Aerodynamic excitation and sound production of blown ...Aerodynamic excitation and sound production...

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Aerodynamic excitation and sound productionof blown-closed free reeds without acoustic coupling:The example of the accordion reed

Denis Ricota)

Laboratoire de Me´canique des Fluides et d’Acoustique, UMR CNRS 5509, Ecole Centrale de Lyon, France

Rene Causse and Nicolas MisdariisInstitut de Recherche et Coordination Acoustique/Musique, UMR CNRS 9912,1 place Igor Stravinsky 75004 Paris, France

~Received 7 February 2003; revised 27 September 2004; accepted 2 December 2004!

The accordion reed is an example of a blown-closed free reed. Unlike most oscillating valves inwind musical instruments, self-sustained oscillations occur without acoustic coupling. Flowvisualizations and measurements in water show that the flow can be supposed incompressible andpotential. A model is developed and the solution is calculated in the time domain. The excitationforce is found to be associated with the inertial load of the unsteady flow through the reed gaps.Inertial effect leads to velocity fluctuations in the reed opening and then to an unsteady Bernoulliforce. A pressure component generated by the local reciprocal air movement around the reed isadded to the modeled aerodynamic excitation pressure. Since the model is two-dimensional, onlyqualitative comparisons with air flow measurements are possible. The agreement between thesimulated pressure waveforms and measured pressure in the very near-field of the reed is reasonable.In addition, an aeroacoustic model using the permeable Ffowcs Williams–Hawkings integralmethod is presented. The integral expressions of the far-field acoustic pressure are also computed inthe time domain. In agreement with experimental data, the sound is found to be dominated by thedipolar source associated by the strong momentum fluctuations of the flow through the reed gaps.© 2005 Acoustical Society of America.@DOI: 10.1121/1.1852546#

PACS numbers: 43.75.Pq@NHF# Pages: 2279–2290

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I. INTRODUCTION

The accordion is based on a very old system: the fmetal reed which dates from the third millenium BC. It wused in a Chinese musical instrument, the tcheng. Nowadthis system can be found in the harmonica and the harnium, for example. The accordion reed is a thin metal plriveted at one end to a support plate. There is a rectangslot in the support plate immediately beneath the reed.aperture is a bit larger than the reed so that the it frevibrates as a cantilevered beam and does not strike agthe support plate. An air flow is generated by the inwardoutward movement of the bellows. With regards to Fletcheclassification,1 the accordion reed is a valve of type~2,1!.In the presence of pre-existing flow in the same directionthe instrument, a steady overpressure applied from thestream side of the reed tends to close the valve and invera steady overpressure applied from downstream of thecauses it to open further. Of course, if we consider a steflow in the opposite direction, the valve is of type~1,2!.However, in the instrument, the accordion reed only operain its blown-closed configuration: the inward movementthe bellows generates an upstream steady overpressurP0

and the outward movement generates a depression ondownstream side of the reed which is equivalent to anstream overpressure. The self-sustained oscillation res

a!Current address: 10 avenue Paul Cezanne 78990 Elancourt, Francetronic mail: [email protected]

J. Acoust. Soc. Am. 117 (4), Pt. 1, April 2005 0001-4966/2005/117(4)/2

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from the unsteady pressure difference between both sidethe reed. But unlike most wind instruments such as wowind and brass,2,3 the time-dependent pressure drop is ndriven by the oscillation of the air column of a resonator.the accordion, we can consider that the reed oscillatmechanism and the emitted sound do not depend onacoustic influence of the upstream and downstream volumThis kind of oscillating valve can be called a dominant ostrong reed.

In classical reed models1,2,4 that have been developed foresonator-coupled reeds, the pressure fluctuationpac(t) iswritten from the acoustic response of the resonator excby an unsteady volume fluxqac(t). By assuming the incom-pressibility of the flow around the reed, the acoustic voluflux at the resonator inlet is matched to the aerodynamicthrough the reed apertureqac(t)ªq(t)}V(t)x2(t)L whereV(t) is the flow velocity,x2(t) is the reed displacement anL is a characteristic length of the orifice. The aerodynamvelocity—and thus the volume flux—is easily calculatfrom the steady-state Bernoulli equation with a pressure dP02pac(t). This modeling approach implicitly supposes ththe pressure of the reed flow at the exit of the reed chanexactly equals the acoustic pressure at the entrance oresonator. This assumption can be justified5 by noting that ajet is formed at the exit of the reed channel because oflarge abrupt transition in cross-sectional area from reed chnel to instrument mouthpiece. All the kinetic energy of tflow is supposed to be dissipated in the spreading turbu

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jet. This hypothesis is also consistent with the modelingthe acoustic source. Indeed, it is clear that only the fluctuing mass is taken for the excitation of the acoustic air cumn. The fluctuating force associated with the periodic mmentum injection can be ignored. Finally, the reedmodeled by a simple mass-spring-damping system. Coquently, a coupled system of three equations~the impedancerelation, the Bernoulli equation and the simple harmoniccillator equation! and three unknown variable@x2(t),q(t),pac(t)# can be written. With the same approacthe influence of the internal acoustic impedance of thesupply system~the player’s lungs, vocal tract and mouth! canbe taken into account.1 Because of the impedance relatiomost calculations are performed in the frequency domain1,2

But the frequency–domain approach is not advantageoustrongly nonlinear equations such as the Bernoulli equatTime–domain solutions6,4 and hybrid methods7 have alsobeen proposed.

For a constant surface of contact between the flowthe reed, the dynamic excitation force is a combinationacoustic pressure and pressure induced by velocity variadue to the variation in the aperture area. The velocinduced pressure fluctuations are found by applying thetionary Bernoulli equation which associates a decrease opressure with an acceleration of the fluid. But, without othphysical mechanisms such as acoustic feedback, thesesure fluctuations cannot induce self-oscillations. Forample, if we suppose a constant volume flow throughreed aperture, it is straightforward to show that the foinduced by fluid velocity variation is in phase with the diplacement. Energy transfer between the constant supply psure and the reed oscillation is not possible if we consionly the stationary Bernoulli equation. Titze8 explained thatan asymmetry of the pressure force during the cycle is nessary for reed excitation. In case of resonator-coupled rethe pressure asymmetry is provided by the acoustic respof upstream and/or downstream volumes. In case of thecordion reed, there is no acoustic feedback. By suppothat the stationary Bernoulli equation well describesaerodynamic flow through the reed aperture, other physmechanisms have been proposed to explain the phase shthe total force with regard to the reed displacement. Forample, a complex evolution of the channel geometry durthe cycle can lead to the asymmetry of the total force. Tapproach is used to model voice production because vfold oscillations are not considered to be driven by tacoustic impedance of lungs and vocal tract. Titze8 used awave motion model to describe the movement of the tissand Pelorsonet al.9 added a second mechanical oscillatorphase delay of the total pressure force is obtained becthe shape variations of the channel are not the same duclosing and opening of the glottis.

The effect of separation and reattachment of the flowthe reed channel has also been studied. Indeed, the flow sration at the inlet of the clarinet reed channel leads to a loacceleration of the flow5 ~vena-contracta effect!. Moreover,the total pressure force can also depend on the possibletachment of the air flow on the reed further in the reed chnel. The vena-contracta factor and the location of the re

2280 J. Acoust. Soc. Am., Vol. 117, No. 4, Pt. 1, April 2005


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tachment point may vary as a function of the reed positand produce a force in phase with the reed velocity. Anample of the modeling of these viscous effects can be foin the study of Pelorsonet al.9 At the outlet of the vocal foldchannel, he studied the movement of the point where theflow separates from the glottis. In this case, the total pressforce fluctuates because the reed surface in contact withunsteady flow fluctuates. Although the consequences offlow separation are qualitatively and quantitatively signcant, it was not shown whether this phenomenon couldduce self-oscillation without the presence of the two-mmodel.

The separation points of the accordion reed flowfixed at the sharp edge of the reed. As shown in Sec. II ofpaper, there is no flow reattachment on the other side ofreed, therefore the surface of the reed immersed in the floconstant. The flow separation/reattachment effects are ncandidate for a possible excitation mechanism of the acdion reed.

If we discard the supposition that the unsteady flow cbe described by the stationary Bernoulli equation, it appethat the energy transfer from the flow to a free reed canobtained by two other physical mechanisms. First, the vorshedding behind a solid obstacle10 induces a periodic forceon the structure. Thanks to their experimental investigatioSaint Hilaireet al.11 concluded that vortex shedding is nresponsible for the excitation of his harmonium reed. Tresult has been also confirmed in a recent study12 for ablown-open reed.

At last, the inertial effect of the upstream flow has beproposed to be the excitation mechanism of free reeds11 inthe absence of acoustic coupling. The verification of tassumption is one of the purposes of the present work.analysis of the flow around the accordion reed~Sec. II!shows that the flow can be modeled as an unsteady potetwo-dimensional flow. In Sec. III, the aerodynamic forcwhich excites the vibration of the reed is calculated underincompressible flow hypothesis. From aerodynamic datamodel for the calculation of the aeroacoustic source basean acoustic analogy is proposed~Sec. IV!. Flow and aeroa-coustic models are solved in the time domain and resultscompared with experimental data~Sec. V!. Since the atten-tion of this paper is turned to the analysis of the exact natof both the excitation force and acoustic source, empirihypotheses or adjustable parameters must not be usedcause of the complexity of flow equations, the calculationanalytical expressions is only possible with a very simplifirepresentation of the accordion reed. Therefore quantitavalues of physical variables are not expected to agree wthe actual ones. Only waveforms and spectra are comparemeasurements.

II. FLUID MECHANICAL DESCRIPTION OF THE FLOW

The geometry of the accordion reed is slightly differefrom the reed geometry used by Saint Hilaireet al.11 in theirflow visualizations. The initial distancex0 between the vi-brating reed and the support plate is smaller for the accdion reed than for the Saint Hilaire arrangement. The reedirectly riveted onto the support plate, thereforex0 is very

Ricot et al.: Oscillation and sound of free reeds


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small and greatly depends on the stationary profile ofreed. In our case, we can suppose thatx0<0.4 mm. Thethickness of the support plate is another difference. Thein the support plate will be called the exhaust channel infollowing ~see Fig. 1!. The length of the exhaust channelthe streamwise direction is equal to the thickness of the sport plate. We see that this length is quite large comparethe width of the gaps. Therefore a flow reattachment ispected to occur on the lateral walls of the exhaust chann

In order to check the validity of Saint Hilaire’s conclusions concerning the inability of wake instabilities to excthe reed and to have more details on the flow structure, flvisualizations using dye stream injections have been cducted in water. The experimental arrangement is givenFig. 2. The accordion reed is mounted on a Plexiglas revoir of about 800 cm3. The steel reed wasl 532 mm inlength andb50.2 mm in thickness and has a mean widthh53.6 mm. In air, the natural frequency of the reed isf 0

5330 Hz. The parametera is the remaining distance between the reed and the exhaust channel walls when the

FIG. 1. Sketch of the Plexiglas support plate and riveted reed. The flofrom right to left. The mathematical axes used in the model~Secs. III andIV ! are also shown.

FIG. 2. The experimental setup for water-flow visualizations. This setualso used for measurements of reed displacement, aerodynamic and acpressures in air flow~Sec. V!. The two large arrows indicate the views of thphotographs shown in Fig. 3 and Fig. 4.

J. Acoust. Soc. Am., Vol. 117, No. 4, Pt. 1, April 2005


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is closed. It means that the width of the exhaust channeh12a. Generally,a is less than 0.2 mm. The notations fothe reed dimensions are given in Fig. 1. The reed mountedits supply cavity is immersed in a discharge tank. The heiof the free surface of water in the discharge tank is matained constant thanks to an overflow pipe. The water wsupplied to the reed cavity through a narrow flexible tufrom a large supply tank. The average supply pressure ofreed ~the difference between the upstream and the dostream pressures! was simply set by varying the distancbetween the free surfaces of water of the supply andcharge tanks. A honeycomb flow straightener ensureseven upstream flow. This system is visible in the backgrouof the photography given later in Fig. 4. Visualization of thflow field was accomplished by the injection, via a capillatube, of a dye stream at various locations upstream ofreed.

For a supply pressure ofP052000 Pa, self-sustained oscillations of the reed can be observed. The mean velocitVof the flow through the reed gaps can be evaluated usingstationary Bernoulli equation. The Reynolds number baon the reed widthh is about Reh5Vh/n'6000 wheren is thekinematic viscosity of water. For self-sustained oscillatioin air at normal playing pressure, the Reynolds number offlow is slightly inferior to that observed in water but has tsame order of magnitude. Thus, we can suppose thatexcitation mechanism and the flow characteristics in waare the same as that in air flow. The frequency of reed oslation under water flow is 142 Hz. The stroboscopic imagthat are not reported here clearly show that the reed oscillon the first mode of the cantilever beam. The great differebetween the oscillation frequency and the natural frequeof the reed in air is a consequence of the added mass eof water ~see Sec. III C!.

Typical flow patterns are shown in Figs. 3~a! and 3~b!.As expected, the upstream flow is laminar. The dye is cvected along well-defined streamlines. The dye streams s

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FIG. 3. Side views~see Fig. 2 and Fig. 5! of the flow pattern for two dyestream injection points situated~a! in front of a reed gap, and~b! in thesupport plate plane. Only a central part of the reed is shown. The sucavity is on the right. In the upstream region, the dye is convected alpotential streamlines. Downstream of the reed, turbulent mixing and dsion are visible.

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that it is consistent to model the upstream flow by potensinks. Moreover, for injection points far enough from thegap, streamlines lie in planesx35constant, wherexi is thecoordinate along the axeei ~see Fig. 1!. Thus, the flow up-stream of the reed seems to be two-dimensional. Thissumption is used in the proposed flow model to describenear-field potential flow. The outflow is turbulent. In photgraphs, we see the mixing and the diffusion of the dyedownstream of the reed. However, the characteristic sizethe turbulent structures are small compared to the reed wh. There are no large coherent vortices that could have bresponsible for the reed excitation. Because of the very smgap sizes compared to the reed width and to the thicknesthe support plate, the downstream flow does not havecharacteristics of a wake. In fact, it appears that the jeattached to the lateral wall of the exhaust channel.

Figure 4 shows the water flow around the reed if tdischarge tank is empty so that the downstream fluid isSelf-oscillations also occur in this configuration. The addmass effect is reduced in comparison with the first confiration because only one side of the reed is immersed inter. Thus, it is not surprising to find a vibration frequency206 Hz that is superior to that obtained when the upstreand downstream fluid is water. The flow view of Fig.shows that the downstream flow can be considered as pjets, which do not interact with the downstream side ofreed. Even though this configuration is unrealistic, it is intesting because it definitively shows that the downstream fldoes not take part in the reed oscillation mechanism.

III. AERODYNAMIC EXCITATION AND REEDOSCILLATION

A. Dynamics of the potential flow

The flow visualizations in water show that only the ustream face of the vibrating plate is in contact with the moing fluid. Therefore the excitation force must be evaluatedintegrating the aerodynamic pressure fluctuation over thestream face. The flow through the reed gaps can be supp

FIG. 4. Front view~see Fig. 2 and Fig. 5! of the water flow around the reeif the water discharge tank is empty. The reed also oscillates in this conration. Water plane-jets are formed from the lateral and tip gaps. The dostream side of the reed~that is visible in the picture! is in a dry region.



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incompressible because the Mach and Helmholtz numbare very small. Even if the downstream jets are turbulenthe exit of the exhaust channel, it can be considered tolaminar and irrotationnal to a certain extent from the gaConsequently, potential expressions are used to modelupstream and downstream flows. In this analysis, the invistwo-dimensional flow is described in a section perpendicuto the length of the reed~planex35constant). This section issupposed to be far enough from the tip of the reed to avthree-dimensional effects. Using the superposition propeof potential flows,13 the total flow could be found by thesummation of the identical potential flows created by the tlateral gaps drawn in Fig. 5. However, in order to obtain tmost simple mathematical expressions, it is reasonablesuppose that the physics of reed excitation can be descrconsidering simply one-gap flow. This very simplified reprsentation of a blown-closed free reed is shown in Fig. 6.

The width of the gap ise(t)5Ax22(t)1a2 if x2(t).0

and e(t)5a if x2(t),0. The flow past the reed is approxmated by assuming that the reed lies in the support pplanex250. The potential of the upstream flow is expressas the sum of a continuous distribution of fluid sinks,13

f1~x1 ,x2 ,t !521

pV~ t !E

0

e~ t !lnSA~x12u!21x2

2

e~ t !/2D du,

~1!

where V(t) is the velocity of the downstream jet. The jevelocity V252V(t)e2 is supposed to be constant acrossgap, along the axee1. The integral of Eq.~1! can be analyti-cally evaluated. The expressions off1(x1 ,x2 ,t) and the ve-locity V1(x1 ,x2 ,t)5grad(f1) are given in Appendix A. Thelogarithmic singularity in the expressions of the potential avelocity at points~0,0! and „e(t),0… are a classical consequence of the inviscid potential model. At these points,

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FIG. 5. Sketch of the flow patterns through the reed gaps in a planex3

5cost. The upstream flow is a potential flow induced by the sinks~thelateral gaps!. Downstream of each gap, the flow is a plane wall-jet, andturns to a turbulent free-jet after separation from the exhaust channel w



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flow separates due to viscous effects. No practical maematical expressions are available for the description oftire flow with separation. Then, for simplicity the downstream flow potential is chosen in order to ensurecontinuity of the potential, velocity and pressure at the cenA of the reed opening:

f2~x1 ,x2 ,t !52V~ t !x211

pq~ t !, ~2!

whereq(t)5V(t)e(t) is the volume flux. This potential describes a laminar jet of velocityV(t). Of course, fartheraway downstream from the reed gap, the flow is a spreadturbulent jet but there is no need for modeling this region

The aerodynamic variable of the potential flow muverify the unsteady Bernoulli equation,13

]f~x,t !

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r05C~ t !, ~3!

whereP(x,t) is the deviation pressure from the atmosphepressurePatm, r0 is the density of the fluid andC(t) is aconstant independent of spatial coordinates but that canpend on time. The value ofC(t) can be evaluated by applying the Bernoulli equation at the centerA of the reed gapwhere x1

A5e(t)/2, x2A50. At this point, we havePA50,

VA52V(t)e2 and]fA /]t5]q/p ]t; then we obtain

C~ t !5V2~ t !

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A second application of Eq.~3! at a pointB located faraway upstream the reed leads to a fundamental equationexpresses the inertial effects of the flow. For the sakesimplicity, we can take the pointB at the positionx1

B

5e(t)/2, x2B@e(t) wherePB5P0 , VB'0. We can write

2K0~ t !]q

]t1

P0

r05

V2~ t !

2, ~5!

FIG. 6. One-gap simplified model: schematic representation of the reeaerodynamic and aeroacoustic modeling.¯ is the control surfaceS that isused in the aeroacoustic source calculation.



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where K0(t) is calculated from the expression of the timderivative of the upstream potential@Eq. ~A3!#. With the par-ticular choice of point B, we obtain K0(t)' ln„2x2

B/e(t)…/p.This differential equation~5! shows that the time depen

dence of the orifice flow velocity results from the inertiproperty of the upstream flow. The choice of the pointBappears to be arbitrary and this point is discussed in Sec

B. Aerodynamic pressure force

The pressure on the upstream face of the reed canfound by a third application of the Bernoulli equation. Thexpression of the aerodynamic pressure for a point suce(t),x1,e(t)1h andx250 is

P~x1,0,t !5r0

2V2~ t !S 12S 1

plnS x12e~ t !

x1D D 2D

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]tx1 lnS x12e~ t !

x1D

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]q

]tlnS x12e~ t !

e~ t !/2 D . ~6!

The total pressure force induced by the one-gap fldynamics on the upstream side of the reed is

F1~ t !52S Ee~ t !

e~ t !1hP~x1,0,t !dx1D e2 ~7!

or

F1~ t !52hr0

2V2~ t !„12A1~ t !…1r0h2A2~ t !

]V

]t

2r0hA3~ t !]q

]t. ~8!

The expressions of adimensional parametersA1(t),A2(t) andA3(t) are given in Appendix B. The first term othe right hand side of Eq.~8! is the classical so-called Bernoulli force in quasi-stationary modeling. In this term,A1(t)is the parameter that represents the pressure fluctuationsociated with volume flux fluctuations due to the reed apture variations. This kind of time-dependent contraction prameter is found in all oscillating reed models.1,5,2,4 It isinteresting to note that the force fluctuations induced bycontraction effect are not responsible for the reed excitamechanism. Indeed, it is possible to show from mathematexpressions given in Appendix B that at the first ordA1(t), is proportional toe2(t). Therefore the force assocated with the aperture variation is in phase with the redisplacement: it does not transfer energy to the reed mot

The Bernoulli force also depends onV2(t). These ve-locity fluctuations, given by Eq.~5!, are generated by theinertial loading of the upstream fluid. The fluid inertia alsadds the second and third terms on the right hand side of~8!. The velocity fluctuations due to fluid inertia and thinertial forces are not in phase with the reed displaceme

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C. Reed motion and reciprocal flow

It has been previously shown that the motion of a frreed14,15 is almost perfectly sinusoidal. Laser vibrometer sytem measurements that are not reported here have alsomade on our accordion reeds and the results are consiwith the previous observations. Thus the reed can be meled by a mass-spring-damping single oscillator system.reed motion leads to a reciprocal fluid current that isdependent on the main unsteady flow. The contribution ofreed surface vibration to the total pressure can be stuusing the radiation impedance of a baffled rectangular pisZm(v)5F2(v)/u(v), whereu(v) is the normal velocity ofthe reed andF2(v) is the reaction force of the fluid back othe driving piston for an angular frequencyv. For v l /c0

!1, wherec0 is the speed of sound, the largest term inZm isits imaginary part, the reactance. Morse and Ingard16 calcu-lated the expression of the reactance of a baffled rectangpiston for one of its faces,

F2~v!52 ivmfu~v!, with

mf58

9pr0lh

l 21 lh1h2

l 1h. ~9!

This forceF2 is in an opposite phase with the reed aceleration and it describes that the fluid entrained byreed side has an apparent mass ofmf . Consequently, theeffective fluid mass 2mf must be added to the acceleratreed massm in the two-dimensional reed motion equation

~m12mf !

l

]2x2

]t21

c

l

]x2

]t1

k

l~x22x0!5F1~ t !, ~10!

wherec andk are the equivalent damping and stiffness of treed. The added mass causes a decrease of the naturaquency of the reed. In air, this effect is very small but inheavier fluid such as water the difference between the intsic natural frequency and the measured natural frequencybe very large. The reed oscillator coefficients have beentermined by a classical experimental procedure basedstatic stiffness evaluation and free decay behavior. Thmeasurements are not detailed in this paper. The oscillmass is determined from the measurement of the naturalquency of the reed in air. The experimental effective mthen includes both the intrinsic massm and the added masof air 2mf . In fact, the study of the reed-induced reciprocflow is only useful for the estimation of the near-field resuant pressure that is added to the aerodynamic pressureP1(t)for comparisons with experimental data~Sec. V!. In the nearfield of the upstream region, the mean fluctuating pressassociated with the reed reactance is

P2~ t !521

lhmf

]2x2

]t2. ~11!

Moreover, it is worth noting that the reciprocal flow asociated with the reed reactance must not be taken intocount for the calculation of the far-field acoustic pressure

In this section, we have written in the time domain tequations of flow and reed motion. If we notice thatV(t) and]V/]t can be expressed as a function ofx2(t) andq(t) and



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their time derivative, we see that the model of reed excitatis a differential system of two coupled nonlinear equatio~5! and ~10! with the unknownsx2(t) and q(t). A fourthorder Runge–Kutta algorithm is used for the numerical stem solving. The nonconstant coefficients of the differensystem are updated at each step of the Runge–Kutta prdure.

IV. AEROACOUSTIC MODEL

As for other wind instruments, the sound producedthe accordion is a consequence of the unsteady flow throthe reed apertures. But unlike resonator-coupled instrumewe cannot deduce the sound from the analysis of the restor response to the inlet unsteady flux. For the accordreed, we can consider that the fluctuating fluid flow actsrectly as a source of sound on the free acoustic mediTherefore we neglect in the model all contributions of refletion or diffraction of acoustic waves on the neighboring sowalls.

A. Lighthill’s acoustic analogy

The problem of sound production by flow was first ivestigated by Lighthill17 in 1952. His theory is based on thcomparison between the exact equations of fluid motion wthe equations of sound propagation in a medium at rest.mass and momentum conservation equations are rewritteform the most general inhomogeneous wave equation,

]2r8

]t22c0

2¹2r85]2Ti j ~y,t !

]yi ]yj~12!

wherer85r2r0 and p85p2Patm. The tensorTi j 5ruiuj

1(p82c02r8)d i j 2t i j is the Lighthill stress tensor whered i j

is the Kronecker delta andt i j is the viscous stress tensoThe summation convention on indicesi and j is used. Forhigh Reynolds, low Mach number flows and in the absenof thermal sources the Lighthill stress tensor canreduced17 to Ti j 'r0uiuj . With this simplification, thesource term of Eq.~12! expresses the variation of the ratemomentum flux which is induced by turbulent processThe upstream sink flow is not turbulent and, arguing thatupstream and downstream acoustic waves are the samwith an opposite sign, it becomes evident that the turbulsources are not important in the generation of an accordreed sound for both the blowing and drawing notes. Hoever, the physics of aeroacoustic sources can only belyzed by using the mathematical expressions of theacoustic field. The far-field acoustic pressure is calculausing the convolution product of Lighthill’s source with thfree-space Green’s function. In this case, the spatial dertives of Lighthill’s tensor with respect to the source positimust be rewritten in terms of derivatives with respect toobserver’s position. By a careful mathematical analysis18,19

of the bounded volume integration of Lighthill’s tensor, maand momentum sources appear as residual terms ofLighthill tensor flux through the surface that bounds the vume. These monopole and dipole terms on the surface adthe quadrupole source generated by turbulence in the covolume. This analysis was first proposed by Curle,18 and



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Ffowcs Williams and Hawkings20 gave a rigorous mathematical framework in order to take into account solidfictitious ~permeable! control surfaces and moving solid bodies.

B. Application of the permeable FfowcsWilliams–Hawkings integral method

The Ffowcs Williams–Hawkings~FW–H! integralmethod20 was first used for the calculation of noise generaboth by turbulence and moving impenetrable surfaces,recently the FW–H equation has been validated as an igral formulation with permeable surfaces.21,22

The derivation of the FW–H formulation is based on tmathematical technique of generalized functions. A geneized function is formed with the aid of Heavisides’s functioH( f ) defined to be unity wheref .0 and zero wheref ,0.The equationf 50 defines the control surfaceS.

The Navier–Stokes equations are written for the genalized densityH( f )r8, momentumH( f )rui , and pressureH( f )p8. As for Lighthill’s equation~12!, an inhomogeneouswave equation can be expressed from the mass and motum conservation equations. Details of the derivation ofFW–H equation from conservation laws are given by Brener and Farassat.23 For a nonmoving permeable surface, tinhomogeneous wave equation is

]2r8 H~ f !

]t22c0

2¹2r8H~ f !

5]2Ti j H~ f !

]yi ]yj2

]

]yiS ~ruiuj1pd i j 2t i j !d~ f !

] f

]yjD

1]

]t S rujd~ f !] f

]yjD . ~13!

Generally, the control surfaceS is supposed to enclosall the volume sources described by Lighthill’s tensorTi j .Then, Ti j H( f )50 for f >0. Moreover, as the free-fieldGreen’s function is used for the calculation of the far-fieacoustic pressure, all the possible flow/acoustic interactand acoustic wave reflections on solid walls must be tainto account as flow variables on the control surface. Bcause of our incompressible approach, all these effectsneglected in the source terms. Finally, neglecting the viscstress tensor in the second term of the right hand side ofFW–H equation, we see that the acoustic sources candivided into a monopoleQ and a dipoleFi that are calculatedon the permeable control surface,

Q~y,t !5ruj

] f

]yj'r0uj

] f

]yj, ~14!

Fi~y,t !52~ruiuj1pd i j !] f

]yj'2~r0uiuj1pd i j !

] f

]yj.

~15!

The free-field solution of Eq.~13! is the convolutionproduct of the source terms with the free-space Green’s fution. After rearrangement of the integral expressions~see Ap-pendix C!, the density fluctuations are given by



r

dute-

l-

r-

en-e-

nsn-res

hebe

c-

r8~x,t !51

4pc02

]

]xiE

SFr0uiuj1pd i j

r Gnj dS

11

4pc02

]

]t ESFr0uj

r Gnj dS, ~16!

where the square brackets denote that the functionsevaluated at the retarded timet5t2r /c0 and nj are thecomponents of the unit outward normal vector to surfaWhenuxu!L!l, whereL is a typical dimension of the control surfaceS and l is a typical wavelength of the soungenerated, a classical simplification procedure18,17 leads tothe final expression of the acoustic pressure,

pac~x,t1uxu/c0!51

4pc0

xi

uxu2

]

]t ES~r0uiuj1pd i j !nj dS

1r0

4puxu]

]t ESujnj dS. ~17!

Obviously, the second integral on the right hand sideEq. ~17! is the expression of the total volume flux througthe control surface that is evidently equal toq(t). The firstsource term of Eq.~17! has a dipolar form and its value ievaluated numerically using an adaptive recursive Simpsrule.

V. EXPERIMENTS AND MODEL VALIDATION

A. Experimental study in air

The experimental setup for air measurements is simto that used for water visualizations~Fig. 2!. The water sup-ply tank is replaced by a low-impedance pressure sourceinsures a constant blowing pressure inside the reservoprobe microphone measures the aerodynamic pressurefew millimeters from the upstream face of the reed. A vaable capacitance transducer is used for the motion ofreed. This transducer is based on a condenser microphthe membrane of which is electrically replaced by the reedwater manometer measures the average supply pressura microphone captures the radiated sound. This microphis located at 25 cm downstream of the reed.

In Fig. 7, displacement, aerodynamic pressure aacoustic pressure are represented for a reed with a nafrequency of f 05795 Hz. The characteristic dimensionsthe valve arel 522.9 mm,h52.5 mm, a50.14 mm, andx0

50.2 mm. The supply pressure isP0540 Pa. As expectedwe see in Fig. 7~a! that the reed motion is composedsinusoidal oscillations around a displaced equilibrium potion. We note that for low playing pressure, the steady dplacement of the reed is small and we can consider thatequilibrium position is aboutx0 . Figure 7~b! shows a sharpaerodynamic pressure spike at the moment when theenters the slot. Since the equilibrium position is not sofrom the planex250, the force induced by this strong presure variation is nearly in phase with the reed velocity. Thit provides energy to the oscillator. Similarly at the reopening, a negative pressure variation occurs but it isviolent. The downstream acoustic signal shown in Fig. 7~c!



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greatly differs from the aerodynamic pressure wavefoThis illustrates the complex conversion mechanism offluid mechanical energy into acoustic energy. Only the shpressure spike associated with the reed closure is commothe two waveforms with an opposite sign. This reverse sresults from the quasi-symmetry of the upstream and dostream flow with regard to the aperture plane. Figure 8 shthe spectrum of the acoustic pressure. In the rich specdistribution of the reed sound, the odd harmonics are pdominant. It is interesting to note that both the waveform a

FIG. 7. Experimental measurements in air flow:~a! reed displacement;~b!upstream aerodynamic pressure;~c! downstream acoustic pressure.¯, clos-ing of the valve; –•–, opening of the valve.

FIG. 8. Spectrum of the measured acoustic pressure.



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nn-sale-d

spectrum of the acoustic signal compare well with signextracted from recordings of accordion notes.24

B. Application of the model

For the calculation, the model variables are initializedzero and the reed position is set tox0 . The differential equa-tion system, Eqs.~5! and~10!, is solved using Eq.~8! for theexcitation force. After the transient, self-sustained sinusooscillations are observed at the natural frequency of the r@see Fig. 9~a!#. Using the computed fluid variablesq(t) andV(t), the aerodynamic pressure at point C of Fig. 6 is coputed from the Bernoulli equation~3! with the complete ex-pression of the potential derivative, Eq.~A3!. For a compari-son with the probe microphone signal, the contributionP2(t)of the reciprocal flow is added to this aerodynamic excitatpressure. The total signal is plotted in Fig. 9~b!. The agree-ment with the measured waveform is good.

Since the downstream flow model is very simplifiecompared to the actual development of turbulent spreadjets, the acoustic pressure is calculated in the upstreamgion. The symmetry property of the produced sound is uto compare the calculated signal to the acoustic measuremperformed in the downstream region. The modeled acoupressure is the external acoustic pressure that would be msured if the reed oscillated in its sucking configuration. Tfluid variables calculated on the control surfaceS defined inFig. 6 are introduced in the integral equation~17!. The resultis plotted in Fig. 9~c!. Even if the pressure spike occurrinjust after the reed opening is smaller than the measured

FIG. 9. Results of the model calculation:~a! reed displacement;~b! aerody-namic pressure at point C;~c! reverse of the acoustic waveform at point D¯, closing of the valve; –•–, opening of the valve.



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the modeled waveform reproduces quite well the experimtal acoustic signal. Moreover, it is shown in Fig. 10 that todd harmonics of the modeled sound spectrum are mpowerful than the even ones, which is a characteristic ofaccordion reed sound. Examples of synthetized soundsavailable on the IRCAM website.25

C. Analysis of the aerodynamic excitation

From the previous comparisons, we can suppose thaphysics of both reed excitation and acoustic emissionwell reproduced by the model. The calculation results canused to highlight some details of the physical mechanism

For example, Fig. 11 shows the components of the adynamic pressure forceF1(t) associated with the Bernoulforce @the first term of Eq.~8!# and inertial force@second andthird terms of Eq.~8!#. This figure clearly demonstrates ththe inertial force is a negligible part of the total force.most reed models,1,5,2,4 inertial forces are neglected. Thsimplification can also be done for the accordion reed moand from Eq.~8!, the average excitation pressure on the ustream face of the reed is now

FIG. 10. Spectrum of the modeled acoustic pressure.

FIG. 11. A comparison of the three terms of the aerodynamic force giveEq. ~8!: –•–, first term;n, second term;1, third term;2, total force.



n-

reere

heree.o-

el-

^P1~ t !&521

hF1~ t !5

r0

2V2~ t !„12A1~ t !…. ~18!

Replacing the fluid velocity in Eq.~18! by its expressiongiven by Eq.~5!, the average pressure becomes

^P1~ t !&52r0K0~ t !„12A1~ t !…]q

]t1P0„12A1~ t !….

~19!

As explained before, the contraction effect associawith A1(t) does not contribute to the excitation mechanisThus, the termP0„12A1(t)… on the right hand side of Eq~19! is not an excitation pressure. Finally, the expressionthe excitation pressure is

^P1~ t !&exc52r0K0~ t !~12A1~ t !!]q

]t. ~20!

This relation looks like an impedance relation giving tpressure as a function of the volume flux. In this case,impedance is inertial. The value of the equivalent ‘‘acoustinertia M (t)5r0K0(t)„12A1(t)… is time-dependent.

Unlike traditional theoretical approaches of oscillatinvalves1 based on acoustic impedances of upstreamdownstream volumes, Eq.~20! is related to the impedance othe orifice. M (t) can be interpreted as the mass ecorrection16 of the upstream side of the gap. The mass ecorrection depends on the detailed form of the potential fland gap geometry. Thus, the time dependence ofM (t) is dueto the variation of gap size. In particular,K0(t) is related tothe total quantity of fluid that takes part in the inertial mechnism. For the calculation presented in this paper, the avevalue of the parameterK0 is 4.5. This corresponds to a verlarge distance of the reference pointB where the pressure isupposed to be constant and the fluid velocity is zero. Tchoice of pointB is arbitrary but first applications of theaerodynamic model show that the influence ofK0 on theresult becomes small above a certain value that dependdimensions and characteristics of the simulated reed.

D. Analysis of the sound production

In addition, the presented model allows us to analyzemechanism of sound production. The monopole and dipsources of Eq.~17! are represented in Fig. 12. It appeaimmediately that the sound of the reed is dominated bydipolar component. This observation is in opposition to tphysics of the reed coupled to a resonator. Indeed, as mtioned in the Introduction, the acoustic source exciting thecolumn of the resonator is modeled as the variation ofvolume flux through the reed gaps. In the case of a free rethe monopolar source associated with unsteady volumeat the orifice is negligible, the acoustic waves are excitedthe violent changes of momentum flux through the orifice

VI. CONCLUSIONS

In this paper, a simplified representation of a blowclosed reed used, for example, in accordions has been sied. The flow around the reed is described using an incopressible potential flow theory and the excitation force in ty



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model is associated with the inertial load of the upstrefluid. The delay in the variation of upstream fluid velocidue to the inertia leads to a velocity fluctuation in the reopening. Unlike for resonator-coupled reeds, the pressuvelocity relationship is not driven by the impedance of ustream and downstream volumes but by an equivalent itial impedance of the gaps. One of the difficulties arises frthe time-dependence of the orifice geometry which leadstime-dependent mass inertia.

This work also provides the first model of the sourcesound generated by a free reed in the absence of a resonDespite the numerous simplifications made in the calcution, the qualitative agreement of the synthesized pressignal with the measured one in terms of waveform and sptrum is quite good. The model is based on the applicationan acoustic analogy that allows the derivation of the acoupressure in the far-field as a function of the aerodynavariables known in the source region. In our case, the actic source is calculated in the time domain but it is apossible to perform the calculation in the frequency doma

ACKNOWLEDGMENTS

The authors wish to thank Agne`s Maurel and PhilippePetitjeans of the Ecole Supe´rieure de Physique et de ChimIndustrielles for helpful discussions about the flow visualiztion arrangement. We also want to thank Alain Terrier aGerard Bertrand of IRCAM for their help in the measurments.

APPENDIX A: CALCULATION OF THE UPSTREAMFLOW

In this appendix, the analytical expressions of the flpotential and velocity for the upstream region are given.

1. Flow potential in the upstream region

The upstream potential is given by Eq.~1!:

f1~x1 ,x2 ,t !521

pV~ t !E

0

e~ t !lnSA~x12u!21x2

2

e~ t !/2D du.

~A1!

FIG. 12. A comparison of the monopole and dipole terms of the acouexpression, Eq.~17!: n, monopole term; –•–, dipole term; —, total aeroa-coustic source.



d–

-r-

a

ftor.-rec-f

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.

-d

The coordinatesx1 andx2 are given in Fig. 6.V(t) is thefluid velocity through the gap ande(t) is the gap size. Aftercalculations, one can obtain the explicit expression ofpotential:

f1~x1 ,x2 ,t !521

pV~ t !S 1

2e~ t !lnS „x12e~ t !…21x2

2

„e~ t !/2…2D

21

2x1 lnS „x12e~ t !…21x2

2

x121x2

2 D 1x2S arctanS x1

x2D

2arctanS x12e~ t !

x2D D D 1

1

pq~ t !, ~A2!

whereq(t)5V(t)e(t) is the volume flux through the gap.

2. Time derivative of the flow potentialin the upstream region

The time derivative of the flow potential given by Eq~A2! is

]f1~x1 ,x2 ,t !

]t5

1

p

]V

]t S 1

2x1 lnS „x12e~ t !…21x2

2

x121x2

2 D2x2S arctanS x1

x2D2arctanS x12e~ t !

x2D D D

11

p

]q

]tS 12 lnSA„x12e~ t !…21x2

2

e~ t !/2D D .

~A3!

3. Components of the velocity vector in the upstreamregion

The two components of the upstream velocity vectorfound by calculating the gradient of the potentialf1 @Eq.~A2!#:

V1,x1~x1 ,x2 ,t !5

]f1

]x15

V~ t !

2plnS „x12e~ t !…21x2

2

x121x2

2 D , ~A4!

V1,x2~x1 ,x2 ,t !5

]f1

]x2

5V~ t !

p S arctanS x12e~ t !

x2D2arctanS x1

x2D D .

~A5!

APPENDIX B: EXPRESSION OF THE AERODYNAMICFORCE

The total force on the upstream face of the reed is givby integration of the aerodynamic pressure@Eq. ~6!# over thereed widthh,

F1~ t !52S Ee~ t !

e~ t !1hP~x1,0,t !dx1D e2 . ~B1!

The force then has the form

ic



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ed

s in

one

’’

i-ind

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F1~ t !52hr0

2V2~ t !„12A1~ t !…1r0h2A2~ t !

]V

]t

2r0hA3~ t !]q

]t, ~B2!

with the parameters

A1~ t !51

hp2 Ee~ t !

e~ t !1hS lnS x12e~ t !

x1D D 2

dx1 , ~B3!

A2~ t !51

h2pE

e~ t !

e~ t !1hx1 lnS x12e~ t !

x1Ddx1 , ~B4!

A3~ t !51

hp Ee~ t !

e~ t !1hlnS x12e~ t !

e~ t !/2 Ddx1 . ~B5!

For an analytical calculation of the integrals, we cnote that the logarithmic singularities in potential and veloity expressions are integrable.

The final expressions of the parametersA1 , A2 , andA3

are given below.

1. Parameter A 1„t …

A1~ t !51

p2 S ln2~h!1S 11e~ t !

h D ln„e~ t !1h…

3 lnS e~ t !1h

h2 D 1e~ t !

hln„e~ t !…lnS e~ t !

h2 D22

e~ t !

hdilogS e~ t !1h

e~ t ! D D , ~B6!

where the dilogarithm function is defined by

dilog~x!5E1

x ln~u!

12udu. ~B7!


A2~ t !51

2p S ln~h!1e~ t !

h„ln~h2!21…1

e2~ t !

h2ln„e~ t !…

2„e~ t !1h…2

h2ln„e~ t !1h…D . ~B8!


A3~ t !51

p S lnS 2h

e~ t ! D21D . ~B9!

APPENDIX C: CALCULATION OF THE ACOUSTICDENSITY FLUCTUATIONS

The free-field Green’s function is

G~x,tuy,t!51

4pc02r

d~g!, with g5t2t1r /c0 ,

r 5ix2yi . ~C1!



-

wherey is a point in the source region andx is the observer’sposition. Then the acoustic density fluctuations are

r8~x,t !H~ f !51

4pc02 ER3

ER

1

r S ]

]yiFi~y,t !d~ f ! D d~g!dt dy

11

4pc02 ER3

ER

1

r

]

]t„Q~y,t !d~ f !…d~g!dt dy.

~C2!

The spatial integral of the first term can be developed as

ER3

1

r S ]

]yiFi~y,t !d~ f ! D d~g!dy

5ER3

]

]yiS 1

rFi~y,t !d~ f !d~g! Ddy

2ER3

Fi~y,t !d~ f !]

]yiS 1

rd~g! Ddy. ~C3!

The first integral of the right hand side is zero and noting t]„d(g)/r …/]yi5]„d(g)/r …/]xi we can write

ER3

1

r S ]

]yiFi~y,t !d~ f ! D d~g!dy

52]

]xiE

R3

1

rFi~y,t !d~ f !d~g!dy

5]

]xiE

S

1

r~r0uiuj1pd i j !njd~g!dS, ~C4!

with

nj51

u“ f u] f

]yj. ~C5!

The second spatial integral of the density fluctuatiocan be also written as

ER3

1

r

]

]t~Q~y,t !d~ f !!d~g!dy5

]

]t ES

1

rr0ujd~g!nj dS.

~C6!

Finally, the temporal integration of the two transformeexpressions of the spatial integrals immediately givesformulation of Eq.~16!.

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al

dstioss

a-

esun

entic

ng

y,’’

d,’’

d

u-on,

re-

A

stic

-

the

Redistr

oscillations of a clarinet using the harmonic balance technique,’’ J. AcoSoc. Am.86, 35–41~1989!.

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t.

,n

d

s

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25http://www.ircam.fr/equipes/instruments/ website: last time viewed byauthors: November 2004.



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