Effect of Changing the Vocal Tract Shape on the Sound Production of the Recorder: An Experimental and Theoretical Study

1.7 MB·

by  R Auvray

Effect of changing the vocal tract shape on the sound production
of the recorder: an experimental and theoretical study.
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R. Auvray1,2) , A. Ernoult1,2), S. Terrien3), P.Y. Lagr´ee1,2), B. Fabre1,2), C. Vergez3)

n
a 1) Sorbonne Universit´es, UPMC Univ. Paris 06, UMR 7190, LAM-Institut Jean Le Rond d’Alembert,
J
75005 Paris, France. [email protected]

1 2) CNRS, UMR 7190, LAM-Institut Jean Le Rond d’Alembert, 75005 Paris, France.
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3) LMA, CNRS, UPR 7051,Aix-Marseille Univ, Centrale Marseille, F-13402 Marseille Cedex 20,

] France.
h
p Abstract
-
s Changing the vocal tract shape is one of the techniques which can be used by the players of
s
a wind instrumentsto modify thequality of thesound. It hasbeen intensely studied in thecase of reed
l instrumentsbuthasreceivedonlylittleattentioninthecaseofair-jetinstruments. Thispaperpresents
c
afirststudyfocusedonchangesinthevocaltractshapeinrecorderplayingtechniques. Measurements
.
s carried outwith recorder playersallow toidentify techniquesinvolvingchanges ofthemouthshapeas
c
wellasconsequencesonthesound. Asecondexperimentperformedinlaboratory mimicsthecoupling
i
s withthevocaltractonanartificialmouth. Thephaseofthetransferfunctionbetweentheinstrument
y
and the mouth of the player is identified to be the relevant parameter of the coupling. It is shown
h
to haveconsequences on the spectral content in terms of energy distribution among theeven and odd
p
harmonics, as well as on the stability of the first two oscillating regimes. The results gathered from
[
the two experiments allow to develop a simplified model of sound production including the effect of

1 changing the vocal tract shape. It is based on the modification of the jet instabilities due to the
v pulsating emerging jet. Two kindsof instabilities, symmetric and anti-symmetric, with respect to the
5
stream axis, arecontrolled bythecouplingwiththevocal tractand theacousticoscillation within the
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pipe, respectively. The symmetry properties of the flow are mapped on the temporal formulation of
5
thesourceterm,predictingachangeintheeven/oddharmonicsenergydistribution. Thepredictions
5
are in qualitative agreement with theexperimental observations.
0
.
1
In the case of flute-like instruments (mainly
0
recorders, flutes and organ pipes), the first control
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1 parameter that comes to mind is the blowing pres-
: 1 Introduction sure. Several studies underlined the importance of
v
i other parameters such as the jet velocity (directly
X
relatedto theblowingpressure),thejetlengthand
Thestudyofmusicalinstrumentsbasedonphysical
r height, and the area of the outcoming flow. These
a observationshasalwaysattempted,tosomeextent,
parameters, or equivalent dimensionless parame-
to include the musician within the sound produc-
ters (Reynolds and Strouhal numbers) are under
tionmechanisms. Themostbasicdescriptionofthe
the controlledof the musician or of the instrument
musician control,such as a punctual or a static in-
makerdependingonthetypeofinstrumentconsid-
jectionofenergythatstartsthe oscillation,oreven
ered.
amodificationofboundaryconditionsthatchanges
the acoustic properties, already constitute a first Concerning the blowing pressure, one would in-
attempt to describe the control. tuitively say that it controls the loudness of the
1
sound including spectral enhancement that might artificial mouth, the fundamental frequency can
occur because of non-linear behaviour. However it shift of 10 cents (100 cents = semitone), but the
has been shownthat the variationof blowing pres- influence on the spectral content is not observed.
sure also directly affects the frequency of the note Chen et al.[7] led experimental measurements on
andwhetherthe instrumentsoundsononeregister a recorder player during performance. They found
or another. that the recorder player controlled his vocal tract
Understandingthe influence ofthe blowingpres- impedance. No evidence of vocaltracttuning with
sure allows the inclusion of it in sound production thenoteplayedwasfound. Theydidn’tobservethe
models [24, 37, 36, 16, 2]. These models consider fluctuatingpartofthesupplypressure. Theyfound
the blowing pressure as a slowly varying control that the tongue position influence the broadband
parameter, i.e. which varies on time scales much component of the sound. They also mentioned an
largerthan anacousticperiod. A recentstudy jus- effect on the magnitudes of the first 10 harmonics.
tifiedthis assumptionby relating“slow”variations The effect of the changing of vocal tract shape
of the blowing pressure(over10s)to musical inter- has been more investigated for reed instruments
pretations [13]. by[31, 8, 9, 10, 25]. A sound synthesis model in-
On the other hand, the pressure can vary onthe cluding the vocal tract has also been proposed by
same time scales as the acoustic period. During Guillemain[26] for the clarinet. In those studies,
the attack transients, the pressuregrowsfrom zero the vocal tract is modelled as a series impedance
to its final value in ten milliseconds (30ms in very addedto the pipe impedance. For flute-like instru-
soft attack)[6, 30, 37] corresponding to the same ments, the coupling through the channel between
order of magnitude as the sound period. The pres- the instrument and the vocal tract can be written
sure release at the end of the notes and the pres- using Bernoulli relation[38, 2], preventing a linear
surevariationsbetweentwoslurrednotesareofthe combination of the impedances of the vocal tract
same time scales. Some models[36, 17] attempted and the instrument.
toconsiderthesescalesofvaryingblowingpressure In the study presented in this paper, the follow-
to model transients. ingquestionsareaddressed. Isthemusicianableto
However, as it has already been observed by control the acoustic coupling between the recorder
Verge[37], S´egoufin[32] and de la Cuadra et al.[19, and the mouth cavity? What is the effect of this
18], the blowing pressure and the resulting jet, are coupling on the sound production ? Lastly, can
prone to oscillate at time scales much smaller than the existing models be modified to include this ef-
theonesthatcanbedirectlycontrolledbyahuman. fect? This paper is a first step in studying the in-
More precisely, the blowing pressure may oscillate fluence of the vocal tract on sound production of
atfrequencieswithinthe rangeofthesoundingfre- flute-like instruments. While the experiments pre-
quency, certainly because of the acoustic coupling sented in this paper were carried on recorders, the
betweenthevocaltractofthemusician(orthefoot model presented may be applied to other flute-like
of organpipe) and the instrument[3]. instruments.
A more recent study, led by de la Cuadra et The paper is structured as follow: section 2
al.[18],revealedmajordifferencesbetweenanovice presents a preliminary study with recorder play-
andanexperiencedflautistconsideringtheblowing ers addressing the first question about the possi-
pressure, the jet length and height, the area of the bility to control the coupling with the vocal tract.
outcomingflow. Althoughitwasnotusedtoquan- A second experiment was developed in laboratory,
titatively describe the differences between the two allowinga better repeatability of the measurement
players,the authors noticed that the blowing pres- (section 3). It used feedback in an artificial play-
sure presents more acoustic frequency components ing system to vary the coupling and to observe its
inthemouthoftheexperiencedflautistthaninthat effect on sound production.
ofthenovice.Nospecificobservationofthespectral Data gathered during these studies providedthe
content was carried in this study. Coltman[11] led basis to modify an existing model of sound pro-
a study of the effect of the mouth resonance on duction,the jet-drive model, by including the effect
the fundamental frequency in the flute. He showed of the vocal tract. The Jet-Drive model is com-
experimentally that, by varying the volume of an monly used to model all flute-like instrument and
2
Afirstinterviewwithaprofessionalplayerandwith
a teacher allowed to establish a protocol for a sec-
ond interview with five other professional recorder
p
players.
As it is a preliminary study, the instructions of
the second interview focused on isolated notes and
musical scales. The recorder players were asked
pac to modify sound features such as the timbre while
playing these exercises. Players were given no in-
Figure 1: Sketch of the modified recorder that allows structions about using the vocal tract shape but
measurement of the acoustic pressure inside the bore wereinvitedto usewhatevertechniques they chose
and the pressure inside the mouth. to effect the changes in timbre. The freedom given
to the players prevents the possibility to carrysta-
tisticalanalysisofdata: thedataareacquiredwith
the modification proposed here may as well be ap-
different musicians, each of them playing different
plied to the modelling of all flute-like instrument.
notes and using different techniques. The aim of
The modified model is presented in section 4 while
thesepreliminarymeasurementswastoidentifythe
section5 presentsthe predictions ofthe modeland
techniques that can be usually used by recorder
discusses its limitations.
players.
2 Preliminary observations 2.2 First results
In these interviews, different techniques involving
This section presents preliminary measurements
the vocal tract have been mentioned by the musi-
carried out on recorder players in order to inves-
cians. Some of them are, according to the player,
tigatethe controllabilityandthe effect ofthe vocal
directly related to a control of the vocal tract
tract shape on the sound production.
shape (“varying the mouth volume”, “opening the
nasal cavity”, etc.). Others are related to a pos-
ture (“gritting the teeth”, etc.). According to the
2.1 Setup and protocol recorder players,these different techniques may be
associated to a change in the timbre or other as-
Two pressure sensors are mounted on an alto
pects of the sound production such as the “projec-
Aesth´erecorder,madebytherecordermakerJean-
tion” or the “directivity”.
Luc Boudreau (figure 1). The first sensor is a The fluctuations p′ of the supply pressure p are
B&Kmicrophonemodel4938mountedthroughthe
regarded as consequences of the oscillation of the
wall, flush with the inner wall and measures the
acousticpressureinthepipep . Theamplitudeof
ac
acoustic pressure pac near the labium. The sec- the fluctuating part p′ is generally between 10 and
ond sensor is a Honeywell pressure sensor model
0.1 percent of the steady part. Assuming that the
176PC14HG1. It allows measurement of the pres- fluctuatingpartofthe supply pressurep′ is asmall
sure p within the mouth of the player thanks to a
perturbation of the steady part, the coupling can
capillarytube passingthroughthe mouthpieceand be linearized [3], and the relation between p′ and
ending in the mouth. The setup followed the ones
p can be described as a transfer function
ac
used by[18, 31, 32, 38, 39].
The discussion showed that there is no common P′(ω)
=G (ω)ejϕ(ω), (1)
playing techniques, shared by professionals players P (ω) p
ac
and teachers interviewed, that emphasizes clearly
the role of the vocal tract. Without any musical with P′ and P the Fourier transform of p′ and
ac
consensus, we could not provide musical instruc- p , respectively, and where the G and ϕ are the
ac p
tionsthatwouldresultinchangesofthevocaltract. gain and the phase of the transfer function.
3
Figure 2: Isolated note performed by a first recorder Figure 3: Isolated note performed by asecond recorder
player (C6, 1046 Hz). He reports changing his mouth player (G5, 784 Hz). He reports opening his nose cav-
volumewhileplaying. From toptobottom: gainof cou- itygradually. Fromtoptobottom: gainofcouplingGp,
pling G , phase of coupling ϕ, amplitude of the eight phase of coupling ϕ, amplitude of the eight first har-
p
first harmonics of p in dB. monics in dB.
ac
A change in the vocal tract shape may lead to Gp = 20dB (p′/pac=0.1),theevenharmonicsin-
−
a change in the gain of coupling. Figure 2 shows crease of about 5 dB. The odd harmonics remain
the gain G and the phase of coupling ϕ on an however constant, or decrease.
p
isolated note (C6, 1046 Hz) performed by a first The change of the spectral content as a function
player who reports “varying his mouth volume”. of the coupling with the vocal tract is not system-
There is no way to check objectively the move- atic. Figure 3 shows the gain and the phase of
ments performed by the recorder player. However, coupling and the amplitude of the first eight har-
a change in the gain and the phase of coupling monicsoftheinneracousticpressureonanisolated
has been observed and corresponds to what the note (G5, 784 Hz) performed by a second recorder
recorder player asserts: a narrowing and widening player who reports opening his nasal cavity grad-
of the mouth. Starting from a “large” mouth vol- ually (gradually opening the velum). In this case,
ume (1 < t < 2 s), the recorder player shrinks it there is no monotonous behaviour of the harmon-
(2.5 < t < 4 s) then brings it back to the initial ics. This may be due to a weaker maximal gain
volume (4.5 < t < 5.5 s). This can be interpreted than in the previous measurement ( 30 dB).
−
as following: as the volume decreases, the mouth This short experimental study confirms that
impedanceattheplayingfrequency(1047Hz)rises recorderplayershave the possibility to use on pur-
substantiallyandthe gainofcouplingG increases pose the vocal tract shape as already asserted by
p
from 60 dB to 20dB.Simultaneously,the phase Chen et al.[7]. It may or may not have measurable
− −
ϕ decreases from 0 to π as the mouth volume de- effect on the harmonics amplitude, depending on
creases. the gain of coupling G , which is a different effect
p
Varyingthe vocaltractshape modifies the prop- than the one highlighted by the study of Chen[7]
erties of the coupling. The properties of air flow whichwasfocusedonthe broadbandcomponentof
through the channel of the instrument are there- thesound. Ifthecouplingaffectsthesteadypart,it
fore expected to be modified, which would in turn isalsoexpectedtoaffecttheunsteadypart,suchas
modify the sound production. The amplitude of the attack transient. As the stability of the differ-
the harmonics of the inner acoustic pressure p ent oscillating regimes of the recorder are sensitive
ac
depends onthe gainandthe phaseofcoupling(see to several parameters, the coupling with the vocal
figure 2). As the gain reaches its maximal value tractmaybecriticalwhilediscussingthestabilities
4
of regimes. sure within the artificial mouth is measured with
The nextsectionpresentsanexperimentalstudy an Endevco pressure sensor (model 8507C-5). The
on the effect of the gain G and the phase ϕ of pressureis controlledby a numericalPID feedback
p
coupling on the sound production. loop implemented with dSpace controller[22]. The
effective pressure is compared to the desired one
every40µs. Thecalculationsneededareperformed
in this time, which lead to a real time setup. The
3 Laboratory experiments
PID is designed to have a global response time of
around 10ms which leads to the filtration of the
Inordertoproducereproduciblemeasurements,an high frequencies.
artificial mouth for recorder[22] has been set to The acoustic coupling with the vocal tract is
mimic the acoustic coupling with the vocal tract. mimicked by the injection of acoustic flux within
One obvious way would be to make different hard- the cavity. The upstream fluctuations are forced
ware vocal tracts, but it limits the possibility of by a feedback loop between the acoustic pressure
different coupling which can be investigated. An p withinthe instrumentandaloudspeakerAura-
ac
alternative way is chosen which allows the simula- sound NSW2 (resonance frequency of 200 Hz, re-
tion of much wider coupling conditions with the sistance of 6 Ω) set within the artificial mouth.
same setup. It is based on the injection of an The acoustic pressure within the instrument is
acoustic flux in the artificial mouth, proportional measured with an Endevco pressure sensor (model
to the acoustic pressure measured within the in- 8507C-2). The feedback loop is managed by the
strument, using a loudspeaker within the artificial samedSpace controllerasthe one usedforthe reg-
mouth. The experimental setup is detailed in the ulation of the slowly varying pressure. The signal
next section. is numerically amplified and phase shifted using a
second-orderall-pass filter. The parameters of this
filter are tuned at the frequency previously mea-
3.1 Experimental setup sured without forcing. The quality factor of the
second-order all-pass filter is equal to 1 while the
For the playing machine study, the upper (nearly centralfrequencyisadjustedtoproducethedesired
cylindrical)partofa BressanZen-Onalto recorder phase shift at the target oscillating frequency. The
isused,anditsconicallowerjointsarereplacedbya modified signal is sent into a power amplifier (Pi-
cylinder of equal diameter, to achieve a near cylin- oneer A107 set with a constant gain) driving the
drical instrument, as used in a previous study[2]. loudspeaker. The artificial mouth and the loud-
The total length is 26.5 cm resulting in a reso- speaker are held by a larger cavity (volume of 0.13
nancefrequencyofapproximately565Hz (between m3)whichpurposeisto absorbthe backwardwave
a C#5(554 Hz) and a D5(587 Hz)). Thus the generatedbytheloudspeaker. Thefluctuatingpart
impedance of the resonator is well known, and the ofthesupplypressureischeckedinordernottoaf-
comparison with simulated results should be eas- fect the regulation of the slowly varying pressure
ier. The other parameters of the instrument are by the PID loop.
thesameasanunmodifiedrecorder(windowlength The phase and amplitude response of the whole
W=4 mm, channel height h= 1mm, inner diame- feedback loop depends on the electrical impedance
ter (cid:31)= 19mm) and are thus optimized to produce of the loudspeaker, on the lag due to numerical
sound at this frequency which lies in the normal treatment as well as on the response of the instru-
range of the instrument. The artificial mouth con- ment itself. The phase and the gain are therefore
sistsinasmallcavityofdiameter44mmandheight measured a posteriori, after measuring the pres-
55 mm yielding a volume V = 5.6 10−5m3. This sures within the mouth and the instrument. They
0
volume is close to one of the experimenter mea- may differ from what was a priori targeted. The
sured with the method of Coltman[11] (measure of phases are estimated with a quadrature phase de-
the weight of the water contained in the mouth). tection algorithm[27, 3] (as used for telecommu-
Air coming from an upstream tank flows into the nication receivers) while the amplitudes are esti-
cavitythroughahole ofdiameter8mm. The pres- matedusing shorttermFouriertransform. Results
5
areplottedasfunctionsoftheaposteriori coupling
parameters.
3.2 Steady sounds and spectral con-
tent
The first of the two blowing conditions used was
chosentostudytheeffectofthecouplingonsteady
sounds, and more precisely on the spectral content
of steady sounds. The artificial mouth is supplied
with a constant flow. Several runs have been per-
formed for sixteen phase shifts by steps of π/8 and
four coupling gains (G = 34, 26 20, 14 dB)
p
− − − −
as wellas for twomeanpressures p = 140Pa,400
h i
Pa. These two values are chosen to provide oscil-
Figure 4: Amplitudes of the first eight harmonics and
lationwithin the first regime,respectively far from un-parityindexI (definedbyEq. (2))asfunctionofthe
and close to the regime change threshold for in- phaseofcouplingϕforthemaximalgain(p′/p =0.2,
ac
creasingpressure regime, respectively far from and u′/v =0.9)andforasupplypressureofhpi=400Pa.
ac
close to the regime change threshold for increas-
ing pressure ( 695 Pa). The former provides a
≃
soundclosetoapuretonewhilethelatterprovides opposite way. The un-parity index I is a good in-
a soundwith a strongsecondharmonic,because of dicatorofthe energydistributionbetweenoddand
the spectral enhancement of the non-linear satura- evenharmonics. Itsummarizesinoneindextherel-
tion due to the increase of pressure. ative evolution of odd and even harmonics. Here,
The spectral content is characterised by the en- it evolves in a range of 30 dB.
ergy distribution discriminating the even and odd Without coupling, the index is around -2 dB for
harmonics. Anun-parityspectrum indexisdefined: thetwosupplypressures. Itsvaluedepends,among
others, on geometrical parameters, such as the off-
N
(a )2 set between the channel axis and the labium[23].
2p+1
I =10logp=1 , (2) Figure 5 shows the un-parity index for the four
PN gains of coupling and the two supply pressures as
 (a2p)2  function of the phase ϕ. The trend described in
 p=1 
 P  the previousparagraphforonegainis thesamefor
that weights the sum of the energy of the odd har- all gains, at least for the supply pressure of 400
monics(a )2 overtheoneoftheevenharmonics Pa. The un-parity indexes are minimal for phases
2p+1
(a )2. By definition, the energy of the fundamen- ϕclosetoπandmaximalforphasesϕclosetozero.
2p
tal is discarded because it is larger than all the The ratio of the odd harmonics over the even in-
other harmonics, and would tend to smooth the creasesasthephasegetsclosetozeroanddecreases
variationofthe index I. Harmonicsfromrank2 to as the phase gets close to π. However, this effect
7 are considered (N = 3), without the fundamen- is less pronounced when the gain of the coupling
tal. Figure 4 shows the evolution of the first eight decreases. Forthe 140Pasupplypressure,noclear
harmonics and the associated un-parity index as trendhasbeenobserved,evenforthemaximalgain.
function of the phase ϕ for the maximal gain G = The two supply pressures correspond to two dif-
p
14 dB and for the mean supply pressure of 400 ferent domains: one with few harmonics, and the
−
Pa. The amplitude of the even harmonics may in- other with much more harmonics. The sensitivity
crease up to 20 dB. It is maximal for a phase ϕ ofthesystemwithrespecttothecouplingwiththe
close to π and minimal for a phase ϕ close to zero. vocal tract seems to depend on the initial energy
Theamplitudesoftheoddharmonicsbehaveinthe distribution, i.e. withoutcoupling,amongthe har-
6
Figure 5: Un-parity index I (defined by Eq. (2)) as
function of the phase of coupling ϕ for all gains (in- Figure 6: Experimental dimensionless frequency f/f1
creasing gains with darker gray: Gp = −14 dB: black, as function of the mean supply pressure hpi for three
Gp = −34 dB: lightest gray) and for two mean supply coupling conditions. From top to bottom: without cou-
pressures (140 Pa et 400 Pa). pling, ϕ=π, ϕ=0.
monics. 565 Hz).
The oscillating frequency results from the bal- Figure 6 shows the oscillatingfrequency as func-
ance between the harmonics amplitudes ruled by tion of the mean supply pressure p for all the
h i
the non-linear mixing of the exciter. As soon as coupling conditions. The operating range of the
therelativestrengthsoftheharmonicsarechanged first regime is significantly affected by the acoustic
because of the coupling with the vocal tract, the coupling with the vocal tract. For a phase ϕ π,
≃
frequency is expected to deviate from its natural the increasing threshold from the first to the sec-
case. For the maximum gain used (Gp = 14dB), ond regime is shifted from 695 Pa to 745 Pa (7 %)
−
the frequency varies of around 7 cents between ϕ for the maximal gain (G = 14 dB). The smaller
p
−
closetoπ (minimum)andϕcloseto0(maximum). gains seem to have only a moderate effect on the
Thisvalueisinthesameorderthantheshiftfound extent of the first regime. For a phase ϕ 0, the
≃
by Coltman[11] (around 10 cents). increasing threshold is reduced, on the contrary,
from 695 Pa to 630 Pa (6.5 %), for the maximal
gain. Foraphaseϕ 0,smallergainshavealsoan
≃
3.3 Effects on regime change thresh- impact on the increasing thresholds.
The decreasing threshold - that from the second
olds
to the first regime - is also modified by the acous-
Thesecondblowingconditionaimsatstudyingthe tic coupling. For a phase ϕ π, the threshold is
≃
effectofthecouplingonthestabilityoftheregimes reducedfrom534Pato 495Pa(7 %), forthe max-
of oscillation. The supply pressure now slowly imal gain. For a phase ϕ 0, the threshold is also
≃
varies with a triangle shape in time between two reducedfrom534Pato 520Pa(3 %), forthe max-
extremalvalues ( p [50,1200]Pa). The range of imal gain. The decreasing threshold is reduced for
h i∈
pressure allows the instrument to sound in its first all the coupling conditions.
twooscillatingregimes. Severalrunshavebeenper- The phase seems to be the relevant parameter
formedwithtwophases(ϕ=0orπ)andthreegains concerning the effect of the changingof vocaltract
ofcoupling(G = 34, 20, 14dB).Thephaseϕ shape on the stability of the regime, the gain hav-
p
− − −
issetbasedonthefrequencyoftheinstrumentjust ing only an effect of emphasis on threshold shifts.
before the transition to the second register (close A phase ϕ close to π extends the hysteresis range
to, but above, the resonance frequency of the pipe while a phase ϕ close to 0 reduces it.
7
This result seems counter-intuitive since the ofthis coupling. This mechanismhasstartedto be
phase close to π has also been identified as a de- studied by Auvray et al.[3].
termining factor in the rise of the even harmonics. The description of the coupling in terms of a
One would expect that an increase in the second transfer function between the mouth and the bore
harmonics within the first regime, close to the fre- pressures (see Eq. (1)) is well suited for the ex-
quency of the first harmonic on the second regime, periments. For the model, an equivalent transfer
would induce a reduction of the increasing thresh- function can be written between the jet velocity
old. This highlights the complexity of the mech- fluctuationsu′ andthe acousticvelocitywithinthe
anisms that rule the regime changes and calls for window v :
ac
deeper investigations.
U′(ω)
=G (ω)ejφ(ω), (3)
u
V (ω)
ac
with U′ and V the Fourier transform of u′ and
4 Model ac
v , respectively. In the present study the gain G
ac u
and the phase φ are considered as the two control
The modified model of sound production incorpo- parameters of the coupling. They are varied inde-
ratingchangingofvocaltractshape uses threedis- pendentlywithouttakingcareoftheirphysicalori-
tinct ingredients: the description of the coupling, gins. Ithighlights the effects of these two variables
thegrowthofinstabilitiesonajetandtheJet-drive on the model.
source model that has been shown to predict quite
In the initial model[2], the jet instabilities are
fairresultsinquasisteadycondition[2]. Themajor
triggered by the transverse acoustic velocity near
difficultyliesincombiningthesedescriptionsbased
the flue exit. This external perturbation is anti-
ondifferenttimescalesandunderdifferentassump-
symmetric and only the anti-symmetric unstable
tions. In other words the quasi-steady model is
mode is then triggered. This has been confirmed
enhanced in order to allow fast fluctuations of the
by several experimental observations[20]. Taking
blowing pressure. account of the fluctuating part of the velocity u′,
anotherexcitationfor the jet instabilities is added.
The effectofthis perturbationonthe jet isstudied
in the next section.
4.1 Modelling the coupling
Moststudies relatedto the effectof the vocaltract
tuning focus on the impedance of the vocal tract, 4.2 Effect of fluctuating velocity on
described in the frequency domain, and put in se- the jet instabilities
ries with the instrument impedance [31, 8, 9, 10,
Evenifitremainsanacademicsituation,thestudy
25]. This description can not be used in flute-like
of the instabilities of an infinite plane jet for in-
instruments models. There is no simple relation
compressibleparallelandnonviscousflowstillpro-
of continuity between the mouth variables (pres-
vides insights on the instability mechanisms that
surepandflow)andthe acousticvariablesintothe
occur during flute operationfor a finite extent and
instrument (p and v ). The impedances are not
ac ac
roughly plane jet.
inseries. Anotherdescriptionuses,inthetemporal
The theoretical jet is assumed to flow along x
domain,asetoftwodifferentialequations[38](mass
direction. The cross-stream profile U(y) along y
andmomentumconservationthroughthefluechan-
direction is assumed to be a Bickley profile:
nel). This offers the advantage to keep the hydro-
dynamic variables: fluctuations u′ of the jet veloc-
u
ity u= u +u′, with u the steady partofthe jet U(y)= h i y , (4)
velocity,harieinducedbhytihecouplingwiththevocal cosh2
b
tract. As the present study is a first approach in
modellingtheeffectofthecoupling,nocarewillbe where u is the centerline velocity and b the half-
h i
takento describeprecisely the physicalmechanism widthwhichisconstantinthecaseofanonviscous
8
flow. A perturbation flow u=ux+vy is added to perturbation of the jet. The coexistence of two
themeanflow. Thelinearstabilityanalysisconsists kinds of perturbations, the anti-symmetric acous-
in finding a propagative solution of the perturba- ticfieldandthesymmetricpulsatingjetflow,leads
tionfieldu=uˆei(ωt−αx). Suchaperturbationflow to the linear superimposed growths of the two un-
is ruled by the equation of Rayleigh[28]: stable modes, anti-symmetricand symmetric, with
growth factors α and phase velocities c of order
i p
ω d2v(y) d2U(y)
U(y) α2u(y) u(y)=0, (5) ofmagnitudepredictedbythelinearstabilityanal-
− α dy2 − − dy2 ysis of the infinite jet flow. The superimposition is
h i(cid:20) (cid:21)
linear during the linear development of the jet.
where ω is a real driving pulsation, α = α +iα
r i
thecomplexwavenumberandc =ω/α thephase
p r
velocity. This equation is subject to the boundary
4.3 Modified Source model
conditions:
As witheveryself-sustainedinstrument, the model
u′( )=u( )=0. (6)
±∞ ±∞ of sound production of a flute-like instrument is
This set of equations can be solved numerically basedona feedbackloopsystemconnectinganon-
[28,30]. ForasymmetricjetprofileU( y)=U(y), linear exciter and a linear resonator[29]. In the
Mattingly and Criminale[28] showed t−hat two un- specific case of a flute-like instrument, the model
stable modes may rise,the antisymmetric v( y)= is refined by adding non-linear losses to the former
v(y) being more unstable than the symm−etric linear acoustic part[21].
v−( y)=v(y). This is illustrated in figure 7, which The modified model presented in this section is
sh−ows the dispersion relations for such symmetric based on the Jet-Drive source model as written by
and anti symmetric perturbations. The parame- Auvray[2]. Only the parts including modifications
ters associated with antisymmetric and symmetric are extensively presented here (exciter), the others
perturbations are respectively indexed by (a) and being onlymentioned(linearandnon-linearacous-
(s). tics).
For a given driving pulsation ω and for a given The air column of the resonator is driven by
setofparameters,i.e. for a givenStrouhalnumber a force term written as a pressure difference ∆p
ωr = ωb/ u , the spatial growth factor αi de- acrossthewindow. Theacousticvelocityvacwithin
scribes howh tihe perturbationis amplified w−hile the the window results from the excitation of the res-
real phase velocity c = ω/α describes at which onator by the difference of pressure through the
p r
velocity the perturbation is convected. Without admittance relation:
anyexternalexcitation,theantisymmetricpertur- V
ac
Y = , (7)
bationis moreunstable thanthe symmetric,corre- ∆P
sponding to the Strouhal number ωb/ u 0.25.
h i ∼ where Vac and ∆P are the Fourier transforms of
As soon as the unstable mode is amplified, the
v and ∆p, respectively. The resonator part, i.e.
ac
present linear description is no longer valid. The
the admittance, is described as a modal acoustic
vorticity of the shear layer accumulates at the in-
admittance:
flection points. However, the structure of the flow
∞
after non-linear development maintains a “mem- jωYn
Y = , (8)
ory” of the structure presented during the linear ω2 ω2+jε ω ω
n=1 n− n n
development. Thus,therelativepositionofvortices X
where Y , ε andω are the amplitude, the damp-
on the two shear layers of the jet is an indication n n n
ing coefficient and the pulsation of the n-th mode,
of which mode has been excited during the initial
respectively. The non-linear losses,due to the con-
part. This is confirmed by an experiment of flow
strictionwhichyieldstheformationofafreejetev-
visualization presented in appendix A.
ery half acoustic period, are written as a pressure
The main hypothesis of the modification of the
drop across the window:
current simplified model is that the fluctuations of
the jet velocity profile due to the acousticcoupling 1 v v
ac ac
∆p = ρ | |, (9)
with the vocal tract consist in a symmetric initial los −2 0 α2
vc
9
Figure7: Top: Shapeoftheperturbationofthejetaddedtothemeanflowforantisymmetric(left)andsymmetric
(right) cases. Bottom: Corresponding real phase velocity c =ω/α , real part α and imaginary part −α of the
p r r i
wave number α as function of the Strouhal number ω = ωb/hui, obtained by numerical solution equation Eq.
r
(5).
with α = 0.6 a vena contracta coefficient. This propagative function:
vc
pressure drop will be added to the source pressure
difference. η(x,t)=e−αi(a)xη0(t−x/c(pa)), (10)
The exciter part is split into two distinct parts:
the birth, growth and convection of the jet insta- with −α(ia) ∼ 0.3/h the anti-symmetric growth
bility from the flue exit to the labium and the rate, h the height from which the jet emerges and
jet/labium interaction as a mass injection from c(pa) 0.4 u theanti-symmetricphasevelocityand
∼ h i
both sides of the labium. where η0(t) is an “initial perturbation” at x = 0.
This expression is only valid a few characteristic
Regarding the antisymmetric perturbations distances h downstream so that the initial pertur-
bation η has no specific physical meaning except
0
Eitherspreadingornot, the jet is describedbya asanempiricalfittothedata. DelaCuadrafound
unique variable: itscenterpositiondenotedη(x,t). for his experiments:
The developmentoflinear anti-symmetricinstabil-
η (t) v (t)
ities from the flue exit to the labium is assumed to 0 = ac , (11)
be well described by the center position of the jet h u
h i
along the same path. As for the perturbation field
where v is the transverse acoustic velocity near
of the linear stability analysis, the center position ac
the separation point (at x = 0). The values of the
is then described as a propagative solution, func-
growthfactor α(a) and the phase velocity c(a) vary
tionoft x/c(a) where c(a) is the phase velocityof i p
− p p from one condition to another but remain of this
the anti-symmetric perturbation. The issue of the
order of magnitude, in agreement with the linear
triggering of the jet perturbation is still an open
stability analysispresentedinthe previoussection.
question,butdifferentauthors[5,15,20]agreethat
Adding the symmetric perturbations
the jet is mainly perturbed at the separationpoint
of the flow (x 0) by the acoustic velocity field. When the velocity profile at the flue exit oscil-
≃
Based on flow visualisations, de la Cuadra[15, 16] lates because of the acoustic coupling with the vo-
proposed an expression of the jet center line as a cal tract, symmetric instabilities are expected to
10