P. F. is grateful to the LPT for itshospitality during an internship sponsored by the ÉcoleNormale Supérieure. We acknowledge partial support fromthe French National Research Agency under the ProgramInvesting in the Future Grant No. ANR-11-IDEX-0003-02associated with the project QEAGE (Quantum Effects inAnalogue Gravity Experiments).

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