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Effect of Changing the Vocal Tract Shape on the SoundProduction of the Recorder: An Experimental and

Theoretical StudyRoman Auvray, Augustin Ernoult, Soizic Terrien, Benoît Fabre, Christophe

Vergez

To cite this version:Roman Auvray, Augustin Ernoult, Soizic Terrien, Benoît Fabre, Christophe Vergez. Effect ofChanging the Vocal Tract Shape on the Sound Production of the Recorder: An Experimentaland Theoretical Study. Acta Acustica united with Acustica, Hirzel Verlag, 2015, 101, pp.317-330.10.3813/AAA.918829. hal-01259292

Effect of changing the vocal tract shape on the sound production

of the recorder: an experimental and theoretical study.

R. Auvray1,2) , A. Ernoult1,2), S. Terrien3), P.Y. Lagree1,2), B. Fabre1,2), C. Vergez3)

1) Sorbonne Universites, UPMC Univ. Paris 06, UMR 7190, LAM-Institut Jean Le Rond d’Alembert,75005 Paris, France. auvray@lam.jussieu.fr2) CNRS, UMR 7190, LAM-Institut Jean Le Rond d’Alembert, 75005 Paris, France.3) LMA, CNRS, UPR 7051, Aix-Marseille Univ, Centrale Marseille, F-13402 Marseille Cedex 20,France.

Abstract

Changing the vocal tract shape is one of the techniques which can be used by the players ofwind instruments to modify the quality of the sound. It has been intensely studied in the case of reedinstruments but has received only little attention in the case of air-jet instruments. This paper presentsa first study focused on changes in the vocal tract shape in recorder playing techniques. Measurementscarried out with recorder players allow to identify techniques involving changes of the mouth shape aswell as consequences on the sound. A second experiment performed in laboratory mimics the couplingwith the vocal tract on an artificial mouth. The phase of the transfer function between the instrumentand the mouth of the player is identified to be the relevant parameter of the coupling. It is shownto have consequences on the spectral content in terms of energy distribution among the even and oddharmonics, as well as on the stability of the first two oscillating regimes. The results gathered fromthe two experiments allow to develop a simplified model of sound production including the effect ofchanging the vocal tract shape. It is based on the modification of the jet instabilities due to thepulsating emerging jet. Two kinds of instabilities, symmetric and anti-symmetric, with respect to thestream axis, are controlled by the coupling with the vocal tract and the acoustic oscillation within thepipe, respectively. The symmetry properties of the flow are mapped on the temporal formulation ofthe source term, predicting a change in the even / odd harmonics energy distribution. The predictionsare in qualitative agreement with the experimental observations.

1 Introduction

The study of musical instruments based on physicalobservations has always attempted, to some extent,to include the musician within the sound produc-tion mechanisms. The most basic description of themusician control, such as a punctual or a static in-jection of energy that starts the oscillation, or evena modification of boundary conditions that changesthe acoustic properties, already constitute a firstattempt to describe the control.

In the case of flute-like instruments (mainlyrecorders, flutes and organ pipes), the first controlparameter that comes to mind is the blowing pres-sure. Several studies underlined the importance ofother parameters such as the jet velocity (directlyrelated to the blowing pressure), the jet length andheight, and the area of the outcoming flow. Theseparameters, or equivalent dimensionless parame-ters (Reynolds and Strouhal numbers) are underthe controlled of the musician or of the instrumentmaker depending on the type of instrument consid-ered.

Concerning the blowing pressure, one would in-tuitively say that it controls the loudness of the

1

sound including spectral enhancement that mightoccur because of non-linear behaviour. However ithas been shown that the variation of blowing pres-sure also directly affects the frequency of the noteand whether the instrument sounds on one registeror another.Understanding the influence of the blowing pres-

sure allows the inclusion of it in sound productionmodels [24, 37, 36, 16, 2]. These models considerthe blowing pressure as a slowly varying controlparameter, i.e. which varies on time scales muchlarger than an acoustic period. A recent study jus-tified this assumption by relating “slow” variationsof the blowing pressure (over 10s) to musical inter-pretations [13].On the other hand, the pressure can vary on the

same time scales as the acoustic period. Duringthe attack transients, the pressure grows from zeroto its final value in ten milliseconds (30ms in verysoft attack)[6, 30, 37] corresponding to the sameorder of magnitude as the sound period. The pres-sure release at the end of the notes and the pres-sure variations between two slurred notes are of thesame time scales. Some models[36, 17] attemptedto consider these scales of varying blowing pressureto model transients.However, as it has already been observed by

Verge[37], Segoufin[32] and de la Cuadra et al.[19,18], the blowing pressure and the resulting jet, areprone to oscillate at time scales much smaller thanthe ones that can be directly controlled by a human.More precisely, the blowing pressure may oscillateat frequencies within the range of the sounding fre-quency, certainly because of the acoustic couplingbetween the vocal tract of the musician (or the footof organ pipe) and the instrument[3].A more recent study, led by de la Cuadra et

al.[18], revealed major differences between a noviceand an experienced flautist considering the blowingpressure, the jet length and height, the area of theoutcoming flow. Although it was not used to quan-titatively describe the differences between the twoplayers, the authors noticed that the blowing pres-sure presents more acoustic frequency componentsin the mouth of the experienced flautist than in thatof the novice.No specific observation of the spectralcontent was carried in this study. Coltman[11] leda study of the effect of the mouth resonance onthe fundamental frequency in the flute. He showedexperimentally that, by varying the volume of an

artificial mouth, the fundamental frequency canshift of 10 cents (100 cents = semitone), but theinfluence on the spectral content is not observed.Chen et al.[7] led experimental measurements ona recorder player during performance. They foundthat the recorder player controlled his vocal tractimpedance. No evidence of vocal tract tuning withthe note played was found. They didn’t observe thefluctuating part of the supply pressure. They foundthat the tongue position influence the broadbandcomponent of the sound. They also mentioned aneffect on the magnitudes of the first 10 harmonics.The effect of the changing of vocal tract shape

has been more investigated for reed instrumentsby[31, 8, 9, 10, 25]. A sound synthesis model in-cluding the vocal tract has also been proposed byGuillemain[26] for the clarinet. In those studies,the vocal tract is modelled as a series impedanceadded to the pipe impedance. For flute-like instru-ments, the coupling through the channel betweenthe instrument and the vocal tract can be writtenusing Bernoulli relation[38, 2], preventing a linearcombination of the impedances of the vocal tractand the instrument.In the study presented in this paper, the follow-

ing questions are addressed. Is the musician able tocontrol the acoustic coupling between the recorderand the mouth cavity? What is the effect of thiscoupling on the sound production ? Lastly, canthe existing models be modified to include this ef-fect? This paper is a first step in studying the in-fluence of the vocal tract on sound production offlute-like instruments. While the experiments pre-sented in this paper were carried on recorders, themodel presented may be applied to other flute-likeinstruments.The paper is structured as follow: section 2

presents a preliminary study with recorder play-ers addressing the first question about the possi-bility to control the coupling with the vocal tract.A second experiment was developed in laboratory,allowing a better repeatability of the measurement(section 3). It used feedback in an artificial play-ing system to vary the coupling and to observe itseffect on sound production.Data gathered during these studies provided the

basis to modify an existing model of sound pro-duction,the jet-drive model, by including the effectof the vocal tract. The Jet-Drive model is com-monly used to model all flute-like instrument and

2

p

pac

Figure 1: Sketch of the modified recorder that allowsmeasurement of the acoustic pressure inside the boreand the pressure inside the mouth.

the modification proposed here may as well be ap-plied to the modelling of all flute-like instrument.The modified model is presented in section 4 whilesection 5 presents the predictions of the model anddiscusses its limitations.

2 Preliminary observations

This section presents preliminary measurementscarried out on recorder players in order to inves-tigate the controllability and the effect of the vocaltract shape on the sound production.

2.1 Setup and protocol

Two pressure sensors are mounted on an altoAesthe recorder, made by the recorder maker Jean-Luc Boudreau (figure 1). The first sensor is aB&K microphone model 4938 mounted through thewall, flush with the inner wall and measures theacoustic pressure pac near the labium. The sec-ond sensor is a Honeywell pressure sensor model176PC14HG1. It allows measurement of the pres-sure p within the mouth of the player thanks to acapillary tube passing through the mouthpiece andending in the mouth. The setup followed the onesused by[18, 31, 32, 38, 39].

The discussion showed that there is no commonplaying techniques, shared by professionals playersand teachers interviewed, that emphasizes clearlythe role of the vocal tract. Without any musicalconsensus, we could not provide musical instruc-tions that would result in changes of the vocal tract.

A first interview with a professional player and witha teacher allowed to establish a protocol for a sec-ond interview with five other professional recorderplayers.

As it is a preliminary study, the instructions ofthe second interview focused on isolated notes andmusical scales. The recorder players were askedto modify sound features such as the timbre whileplaying these exercises. Players were given no in-structions about using the vocal tract shape butwere invited to use whatever techniques they choseto effect the changes in timbre. The freedom givento the players prevents the possibility to carry sta-tistical analysis of data: the data are acquired withdifferent musicians, each of them playing differentnotes and using different techniques. The aim ofthese preliminary measurements was to identify thetechniques that can be usually used by recorderplayers.

2.2 First results

In these interviews, different techniques involvingthe vocal tract have been mentioned by the musi-cians. Some of them are, according to the player,directly related to a control of the vocal tractshape (“varying the mouth volume”, “opening thenasal cavity”, etc.). Others are related to a pos-ture (“gritting the teeth”, etc.). According to therecorder players, these different techniques may beassociated to a change in the timbre or other as-pects of the sound production such as the “projec-tion” or the “directivity”.The fluctuations p′ of the supply pressure p are

regarded as consequences of the oscillation of theacoustic pressure in the pipe pac. The amplitude ofthe fluctuating part p′ is generally between 10 and0.1 percent of the steady part. Assuming that thefluctuating part of the supply pressure p′ is a smallperturbation of the steady part, the coupling canbe linearized [3], and the relation between p′ andpac can be described as a transfer function

P ′(ω)

Pac(ω)= Gp(ω)e

jϕ(ω), (1)

with P ′ and Pac the Fourier transform of p′ andpac, respectively, and where the Gp and ϕ are thegain and the phase of the transfer function.

3

Figure 2: Isolated note performed by a first recorderplayer (C6, 1046 Hz). He reports changing his mouthvolume while playing. From top to bottom: gain of cou-pling Gp, phase of coupling ϕ, amplitude of the eightfirst harmonics of pac in dB.

A change in the vocal tract shape may lead toa change in the gain of coupling. Figure 2 showsthe gain Gp and the phase of coupling ϕ on anisolated note (C6, 1046 Hz) performed by a firstplayer who reports “varying his mouth volume”.There is no way to check objectively the move-ments performed by the recorder player. However,a change in the gain and the phase of couplinghas been observed and corresponds to what therecorder player asserts: a narrowing and wideningof the mouth. Starting from a “large” mouth vol-ume (1 < t < 2 s), the recorder player shrinks it(2.5 < t < 4 s) then brings it back to the initialvolume (4.5 < t < 5.5 s). This can be interpretedas following: as the volume decreases, the mouthimpedance at the playing frequency (1047 Hz) risessubstantially and the gain of coupling Gp increasesfrom −60 dB to −20dB. Simultaneously, the phaseϕ decreases from 0 to π as the mouth volume de-creases.

Varying the vocal tract shape modifies the prop-erties of the coupling. The properties of air flowthrough the channel of the instrument are there-fore expected to be modified, which would in turnmodify the sound production. The amplitude ofthe harmonics of the inner acoustic pressure pacdepends on the gain and the phase of coupling (seefigure 2). As the gain reaches its maximal value

Figure 3: Isolated note performed by a second recorderplayer (G5, 784 Hz). He reports opening his nose cav-ity gradually. From top to bottom: gain of coupling Gp,phase of coupling ϕ, amplitude of the eight first har-monics in dB.

Gp = −20 dB (p′/pac=0.1), the even harmonics in-crease of about 5 dB. The odd harmonics remainhowever constant, or decrease.

The change of the spectral content as a functionof the coupling with the vocal tract is not system-atic. Figure 3 shows the gain and the phase ofcoupling and the amplitude of the first eight har-monics of the inner acoustic pressure on an isolatednote (G5, 784 Hz) performed by a second recorderplayer who reports opening his nasal cavity grad-ually (gradually opening the velum). In this case,there is no monotonous behaviour of the harmon-ics. This may be due to a weaker maximal gainthan in the previous measurement (−30 dB).

This short experimental study confirms thatrecorder players have the possibility to use on pur-pose the vocal tract shape as already asserted byChen et al.[7]. It may or may not have measurableeffect on the harmonics amplitude, depending onthe gain of coupling Gp, which is a different effectthan the one highlighted by the study of Chen[7]which was focused on the broadband component ofthe sound. If the coupling affects the steady part, itis also expected to affect the unsteady part, such asthe attack transient. As the stability of the differ-ent oscillating regimes of the recorder are sensitiveto several parameters, the coupling with the vocaltract may be critical while discussing the stabilities

4

of regimes.The next section presents an experimental study

on the effect of the gain Gp and the phase ϕ ofcoupling on the sound production.

3 Laboratory experiments

In order to produce reproducible measurements, anartificial mouth for recorder[22] has been set tomimic the acoustic coupling with the vocal tract.One obvious way would be to make different hard-ware vocal tracts, but it limits the possibility ofdifferent coupling which can be investigated. Analternative way is chosen which allows the simula-tion of much wider coupling conditions with thesame setup. It is based on the injection of anacoustic flux in the artificial mouth, proportionalto the acoustic pressure measured within the in-strument, using a loudspeaker within the artificialmouth. The experimental setup is detailed in thenext section.

3.1 Experimental setup

For the playing machine study, the upper (nearlycylindrical) part of a Bressan Zen-On alto recorderis used, and its conical lower joints are replaced by acylinder of equal diameter, to achieve a near cylin-drical instrument, as used in a previous study[2].The total length is 26.5 cm resulting in a reso-nance frequency of approximately 565 Hz (betweena C#5(554 Hz) and a D5(587 Hz)). Thus theimpedance of the resonator is well known, and thecomparison with simulated results should be eas-ier. The other parameters of the instrument arethe same as an unmodified recorder (window lengthW=4 mm, channel height h= 1mm, inner diame-ter = 19mm) and are thus optimized to producesound at this frequency which lies in the normalrange of the instrument. The artificial mouth con-sists in a small cavity of diameter 44 mm and height55 mm yielding a volume V0= 5.6 10−5m3. Thisvolume is close to one of the experimenter mea-sured with the method of Coltman[11] (measure ofthe weight of the water contained in the mouth).Air coming from an upstream tank flows into thecavity through a hole of diameter 8 mm. The pres-

sure within the artificial mouth is measured withan Endevco pressure sensor (model 8507C-5). Thepressure is controlled by a numerical PID feedbackloop implemented with dSpace controller[22]. Theeffective pressure is compared to the desired oneevery 40µs. The calculations needed are performedin this time, which lead to a real time setup. ThePID is designed to have a global response time ofaround 10ms which leads to the filtration of thehigh frequencies.The acoustic coupling with the vocal tract is

mimicked by the injection of acoustic flux withinthe cavity. The upstream fluctuations are forcedby a feedback loop between the acoustic pressurepac within the instrument and a loudspeaker Aura-sound NSW2 (resonance frequency of 200 Hz, re-sistance of 6 Ω) set within the artificial mouth.The acoustic pressure within the instrument ismeasured with an Endevco pressure sensor (model8507C-2). The feedback loop is managed by thesame dSpace controller as the one used for the reg-ulation of the slowly varying pressure. The signalis numerically amplified and phase shifted using asecond-order all-pass filter. The parameters of thisfilter are tuned at the frequency previously mea-sured without forcing. The quality factor of thesecond-order all-pass filter is equal to 1 while thecentral frequency is adjusted to produce the desiredphase shift at the target oscillating frequency. Themodified signal is sent into a power amplifier (Pi-oneer A107 set with a constant gain) driving theloudspeaker. The artificial mouth and the loud-speaker are held by a larger cavity (volume of 0.13m3) which purpose is to absorb the backward wavegenerated by the loudspeaker. The fluctuating partof the supply pressure is checked in order not to af-fect the regulation of the slowly varying pressureby the PID loop.The phase and amplitude response of the whole

feedback loop depends on the electrical impedanceof the loudspeaker, on the lag due to numericaltreatment as well as on the response of the instru-ment itself. The phase and the gain are thereforemeasured a posteriori, after measuring the pres-sures within the mouth and the instrument. Theymay differ from what was a priori targeted. Thephases are estimated with a quadrature phase de-tection algorithm[27, 3] (as used for telecommu-nication receivers) while the amplitudes are esti-mated using short term Fourier transform. Results

5

are plotted as functions of the a posteriori couplingparameters.

3.2 Steady sounds and spectral con-

tent

The first of the two blowing conditions used waschosen to study the effect of the coupling on steadysounds, and more precisely on the spectral contentof steady sounds. The artificial mouth is suppliedwith a constant flow. Several runs have been per-formed for sixteen phase shifts by steps of π/8 andfour coupling gains (Gp= −34, −26 −20, −14 dB)as well as for two mean pressures 〈p〉= 140 Pa, 400Pa. These two values are chosen to provide oscil-lation within the first regime, respectively far fromand close to the regime change threshold for in-creasing pressure regime, respectively far from andclose to the regime change threshold for increas-ing pressure (≃ 695 Pa). The former provides asound close to a pure tone while the latter providesa sound with a strong second harmonic, because ofthe spectral enhancement of the non-linear satura-tion due to the increase of pressure.

The spectral content is characterised by the en-ergy distribution discriminating the even and oddharmonics. An un-parity spectrum index is defined:

I = 10 log

N∑

p=1(a2p+1)

2

N∑

p=1(a2p)

2

, (2)

that weights the sum of the energy of the odd har-monics (a2p+1)

2 over the one of the even harmonics

(a2p)2. By definition, the energy of the fundamen-

tal is discarded because it is larger than all theother harmonics, and would tend to smooth thevariation of the index I. Harmonics from rank 2 to7 are considered (N = 3), without the fundamen-tal. Figure 4 shows the evolution of the first eightharmonics and the associated un-parity index asfunction of the phase ϕ for the maximal gain Gp=−14 dB and for the mean supply pressure of 400Pa. The amplitude of the even harmonics may in-crease up to 20 dB. It is maximal for a phase ϕclose to π and minimal for a phase ϕ close to zero.The amplitudes of the odd harmonics behave in the

Figure 4: Amplitudes of the first eight harmonics andun-parity index I (defined by Eq. (2)) as function of thephase of coupling ϕ for the maximal gain (p′/pac = 0.2,u′/vac = 0.9) and for a supply pressure of 〈p〉 = 400 Pa.

opposite way. The un-parity index I is a good in-dicator of the energy distribution between odd andeven harmonics. It summarizes in one index the rel-ative evolution of odd and even harmonics. Here,it evolves in a range of 30 dB.

Without coupling, the index is around -2 dB forthe two supply pressures. Its value depends, amongothers, on geometrical parameters, such as the off-set between the channel axis and the labium[23].

Figure 5 shows the un-parity index for the fourgains of coupling and the two supply pressures asfunction of the phase ϕ. The trend described inthe previous paragraph for one gain is the same forall gains, at least for the supply pressure of 400Pa. The un-parity indexes are minimal for phasesϕ close to π and maximal for phases ϕ close to zero.The ratio of the odd harmonics over the even in-creases as the phase gets close to zero and decreasesas the phase gets close to π. However, this effectis less pronounced when the gain of the couplingdecreases. For the 140 Pa supply pressure, no cleartrend has been observed, even for the maximal gain.

The two supply pressures correspond to two dif-ferent domains: one with few harmonics, and theother with much more harmonics. The sensitivityof the system with respect to the coupling with thevocal tract seems to depend on the initial energydistribution, i.e. without coupling, among the har-

6

Figure 5: Un-parity index I (defined by Eq. (2)) asfunction of the phase of coupling ϕ for all gains (in-creasing gains with darker gray: Gp = −14 dB: black,Gp = −34 dB: lightest gray) and for two mean supplypressures (140 Pa et 400 Pa).

monics.The oscillating frequency results from the bal-

ance between the harmonics amplitudes ruled bythe non-linear mixing of the exciter. As soon asthe relative strengths of the harmonics are changedbecause of the coupling with the vocal tract, thefrequency is expected to deviate from its naturalcase. For the maximum gain used (Gp = −14dB),the frequency varies of around 7 cents between ϕclose to π (minimum) and ϕ close to 0 (maximum).This value is in the same order than the shift foundby Coltman[11] (around 10 cents).

3.3 Effects on regime change thresh-

olds

The second blowing condition aims at studying theeffect of the coupling on the stability of the regimesof oscillation. The supply pressure now slowlyvaries with a triangle shape in time between twoextremal values (〈p〉 ∈ [50, 1200] Pa). The range ofpressure allows the instrument to sound in its firsttwo oscillating regimes. Several runs have been per-formed with two phases (ϕ= 0 or π) and three gainsof coupling (Gp= −34, −20, −14 dB). The phase ϕis set based on the frequency of the instrument justbefore the transition to the second register (closeto, but above, the resonance frequency of the pipe

Figure 6: Experimental dimensionless frequency f/f1as function of the mean supply pressure 〈p〉 for threecoupling conditions. From top to bottom: without cou-pling, ϕ = π, ϕ = 0.

565 Hz).

Figure 6 shows the oscillating frequency as func-tion of the mean supply pressure 〈p〉 for all thecoupling conditions. The operating range of thefirst regime is significantly affected by the acousticcoupling with the vocal tract. For a phase ϕ ≃ π,the increasing threshold from the first to the sec-ond regime is shifted from 695 Pa to 745 Pa (7 %)for the maximal gain (Gp = −14 dB). The smallergains seem to have only a moderate effect on theextent of the first regime. For a phase ϕ ≃ 0, theincreasing threshold is reduced, on the contrary,from 695 Pa to 630 Pa (6.5 %), for the maximalgain. For a phase ϕ ≃ 0, smaller gains have also animpact on the increasing thresholds.

The decreasing threshold - that from the secondto the first regime - is also modified by the acous-tic coupling. For a phase ϕ ≃ π, the threshold isreduced from 534 Pa to 495 Pa (7 %), for the max-imal gain. For a phase ϕ ≃ 0, the threshold is alsoreduced from 534 Pa to 520 Pa (3 %), for the max-imal gain. The decreasing threshold is reduced forall the coupling conditions.

The phase seems to be the relevant parameterconcerning the effect of the changing of vocal tractshape on the stability of the regime, the gain hav-ing only an effect of emphasis on threshold shifts.A phase ϕ close to π extends the hysteresis rangewhile a phase ϕ close to 0 reduces it.

7

This result seems counter-intuitive since thephase close to π has also been identified as a de-termining factor in the rise of the even harmonics.One would expect that an increase in the secondharmonics within the first regime, close to the fre-quency of the first harmonic on the second regime,would induce a reduction of the increasing thresh-old. This highlights the complexity of the mech-anisms that rule the regime changes and calls fordeeper investigations.

4 Model

The modified model of sound production incorpo-rating changing of vocal tract shape uses three dis-tinct ingredients: the description of the coupling,the growth of instabilities on a jet and the Jet-drivesource model that has been shown to predict quitefair results in quasi steady condition[2]. The majordifficulty lies in combining these descriptions basedon different time scales and under different assump-tions. In other words the quasi-steady model isenhanced in order to allow fast fluctuations of theblowing pressure.

4.1 Modelling the coupling

Most studies related to the effect of the vocal tracttuning focus on the impedance of the vocal tract,described in the frequency domain, and put in se-ries with the instrument impedance [31, 8, 9, 10,25]. This description can not be used in flute-likeinstruments models. There is no simple relationof continuity between the mouth variables (pres-sure p and flow) and the acoustic variables into theinstrument (pac and vac). The impedances are notin series. Another description uses, in the temporaldomain, a set of two differential equations[38] (massand momentum conservation through the flue chan-nel). This offers the advantage to keep the hydro-dynamic variables: fluctuations u′ of the jet veloc-ity u = 〈u〉+u′, with 〈u〉 the steady part of the jetvelocity, are induced by the coupling with the vocaltract. As the present study is a first approach inmodelling the effect of the coupling, no care will betaken to describe precisely the physical mechanism

of this coupling. This mechanism has started to bestudied by Auvray et al.[3].The description of the coupling in terms of a

transfer function between the mouth and the borepressures (see Eq. (1)) is well suited for the ex-periments. For the model, an equivalent transferfunction can be written between the jet velocityfluctuations u′ and the acoustic velocity within thewindow vac:

U ′(ω)

Vac(ω)= Gu(ω)e

jφ(ω), (3)

with U ′ and Vac the Fourier transform of u′ andvac, respectively. In the present study the gain Gu

and the phase φ are considered as the two controlparameters of the coupling. They are varied inde-pendently without taking care of their physical ori-gins. It highlights the effects of these two variableson the model.In the initial model[2], the jet instabilities are

triggered by the transverse acoustic velocity nearthe flue exit. This external perturbation is anti-symmetric and only the anti-symmetric unstablemode is then triggered. This has been confirmedby several experimental observations[20]. Takingaccount of the fluctuating part of the velocity u′,another excitation for the jet instabilities is added.The effect of this perturbation on the jet is studiedin the next section.

4.2 Effect of fluctuating velocity on

the jet instabilities

Even if it remains an academic situation, the studyof the instabilities of an infinite plane jet for in-compressible parallel and non viscous flow still pro-vides insights on the instability mechanisms thatoccur during flute operation for a finite extent androughly plane jet.The theoretical jet is assumed to flow along x

direction. The cross-stream profile U(y) along y

direction is assumed to be a Bickley profile:

U(y) =〈u〉

cosh2y

b

, (4)

where 〈u〉 is the centerline velocity and b the half-width which is constant in the case of a non viscous

8

flow. A perturbation flow u = ux+ vy is added tothe mean flow. The linear stability analysis consistsin finding a propagative solution of the perturba-tion field u = uei(ωt−αx). Such a perturbation flowis ruled by the equation of Rayleigh[28]:

[

U(y)− ω

α

]

[

d2v(y)

dy2− α2

u(y)

]

− d2U(y)

dy2u(y) = 0, (5)

where ω is a real driving pulsation, α = αr + iαi

the complex wave number and cp = ω/αr the phasevelocity. This equation is subject to the boundaryconditions:

u′(±∞) = u(±∞) = 0. (6)

This set of equations can be solved numerically[28, 30]. For a symmetric jet profile U(−y) = U(y),Mattingly and Criminale[28] showed that two un-stable modes may rise, the antisymmetric v(−y) =−v(y) being more unstable than the symmetricv(−y) = v(y). This is illustrated in figure 7, whichshows the dispersion relations for such symmetricand anti symmetric perturbations. The parame-ters associated with antisymmetric and symmetricperturbations are respectively indexed by (a) and(s).For a given driving pulsation ω and for a given

set of parameters, i.e. for a given Strouhal numberωr = ωb/〈u〉, the spatial growth factor −αi de-scribes how the perturbation is amplified while thereal phase velocity cp = ω/αr describes at whichvelocity the perturbation is convected. Withoutany external excitation, the anti symmetric pertur-bation is more unstable than the symmetric, corre-sponding to the Strouhal number ωb/〈u〉 ∼ 0.25.As soon as the unstable mode is amplified, thepresent linear description is no longer valid. Thevorticity of the shear layer accumulates at the in-flection points. However, the structure of the flowafter non-linear development maintains a “mem-ory” of the structure presented during the lineardevelopment. Thus, the relative position of vorticeson the two shear layers of the jet is an indicationof which mode has been excited during the initialpart. This is confirmed by an experiment of flowvisualization presented in appendix A.The main hypothesis of the modification of the

current simplified model is that the fluctuations ofthe jet velocity profile due to the acoustic couplingwith the vocal tract consist in a symmetric initial

perturbation of the jet. The coexistence of twokinds of perturbations, the anti-symmetric acous-tic field and the symmetric pulsating jet flow, leadsto the linear superimposed growths of the two un-stable modes, anti-symmetric and symmetric, withgrowth factors αi and phase velocities cp of orderof magnitude predicted by the linear stability anal-ysis of the infinite jet flow. The superimposition islinear during the linear development of the jet.

4.3 Modified Source model

As with every self-sustained instrument, the modelof sound production of a flute-like instrument isbased on a feedback loop system connecting a non-linear exciter and a linear resonator[29]. In thespecific case of a flute-like instrument, the modelis refined by adding non-linear losses to the formerlinear acoustic part[21].The modified model presented in this section is

based on the Jet-Drive source model as written byAuvray[2]. Only the parts including modificationsare extensively presented here (exciter), the othersbeing only mentioned (linear and non-linear acous-tics).The air column of the resonator is driven by

a force term written as a pressure difference ∆pacross the window. The acoustic velocity vac withinthe window results from the excitation of the res-onator by the difference of pressure through theadmittance relation:

Y =Vac

∆P, (7)

where Vac and ∆P are the Fourier transforms ofvac and ∆p, respectively. The resonator part, i.e.the admittance, is described as a modal acousticadmittance:

Y =

∞∑

n=1

jωYn

ω2n − ω2 + jεnωnω

, (8)

where Yn, εn and ωn are the amplitude, the damp-ing coefficient and the pulsation of the n-th mode,respectively. The non-linear losses, due to the con-striction which yields the formation of a free jet ev-ery half acoustic period, are written as a pressuredrop across the window:

∆plos = −1

2ρ0

vac|vac|α2vc

, (9)

9

Figure 7: Top: Shape of the perturbation of the jet added to the mean flow for anti symmetric (left) and symmetric(right) cases. Bottom: Corresponding real phase velocity cp = ω/αr, real part αr and imaginary part −αi of thewave number α as function of the Strouhal number ωr = ωb/〈u〉, obtained by numerical solution equation Eq.(5).

with αvc = 0.6 a vena contracta coefficient. Thispressure drop will be added to the source pressuredifference.

The exciter part is split into two distinct parts:the birth, growth and convection of the jet insta-bility from the flue exit to the labium and thejet/labium interaction as a mass injection fromboth sides of the labium.

Regarding the antisymmetric perturbations

Either spreading or not, the jet is described by aunique variable: its center position denoted η(x, t).The development of linear anti-symmetric instabil-ities from the flue exit to the labium is assumed tobe well described by the center position of the jetalong the same path. As for the perturbation fieldof the linear stability analysis, the center positionis then described as a propagative solution, func-

tion of t−x/c(a)p where c

(a)p is the phase velocity of

the anti-symmetric perturbation. The issue of thetriggering of the jet perturbation is still an openquestion, but different authors[5, 15, 20] agree thatthe jet is mainly perturbed at the separation pointof the flow (x ≃ 0) by the acoustic velocity field.Based on flow visualisations, de la Cuadra[15, 16]proposed an expression of the jet center line as a

propagative function:

η(x, t) = e−α(a)i

xη0(t− x/c(a)p ), (10)

with −α(a)i ∼ 0.3/h the anti-symmetric growth

rate, h the height from which the jet emerges and

c(a)p ∼ 0.4〈u〉 the anti-symmetric phase velocity andwhere η0(t) is an “initial perturbation” at x = 0.This expression is only valid a few characteristicdistances h downstream so that the initial pertur-bation η0 has no specific physical meaning exceptas an empirical fit to the data. De la Cuadra foundfor his experiments:

η0(t)

h=

vac(t)

〈u〉 , (11)

where vac is the transverse acoustic velocity nearthe separation point (at x = 0). The values of the

growth factor α(a)i and the phase velocity c

(a)p vary

from one condition to another but remain of thisorder of magnitude, in agreement with the linearstability analysis presented in the previous section.

Adding the symmetric perturbations

When the velocity profile at the flue exit oscil-lates because of the acoustic coupling with the vo-cal tract, symmetric instabilities are expected to

10

arise. As for the anti-symmetric instability thatis assumed to modify only the jet center position,the symmetric instability is assumed to modify onlythe half width of the jet b by adding a perturba-tion b′(x, t). As for the anti-symmetric case theperturbation is assumed to be described by a prop-

agative solution function of t− x/c(s)p where c

(s)p is

now the phase velocity of the symmetric unstablemode. Similarly, the half width perturbation canbe written:

b′(x, t) = e−α(s)i

xb0(t− x/c(s)p ), (12)

with α(s)i and c

(s)p the growth factor and the phase

velocity of the symmetric mode and b0(t) an “initialperturbation” of the jet thickness. The initial per-turbation can be written, in first approximation, inthe same form as the anti-symmetric perturbation:

b0(t)

h= σ

u′(t)

〈u〉 , (13)

with σ a proportionality coefficient and u′ the fluc-tuating part of the centerline velocity of the jetdescribed in the previous section. The parameterσ can be roughly estimated by assuming momen-tum conservation between an oscillating Poiseuilleprofile at the flue exit (x = 0) and a Bickley pro-file established downstream, as well as conservationof the centerline velocity (see appendix B). Theproportionality coefficient σ remains a sensitive pa-rameter that further flow visualisations could focuson.Finally, the total jet velocity profile at the labium

is obtained by combining the transverse displace-ment due to the anti-symmetric perturbation andthe modulation of the thickness due to the symmet-ric perturbation:

UW (y, t) =〈u〉

cosh2y − η(W, t)

b+ b′(W, t)

. (14)

Once it reaches the labium, the jet is assumedto split into two flows Qin and Qout going towardthe interior of the instrument and outwards, re-spectively. Each flux can be split into a steadyand a fluctuating components: Qin = 〈Qin〉+Q′

in

and Qout = 〈Qout〉 + Q′out. The flux are as-

sumed to be injected at specific points, behind thelabium, and are separated by an acoustic distance

δd = (4√2hW )/π, which is also a sensitive param-

eter of the model[38]. This distance being smallcompared to the acoustic wavelength, the force ap-plied on the air column is due to the accelerationof the small mass of the assumedly incompressibleair which is contained between the two injectionpoints. The source term is then written as a pres-sure difference across the window:

∆psrc = − ρδdWH

d

dt

(

Qin −Qout

2

)

, (15)

with W and H the length and the width of the win-dow, respectively. The difference of fluxes dependson the velocity profile at the labium:

Qin(t)−Qout(t) =

H

[∫ y0

−∞

UW (y, t)dy −∫ ∞

y0

UW (y, t)dy

]

, (16)

with y0 the vertical offset between the channel cen-terline and the labium. The flux Q′

in directly de-pends on the transverse position η and the widthof the jet b at the labium through Eq. (14).Using Eqs. (14) (15) and (16) yields the modified

source term

∆psrc(t) =

ρδd〈u〉W

d

dt

[

(b + b′(W, t)) tanhη(W, t)− y0b+ b′(W, t)

]

. (17)

The total difference of pressure across the windowis given by combining Eqs. (9) and (17):

∆p = ∆plos +∆psrc. (18)

4.4 Resolution of the model

While there are several parameters that can beadjusted within the model, we will only focus onthose related to the change of vocal tract shape.Some studies propose an investigation of the otherparameters[2, 12].The control parameters of the simplified model

are then the mean or slowly varying blowing pres-sure 〈p〉 (or equivalently slowly varying jet velocity〈u〉) and the parameters related to the acoustic cou-pling with the vocal tract. These have been iden-tified to be the gain Gp and the phase ϕ between

11

the fluctuating part p′ of the blowing pressure andthe acoustic pressure pac (see Eq. (1)). It is almostequivalent to consider instead the gain Gu and thephase φ between the fluctuating part u′ of the jetvelocity and the acoustic velocity vac (see Eq. (3)).The gain will be varied using the proportionalitycoefficient σ involved in Eq. (13).

The phases φ and ϕ are analytically linked. Intothe pipe, the acoustic wave is a standing wave. Thephase between the acoustic pressure pac and theacoustic velocity vac is π/2. The linearization ofthe mass conservation between the mouth cavityand the flue exit yields a phase shift of π/2 betweenthe fluctuating part p′ of the blowing pressure andthe fluctuating part u′ of the jet velocity[3]. Thesetwo phase shifts lead to the relation φ = ϕ+ π.

The numerical resolution is performed by thesame algorithm used by Auvray et al.[1]. It is astep by step time domain resolution. The sam-ple rate is known to be sensitive for the numericalprocedure[33]. The sample rate Sr= 0.1 GHz istaken as high as possible while keeping an accept-able computation time.

The auto-oscillation is initiated by injecting avery low amplitude wide band noise. The systemthen locks on the regime of oscillation correspond-ing to the blowing condition (mean jet velocity).Two kinds of blowing conditions are provided inorder to tackle two issues: one concerning the spec-tral content, the other the stability of the regimesof oscillation.

For the former, the time of simulation Ts is short(2 s) and the mean jet velocity is constant. The sys-tem is solved over several runs for different couplingconditions varying both the gain and the phase ofthe acoustic coupling.

Modalω1= 3547 rad/s ε1= 3.97 10−2 Y1= 1.38 10−3

ω2= 2.023 ω1 ε2= 3.18 10−2 Y2= 1.21 10−3

ω3= 3.066 ω1 ε3= 2.85 10−2 Y3= 9.81 10−4

Antisym. c(a)p = 0.4〈u〉 α

(a)i

= 0.3/h

Symmetric c(s)p = 0.8〈u〉 α(s)i

= 0.2/h

Comput.

Ts = 2s or 20s Sr = 0.1GHz c0 = 340 m/sρ0=1.2kg/m3 αvc = 0.6 y0 = 0.1mmW = 4mm H = 12mm h = 1mmb = 2h/5 σ = 0 ↔ 1〈u〉 = 20m/s or 30m/s or 1 ↔ 56m/s

Table 1: Parameters used for the computation. Themodal parameters of the pipe are taken from Auvray etal.[2].

For the latter, the time of simulation is longer (20s) and the mean jet velocity is slowly varying witha triangle shape in time between its two extremevalues (1 m/s and 56 m/s). Table 1 indexes all thesimulation parameters.

5 Simulation results and dis-

cussion

5.1 Simulated steady sounds

In the simulation, the main control parameter isthe phase shift φ between the fluctuating velocityu′ and the acoustic velocity vac. In order to com-pare the simulation with the experimental data,the results are presented as function of the phaseϕ = φ+ π between the fluctuating supply pressurep′ and the acoustic pressure pac.

The modal decomposition of the resonator is re-stricted to three modes in the simulation. As thehigher order harmonics are naturally weaker, thishas only little impact on the global simulation: onlythe first harmonics significantly contribute to per-turb the jet, at least during the steady state. Theun-parity index is approximated, for the simula-tion, by

I ≃ 20 loga3a2

, (19)

with a2 and a3 the amplitudes of the second andthird harmonics estimated within the steady partof the simulated sound.

As in the experiments, setting a priori the cou-pling parameters does not allow to predict at whichphase and gain the coupling really occurs. Inparticular, for large gain of coupling, the simula-tion does not provide stable oscillation on the firstregime whereas it was stable without coupling.

The order of magnitude of the un-parity indexwithout coupling is strongly underestimated: −10dB for the simulated index (see figure 8) when themeasured one is −2 dB (see figure 5). Neverthe-less, the simulation still predicts the trend exper-imentally observed. Firstly, the variation of thespectral content depends on the phase of couplingϕ. The un-parity index shows minimal value for

12

−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0−14

−13

−12

−11

−10

−9

−8

−70.7 0.5 0.3 0.1 −0.1 −0.3 −0.5 −0.7 −0.9 −1.1 −1.3

Gu = -34 dB

Gu = -20 dB

Gu = -10 dB

Gu = - 6 dB

φb/η/π

φ/π

I(d

B)

−2 −1.8 −1.6 −1.4 −1.2 −1 −0.8 −0.6 −0.4 −0.2 0−9

−8.5

−8

−7.5

−7

−6.50.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1.2

Gu = -34 dB

Gu = -20 dB

Gu = -10 dB

Gu = - 6 dB

φb/η/π

φ/π

I(dB

)Figure 8: Simulated un-parity index I as function of the phase of coupling ϕ for several gains of coupling u′/vacand for two jet mean velocities 〈u〉 = 20 m/s (right) and 〈u〉 = 30 m/s (left). The phase shift ϕb′/η between themodulation of the jet width b′ and the jet center position η is also indicated.

ϕ = π and maximal value for ϕ = 0, as in the ex-periments. The minimum is reached for a phaseshift ϕb′/η close to zero.

This can be intuitively understood by evaluat-ing the flux going toward the interior of the instru-ment (below the labium) and outwards (above thelabium) according to the phase shift ϕb′/η. For twooscillations in phase at the labium (ϕb′/η=0), assketched in figure 9 (left), the injection of flux abovethe labium is larger than the one below it. Thesymmetry of the source is strongly broken. This

Figure 9: Exaggerated sketch of the superimposition ofthe anti-symmetric and symmetric modes of jet, excitedby the acoustic velocity vac and the fluctuating jet ve-locity u′, respectively, for two phases of the oscillation(top and bottom) and for two phase shifts ϕb′/η = 0(left) and ϕb′/η = π (right).

is the same for two oscillations out of phase (fig-ure 9, right). However, when the oscillations arein quadrature, the effect of the symmetry breakis damped: the maximal and minimal thickness ofthe jet due to the modulation occur when the cen-ter position of the jet is zero. In this position, ifthe offset is zero (y0 = 0mm), the problem is sym-metrical and the modulation of the thickness havea minimal effect. In the model used, the offset isnot zero (y0 = 0.1mm): the minimal effect of thethickness modulation occurs for ϕb′/η/π ≈0.7 and0.6 and not for ϕb′/η/π=0.5 (see figure 8).

Secondly, the un-parity index also depends on thegain of coupling, as experimentally observed. Thehigher the gain of coupling, the more pronouncedthe effect. The minimum of the un-parity indexseems to depend on the jet mean velocity. It is diffi-cult to compare with the experimental data, whereit is not the case, at least for the two measuredmean pressures. However, the simulation qualita-tively agrees with the phase ϕ corresponding to ex-tremal values of the un-parity index and with theeffect of the gain of coupling. This calls for a moreexhaustive measurement campaign with simultane-ous electroacoustic simulated coupling and flow vi-sualisation to investigate the effect of the relativephase ϕb′/η of both symmetric and anti-symmetricinstabilities.

13

Figure 10: Simulated dimensionless frequency f/f1 asfunction of the reduced jet velocity 〈u〉/Wf1 for differ-ent coupling conditions. From top to bottom: no cou-pling, u′/vac = 0.1 and ϕ = π and u′/vac = 0.1 andϕ = 0. For clarity purpose, aeolian regimes have beendiscarded.

5.2 Simulation of regime change

thresholds

The Jet-Drive model is known to predict numer-ous unrealistic aeolian regimes[2]. The stability ofthe oscillating regimes is only discussed regardingthe changes between the two main regimes. Theregime change mechanisms are not fully understoodyet[35]. The critical parameters of the model havea direct impact on the stability of the differentregimes[2]. For small gains of coupling, no varia-tions of the regime change thresholds are predictedby the simulation. However, for large gains, theincreasing threshold depends on the phase ϕ. Fig-ure 10 shows the oscillating frequency as functionof the reduced jet velocity 〈u〉/Wf1 for differentphases ϕ, when the gain is Gu = −20 dB. If thedecreasing thresholds are unchanged regarding thecoupling conditions, the increasing thresholds dodepend on the phase. The prediction qualitativelyagrees with the experimental observations: an in-crease in the hysteresis range occurs for a phase ϕclose to π while a reduction occurs for a phase ϕclose to zero.

Every change in the harmonics, in amplitude andphase, seems to have a direct impact on the sta-bility of the regimes. No clear conclusion about

the regime change mechanism can be drawn fromthese results. These first simulated results call foranother study focusing on the relation between thespectral content and the regime stability, which isout of the scope of the present paper.

5.3 Limitations of the model

Mismatches between the predictions and the mea-surements in terms of spectral content (un-parityindex I) and regimes stability (regime changethresholds) can be due to a wrong estimation of thenumerical values of the parameters of the model aswell as the intrinsic assumptions of the model itself.The set of parameters used here was chosen to

match the frequency behaviour (pitch and regimechange threshold). Therefore it is natural that dis-crepancies arise concerning the prediction of thespectral content. The main limitation concerns thenumber of modes considered. Three modes to de-scribe the admittance may not be enough to de-scribe the filter behaviour of the resonator. Thenumber of modes should be increased and their nu-merical values should be checked[34, 14]. Anotherparameter that has a great impact on the spectralcontent is the offset y0 between the channel axisand the labium as shown by Fletcher[23] and dis-cussed by Auvray [4]. Deviation between the realoffset and the simulation value y0 = b/4 may bethe cause of the discrepancy. The other elementsthat affect how the jet flow is split at the labiumwill have an effect on the spectral content of thesource. This is the case of the velocity profile, itsshape and its thickness. The relation between hand b is not straightforward and requires to knowthe exact velocity profiles at the flue exit and at apoint further downstream. The value of b may dif-fer because the velocity profile at the flue exit maynot be a Poiseuille profile and the velocity profile atthe labium may not be a Bickley profile. In futurework, the new parameters of the model should bevaried to make it match the experimental data.Some limitations of the model may also be due to

its strong assumptions. The description of the hy-drodynamics as a one-dimensional pulsating vari-able may represent a crude assumption comparedto a two-dimensional pulsating flow within thechannel. However this remains a first order ap-proximation, like the general level of accuracy of

14

the whole model. Furthermore, much care has beentaken to describe properly the emerging flow undersuch excitation.The description of the jet instability uses a linear

theory for infinite plane jet with a symmetric veloc-ity profile. In the case of flute-like instruments, thejet is of finite extent and may not be plane. Thevelocity profile at the flue exit depends on the his-tory of the flow. In most flute-like instruments, thechannel does not show any particular symmetric re-lation. The profile at the flue exit is not expectedto be symmetric. However it may not be totallydisordered. The deviation from a symmetric pro-file can be seen as a slight perturbation of an idealcase. The maximum value of the amplification alsodepends on the jet velocity profile and thus on thegeometry of the channel and also on the way the jetseparates from the walls. The amplification is verysensitive to the existence and the shape of cham-fers [15, 5]. The present model is however not sorefined. The growth factor and the phase velocityremain sensitive parameters the effects of which onthe sound production have already been studied[2].Furthermore, the use of the linear theory has

been justified by experimental studies. Flow vi-sualisation experiments estimate the growth αihto be in the range [0.1 0.5] for Strouhal numbersωh/2π〈u〉 in the range [0 1], with h the height ofthe channel from which the jet emerges [15]. Theorder of magnitude of the spatial growth factor −αi

and real phase velocity cp are in qualitative agree-ment with the linear theory. A slight dependencyof the amplitude on the Strouhal number has beenfound, but on a smaller range of amplitude.To have more valid values of these parameters,

flow visualisation of superimposed symmetric andanti-symmetric instabilities should be conducted.That would provide an experimental estimation ofthe instabilities parameter. This approximationshould be more suitable than a theoretical estima-tion which relies on strong hypothesis. Further-more, an electroacoustic coupling experiment couldbe conducted simultaneously to flow visualisationto address the issue of the relative phase of thejet width and the jet centerline position, which hasbeen shown to be crucial. The energy distributionamong the harmonics seems to have an effect on thestability of the oscillating regimes, but the relativephase of the harmonics seems to be more decisivein the stability of the regimes.

6 Conclusions and perspec-

tives

This paper represents a study of the effect of chang-ing the vocal tract shape on the sound productionof flute-like instruments. The phase of the cou-pling appears to be the relevant parameter thathas an effect on the spectral content in terms ofodd/even harmonics energy distribution and on theregime change thresholds. These experimental ob-servations are qualitatively confirmed by the resultsof simulation using a simplified model of couplingbased on the modification of the jet instability.

The identification of the effect of the acousticcoupling allows the development of new protocols inorder to study how this control parameter is tunedby recorder player and how this is related to musicalintention. This could be performed along with per-ceptual listening tests to identify the precise pointsto focus on.

Changing vocal tract shape may also have an im-portant impact during the onset of the oscillationand the attack transient, and therefore on the per-ception of such musical sounds. The model pre-sented here is based on a steady-state analysis. Itis therefore not suited for a study of the transients.All the variations (rise of pressure, fluctuations ofpressure and jet velocity due to the acoustic cou-pling, propagation of the first acoustic wave withinthe pipe) occurs at the same time scale, the acousticperiod. Any attempt to model the coupling duringthe transient should avoid the description in thefrequency domain and should focus on the birth ofthe jet and the anti-symmetric and symmetric in-stabilities.

Acknowledgement

The authors would like to thank the recorderplayers, teachers and makers Joel Arpin, PhilippeBolton, Clemence Comte, Francoise Defours, Eti-enne Holmblat, Sarah Lefeuvre and Anne Leleu,who made this study possible.

15

A Flow visualisation of sym-

metric and anti-symmetric

“free” jet

This appendix presents a Schlieren flow visualisa-tion experiment of the superimposition of symmet-ric and antisymmetric jet instabilities.

A pure CO2 jet is released with a centerline veloc-ity of 〈u〉= 1.5 m/s within an acoustic field drivenby two loudspeakers mounted face to face. The ex-perimental set-up is similarly to the one used byDe la Cuadra[17]. The symmetry property of theacoustic excitation is gradually varied by graduallyphase shifting the two loudspeakers by a value ∆ϕ.The jet is excited at a frequency of 240 Hz.

When the two loudspeakers are in phase (∆ϕ=0,figure 11(a)), the acoustic field at the flue chan-nel exit is symmetric, yielding the rise of the sym-metric instability only. When the two loudspeakersare out of phase (∆ϕ=π, figure 11(d)), the acousticfield is anti-symmetric, yielding the rise of the anti-symmetric instability only. For transitional phaseshifts (∆ϕ=π/3, 2π/3, 4π/3, 5π/3, figures 11(b),11(c), 11(e), 11(f)), the acoustic field symmetry isa combination of both previous cases and both sym-metric and anti-symmetric jet instabilities are ex-cited.

The superimposition is linear during the initiallinear development of the instabilities, i.e. on thefirst 1/8 of the pictures. When the non-linear be-haviour of the instabilities arises, vortices appearon the upper and lower shear layers. Their loca-tions depend on the former state of the jet duringthe linear development.

B Estimation of the sensitive

parameter σ

It is possible to estimate the parameter σ, byassuming the momentum conservation between aPoiseuille profile at the flue exit and a Bickley pro-file established downstream as well as the conser-

vation of the central velocity which gives

h/2∫

−h/2

ρ0(〈u〉+ u′(t))2

(

1−(

y

h/2

)2)2

dy =

∞∫

−∞

ρ0

( 〈u〉cosh2(y/b(t))

)2

dy, (20)

leading to a relation between the thickness of theBickley profile and the evolution of the jet velocity

b(t) =2h

5

(

1 +u′(t)

〈u〉

)2

(21)

=2h

5+

2h

5

(

2u′(t)

〈u〉 +

(

u′(t)

〈u〉

)2)

.(22)

In Eq. (22) the thickness can be seen as a meanvalue and an ”hypothetical initial perturbation”

b0(t) =2h

5

(

2u′(t)

〈u〉 +

(

u′(t)

〈u〉

)2)

. (23)

In the case of small perturbation, these expressioncan be linearised

b0(t)

h= σ

u′(t)

〈u〉 , (24)

where σ = 4/5. At cost of approximations andassumptions, it is possible to find a linear relationbetween fluctuation of jet thickness and fluctuationof jet velocity. Even if this reasoning is a good argu-ment for the validity of the linear relation betweenthese two variables, the numerical value of the pro-portionality coefficient can’t be estimated this wayand still remains a sensitive point of the model.

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