excitation mechanisms in woodwind and brass instruments

Excitation Mechanisms in Woodwind and Brass Instrumentsby N. H. Fletcher

Department of Physics, University of New England, Armidale N.S.W. 2351, Australia

SummaryA general analysis is given of the behaviour of musical instruments driven by a reed mechanism.

Attention is concentrated on the reed mechanism itself and a clear distinction is drawn betweenthe behaviour of reeds striking inwards (as in the clarinet, oboe and organ pipe) and reeds strikingoutwards (as in the trumpet and other brass instruments). Impedance curves are calculated forreed generators of each type and it is shown that sounding of the instrument requires that theacoustic admittance of the reed, as seen from inside the mouthpiece, should have a negative realpart of larger magnitude than the real part of the pipe admittance. This implies that there is aminimum permissible blowing pressure for each reed configuration. A reed striking inwards mustoperate at a frequency below the reed resonance and below but close to the frequency of an impe-dance maximum for the pipe. A reed striking outwards must operate at a frequency above andclose to the reed resonance and above and close to an impedance maximum for the pipe. Briefconsideration is given to other matters, including non-linearities and their role in limiting oscilla-tion amplitude and in generating harmonics.

Anregungsmechanismen in Holzblas- und Blechblasinstrumenten

ZusammenfassungDas Verhalten von Musikinstrumenten, die durch einen Zungenmechanismus angeregt werden,

wird in allgemeiner Form analysiert. Dabei wird dem Zungenmechanismus selbst besondere Auf-merksamkeit gewidmet. Es wird streng zwischen dem Verhalten nach innen arbeitender Zungen(wie bei der Klarinette, Oboe und Orgelpfeife) und dem nach auBen arbeitender (wie bei derTrompete und anderen Blechblasinstrumenten) unterschieden. Fur Zungengeneratoren beider Ty-pen werden Impedanzkurven berechnet und es wird gezeigt, daB zur Schallabstrahlung des In-struments die vom Innern des Mundstiicks aus gesehene Zungenadmittanz einen negativen Real-teil haben mufi, der betragsmaBig groBer als der Realteil der Rohrimpedanz sein sollte. Dies im-pliziert einen minimalen zulassigen Blasdruck fur jede Zungenkonfiguration. Eine nach innenarbeitende Zunge muB bei Frequenzen unterhalb der Zungenresonanz und kurz unterhalb einerFrequenz, fur die die Rohrimpedanz ein Maximum aufweist, betrieben werden, eine nachauBen arbeitende Zunge bei Frequenzen kurz oberhalb der Zungenresonanz und kurz oberhalbeines Impedanzmaximums des Rohrs. In kurzer Form werden auch andere Probleme, z.B. Nicht-linearitaten und ihre Rolle bei der Begrenzung der Schwingungsamplitude und bei der Erzeu-gung von Harmonischen betrachtet.

Les mecanismes d'excitation du son dans les instruments a vent (bois et cuivres)

SommaireOn a precede a une analyse generale du fonctionnement des instruments de musique a anche, encentrant Fattention sur le mecanisme de l'anche elle-meme. Une claire distinction est etablie entrele comportement des anches travaillant vers l'interieur, comme dans la clarinette, le hautbois et letuyau d'orgue, et celui des anches travaillant vers Fexterieur, comme dans la trompette et lesautres cuivres. On a calcule les courbes d'impedance pour des generateurs a anches de chaquetype et montre que remission sonore ne peut avoir lieu que si l'admittance acoustique de l'anche,vue de l'interieur de l'embouchure, possede une partie reelle negative d'amplitude absolue supe-rieure a la partie reelle de l'admittance du tuyau. Cette condition implique qu'il existe une valeurminimale admissible pour la pression du souffle sur chaque configuration d'anche. Une anchetravaillant vers l'interieur doit fonctionner a une frequence inferieure a celle de sa resonance etinferieure egalement, mais de peu, a la frequence du maximum d'impedance du tuyau. Une anchetravaillant vers l'exterieur doit operer a une frequence un peu superieure a celle de sa resonanceet un peu superieure egalement a celle du maximum d'impedance du tuyau. D'autres questionssont examinees plus brievement, comme celle des non-linearites et de leur role dans la limitationdes oscillations et dans la generation des harmoniques.

l. introduction century, and very considerable advances in under -The behaviour of musical wind instruments in standing have been made. A recent informal book

which the sound is excited either by an air-driven by Benade [1], a more formal treatment by Neder-reed or by an air-driven vibration of the player's veen [2] and a collection of reprints edited by Kentlips has been the subject of study for more than a [3] serve to summarize the present situation.

64 N. H. FLETCHER: EXCITATION IN WOODWIND AND BRASS INSTRUMENTSACUSTICAVol. 43 (1979)

Somewhat surprisingly, though the theory of wavepropagation and resonance effects in the bores andfinger holes of such instruments has been examinedin detail, comparatively little attention has beengiven to the actual mechanism by which the soundis produced and coupled to the tube of the instru-ment. There are, in fact, several notable papers onreed mechanisms [4], [5], [6], [7] but the wholepicture is not entirely clear, particularly in relationto the lip-driven brass instruments.

It is the purpose of the present paper to attemptto remedy this situation to some extent, not bytreating a particular example in great detail butrather by developing a more general theory andinvestigating its implications for particular classesof instruments. Many of the results are not original,but they do not appear to have been presented inthis form before.

2. Phase relations

The first explicit discussion of reed-driven windinstruments seems to be that of Helmholtz [4], whodistinguished two cases as follows.

"The action of reeds differs essentially accordingas the passage which they close is opened when thereed moves against the wind towards the windchest,or moves with the wind towards the pipe. I shall saythat the first s t r ike inwards , and the seconds t r ike o u t w a r d s . The reeds of the clarinet, oboe,bassoon, and organ all strike inwards. The humanlips in brass instruments, on the other hand, arereeds striking outwards".

Helmholtz then analyzes the two cases in somedetail. Because his results are a useful introductionwe repeat them here but employ a rather differentnotation and method of development. The analysisis correct, as far as it goes, but omits a number ofimportant features of the real problem.

Suppose the pressure in the mouthpiece of theinstrument is p exp(jco^) and that the coordinatedescribing the reed opening towards the air supplyis £, so that the size of the opening varies like

io + £ e x P j (at + <p) (1)

where £o must be taken as positive for a reed strik-ing inwards and negative for a reed striking out-wards as shown in Fig. 1. Because the reed is adamped mechanical resonator driven in the £direction by the mouthpiece pressure, we can write

rar (—c a)T = sxp (2)

Fig. 1. Simple realizations of reed valves(a) striking inwards with |o > 0,(b) striking outwards with £o < 0.All are assumed to be blown from the left and to communi-cate with a musical horn on the right.

frequency and xv its damping coefficient. From thisequation we immediately find for the phase angle cp,

tan <p = xT a> a>rl((o2 — cov2) (3)

and cp varies from 0 for co <| cor to n for a> > a>T-If we suppose the magnitude of the blowing

pressure to be much greater than p, then the phaseof the air flow into the instrument is <p for a reedstriking inwards and 99 — n for a reed striking out-wards.Now the input impedance of a cylindrical pipe oflength I and cross-section S which is damped pre-dominantly by wall effects has, as is shown in theAppendix, the form

1 + jHt&nkl*- = 'ir H + j tan kl

(4)

where rar and sT are respectively the effective movingmass and the area of the reed, coT is its resonance

over that part of the frequency range where theheight of the impedance maxima is H relative to thereference level QC/S. Q is the density of air, c thespeed of sound in air and k = cofc. This form of theresult is useful since, by making both I and H to beslowly-varying functions of co, it adequately repre-sents the input impedance of a real musical instru-ment. If we write Z as \Z\ exp (j 6) then

tan 6 = [{H2 — 1)[2H] sin 2 Id (5)

and the phase of the acoustic flow relative to thepressure is — 6.

Now for the instrument to function in a steadymanner, the phase of the acoustic flow must beconstant throughout and therefore tan 0 = —tan 6,which relation accommodates the requirements forboth reed geometries. Using eqs. (3) and (5) this

ACUSTICAVol. 43 (1979) N. H. FLETCHER: EXCITATION IN WOODWIND AND BRASS INSTRUMENTS 65

implies

—co2) (6)

which is essentially the condition derived by Helm-holtz. In order that the reed be a generator ratherthan a load, it is necessary that both sides of eq. (6)be positive for reeds striking inwards and negativefor reeds striking outwards. In the first case, there-fore, which applies to woodwind instruments, wemust have co < coT, while for brass instruments withlip drive co > coT. The necessary requirement thatthe left side of eq. (4) be no greater than unityimplies only that the fractional shift from reedresonance, (a>r — co)jcoT, be greater than about xTjHwhich is typically of order 10~2.

Because eq. (6) contains none of the dynamics ofthe problem, it is necessary to appeal to otherarguments to proceed further. If the reed is not veryheavy, so that it is influenced by the pipe, and co isnot nearly equal to cor, then eq. (6) implies that sin2kl = ^ e where e is a small positive quantityand the plus sign applies to woodwind and the minusto brass instruments. The solutions to this are co ^nnc\1l or I na nX/4, n = 1, 2, 3, .. . , but those witheven values of n must be ruled out because theycorrespond to situations with a pressure node at thereed so that no drive can take place. When the non-zero value of e is included we find

co = (2w + 1) TZC/21 =F <5 = eop =F 6 (7)

where d is a small quantity determined by e and nowthe minus sign applies to woodwind and the positivesign to brass instruments. The cop are resonancefrequencies for a pipe stopped at the end where thereed is fitted.

3. Flow dynamics

The next extension of the theory is due to Backus[5] who included a treatment of air flow dynamics ina clarinet-like system but simplified matters, con-sistently with the conclusions drawn from eq. (6),by taking coT = oo, which is equivalent to neglect-ing the mass of the reed. This is largely justified forthe actual case of a clarinet but restricts the gene-rality of the results. This particular assumption wasrelaxed in later treatments, also for a clarinet-likesystem, by Worman [6] and by Wilson and Beavers[7]. All these workers, except Worman, were con-cerned with small-amplitude oscillations just abovethe self-excitation threshold so that the theories areessentially linear. They do however, give some infor-mation about excitation of the various pipe modesand, as for the predictions of Helmholtz's theory,

agree with experiment when checked againstappropriately circumscribed cases.

The primary innovation of these treatments re-lates to the dynamics of flow past the reed and inparticular to inclusion of the effect of mouthpiecepressure on this flow. We shall develop this rathermore generally so that it applies to reeds striking ineither direction.

Suppose that the blowing pressure applied to thereed is po and the pressure in the mouthpiece is p.Then in a mechanically static situation the volumeflow U through the reed will be given by theBernoulli result

U = \tj\b[2(po — p)lo\1^ (8)

where |£| and b define the dimensions of the reedopening and Q is the air density. While this resultis adequate for simple reed geometry, many practicalsituations are more complex, so we write

U = D\^\a(po — p)P (9)

where a ^ l and ft & ^. In fact Bacus [5] has shownempirically that for a typical clarinet reed andmouthpiece <x ^ 4/3, ft ?& 2/3.

The total equation of flow must take into accountthe fact that £ depends on p and also that there is amass-like load

M(g) = Qa[\t;\b (10)

associated with the air in the reed tip channel (orlip aperture) whose length we take as a. We cantherefore write, using eq. (9),

(11)M(£)(dUldt)

where we have written D' for D~1I^. We might alsoadd a small resistive term R (£) U to account forviscous losses in the reed tip channel but, since theeffect of such a term is obvious, we omit it in theinterests of simplicity. Because eq. (11) is clearlynon-linear we can no longer simply add complexexponentials as in eq. (1) but must take

-> lo + cos (ncot +where the sums here and subsequently are overn = 1, 2, 3, ... Again £o > 0 for an inward strikingand £o < 0 for an outward striking reed. Similarlywe write

U -> UQ + J Un cos (ncot + ipn) (13)n

and, taking atmospheric pressure as reference,

V ~>^2<Vn cos {ncot + £n). (14)

66 N. H. FLETCHER: EXCITATION IN WOODWIND AND BRASS INSTRUMENTS ACUSTICAVol. 43 (1979)

tan (cpn — £w) =

where 0 5̂ q>n — £n ^ n, and

pn = «rlmr£w[(cor2 — W2

The equation obeyed by £ is, by analogy with eq. (2),

d2 d 1+ < U T " > < (15)

X cos (ncot + <pn) = ^pn cos (neat -f £M)n

and the mean reed opening £o is given in terms of the"unblown" opening £o' by

lo = £o' — («t/% <*>r2) Po • (16)

We have neglected non-linear terms in eq. (15),which is a reasonable approximation unless the reedopening actually closes. We then immediately havethe results

2ft>2 — cor2) (17)

(18)

Clearly the reed deflection is essentially pnsTlmTcoT2

for nco <̂ coT, peaks to l/xr times this value at theresonance frequency cor, and then decreases aspnsTj'mrn

2o)2' for higher frequencies. mrcor2/sr is just

the static stiffness of the reed.Eqs. (14) to (16) are in fact not complete and it is

worthwhile pointing out what has been omitted inaddition to the resistive term referred to in eq. (11).In eq. (14) we have assumed that the static compo-nent of the mouthpiece pressure p is simply atmo-spheric pressure whereas we should really makeallowance for the finite resistance offered by themouthpiece and instrument bore to steady flow. Theonly effect is a small reduction in the effective blow-ing pressure po which could easily be included butwhich we neglect for simplicity. The second simpli-fication is that we have omitted from the right handside of eq. (15) a small term of the form — e(£7/£)2

which describes the action of the flow in producinga Bernoulli pressure in the smallest constriction ofthe reed opening. The significance of this termdepends on the reed geometry but is usually verysmall compared with the direct effect of mouth-piece pressure. In the interests of simplicity and toavoid introduction of the extra parameter e we omitthis term from the subsequent discussion. Thirdlywe have assumed the impedance of the wind channelleading to the reed to be zero, so that the blowingpressure is always po. Neglect of these effects meansthat the reed will not oscillate in the absence of aresonator, which is not strictly true of real reeds.

Substituting eqs. (12), (13) and (14) in eq. (11),

and assuming ^ £« < l£o| a n ( i 2 ^n ^ ^° ' w e

can collect up terms for various Fourier components.The zero-frequency equation differs only in secondorder from eq. (9) with p = 0. The general equation

for the nth. harmonic can be expressed as

pn cos Cn — 1

, Qa

1 Un

cos <pw — - — cos xpn +

>Un sin wn —-r^\-n-\-4p \p

X I Z 2̂ COS VPm + (pn-tn) +

Z ^2 C o s W™ — <Pm±n)

0

JL/1— A#1 a

(19)

COS (...) + ...

gacosin (. . .) +

4- higher terms

together with a similar expression with cos -> sinand sin->—cos throughout. These expressionsare written out only to second order and only thefirst of the second order terms is given explicitly,the others following a similar pattern. Clearly thenon-linearity of the flow couples together all thepartials of the oscillation, and it is this couplingwhich justifies our original assumption that thesepartials are all harmonically related [8].

First of all let us consider just the linear terms,which will be the most important under conditionsof small excitation. These give, using eq. (18),

Pn[(PoOclpt;oKn) COS (pn — COS= Un[(polpU0)cosy>n— (20)

together with a similar equation with cos -> sinand sin -> — cos. These are generalizations of thecorresponding equations of Backus [5], with allow-ance made for the finite resonance frequency of thereed, as was done by Worman [6] and by Wilsonand Beavers [7], and with the sign of £o takenexplicitly into account.

Since the modes are uncoupled in the linearapproximation, we can take the phase £n = 0. Ifnecessary this phase can be restored later. We alsodrop the multiplier n and write simply a> for thefrequency.

From eq. (20) we then find

—J)(l-B)-co (qafb \£0\)

where

A =

B =

( }

(22)

(23)


and we take \p to lie between —TT/2 and 7z/2.Wecan also define the reed generator admittance YT,as measured from inside the mouthpiece, as

YT = — U/p. (24)

We shall later find this to be a more useful quantitythan the generator impedance ZT = Y~x. At fre-quency co this admittance has phase angle \p andmagnitude

ry 1 v _ (25)(polfiUo) cosy — co(ga/& |£o|) s i n ^

where, because of the restriction placed on xp, [ YT]is allowed to be negative. Note that YT does notbecome zero when B = 1 because of a balancingzero in the denominator. Note also that eqs. (21)and (25) cease to be valid in the limit po -> 0,because the Bernoulli-like flow law (8) or (9) thenno longer applies.

4. Instrument performance

We can see the implications of these results with-out great difficulty. Suppose that there is anacoustic pressure p with frequency co in the mouth-piece cavity. This causes an acoustic flow

through the reed into the instrument and suppliesacoustic energy to it at a rate p2 Re (— YT) where p2

is the real mean-square value of p. At the sametime p causes an acoustic flow £7p = p F p in thepipe which dissipates energy at the rate p2 Re (Fp).Clearly if more energy is supplied than is dissipated,that is if

Re(7r + yp) <0 (26)

then the oscillations will grow and the instrumentwill sound, the loudness of the sound being deter-mined, to a first approximation, by the extent towhich the inequality is satisfied. Thus Yr must havea negative real part which is as large as possible andYp a positive real part which is as small as possible,implying that the instrument should operate neara pipe impedance maximum.

Of course for a steady sound UT must equal £7p

in magnitude and phase and this can, in general, beachieved only through the agency of non-lineareffects which reduce Yr at large amplitudes. In thesteady state then, we must have

Re(F r + Yp) = 0 (27)

Im(7r+rp) = 0 (28)

the first condition being reached largely throughadjustment of the amplitude of the oscillation andthe second through adjustment of its frequency.

The behaviour of YT is given by eqs. (21) and (25)and is most easily appreciated if we neglect the smallterms describing the inductive loading contributedby the small mass of air in the reed or lip channel.These are the final terms in the numerator anddenominator of eq. (21) and in the denominator ofeq. (25). Clearly the sign of the real part of YT isdetermined by the sign of (1 — B) and, if this is tobe negative, we must have, from eq. (23) either

fo > 0, co > coT, poc > po > po* (29)

or£o < 0, co < coT, po > (30)

These are exactly the frequency conditions foundbefore but it is now required in addition that theblowing pressure po must be greater than a criticalvalue po* given by

P0* = K/« r) (0£o/«) [(«r2 — CO+ (cocoTxr)

2]l{coT2 —

2)2

(31)

The existence of this critical pressure has been wellproven in the case of woodwind instruments forwhich the reed strikes inwards. It is less well knownfor the case of lip-driven brass instruments. We alsosee that, if we start from a condition of fixed un-blown lip or reed opening £o' then, from eq. (16),there is an upper limit

poc = lo'mrcor2/5r (32)

in the case of the inward-striking reed, above whichit is completely closed and no sound generation cantake place. There is no such upper limit for lip-driven brass instruments.

It is useful now to plot polar diagrams for the reedadmittance YT as a function of blowing pressure po-This is done in Fig. 2, using the physical parametersgiven in Table I and restoring the small termsneglected in the discussion immediately above. Itis clear that a lip driven tube can always be madeto sound, irrespective of the quality of its resonances,provided a sufficiently high blowing pressure is used.

Table I.

Assumed values of parameters.

Unblown reed openingEffective reed massEffective reed areaReed resonance frequencyReed parameters

|o ' = ±0.5 mmmr = 0.1 gsr = 0.5 cm2

cor = 3140 rad s"1 -> 500 Hz« = 1, p = 0.5

Effective reed channel width b = 5 mmEffective reed channel length a = 2 mmReed damping parameter xT = 0 . 1Blowing pressure po = 5 kPaClosing pressure (£o' > 0) poc = 10 kPa

Pressure Conversion: 1 kPa = 10 cm water gauge.


1.0

®

•

1

\ )I0.01

-1.0

-0.2

-0.3-0.3 -0.2 -0.1

Rem-0.1 0.2

Fig. 2. The real part (Re) and imaginary part (Im) of thereed admittance Yr, with blowing pressure po in kilopascals(1 kPa = 10 cm water gauge) as parameter, for the cases(a) |o > 0, co = 0.9 cor and(b) £0 < 0, a) = 1.1 cyr.In case (a) the reed closes for po ^ poc and in each case po*is the critical blowing pressure for generator action. Assum-ed values of other parameters are given in Table I. The unitsfor Yr are reciprocal S. I. acoustic ohms (i.e. m3 Pa~x s-1) xxlO-6.

For a normal reed driven tube with £o > 0, theimaginary part of Yr is positive so that, from eq.(28), the imaginary part of F p must be negative.From eqs. (4) and (5) this implies that d and sin 2klmust be positive which in turn implies that theoscillation frequency co must be less than the neigh-bouring pipe resonance frequency cop. The oppositeis true for an outward-striking lip reed.

We can go a little further and plot the way inwhich the real part of the admittance YT varies withfrequency for a blowing pressure greater than thecritical value po*. This is done in Fig. 3 for severaldifferent values of the reed damping xT. Remember-ing that we are still treating the system in the linear

approximation only, we can now make some addi-tional assertions.

Considering first the normal reed drive for which£o > 0 we see from Fig. 3 a confirmation that comust be less than the resonant frequency cor ifRe (YT) is to be negative. If the reed damping issmall then Re (Yr) has its greatest negative magni-tude just below the resonance so that, if the tubehas impedance peaks of reasonable height near thisfrequency, this near-resonant mode will be favoured.This is the situation in the reed pipes of pipe organs.It is not, however, the playing mode desired in wood-wind instruments, for which a>T is generally chosento be well above the frequency of the first few tubemodes. If the reed is highly damped by the playerslips, the negative reed admittance is nearly constant

-0.10 L

Fig. 3. The real part (Re) of the reed admittance YT as afunction of frequency w for po = 5 kPa (about 3 po*) and(a) |o > 0,(b) |o < 0.The parameter is xT, the damping coefficient of the reed.Other parameters have the values given in Table I. Unitsare as in Fig. 2.


for co < cor and it is the lowest pipe modes whichare excited, as is desired. Little adjustment of lipposition on the reed is then required to producedifferent notes, these being determined by the fre-quency of the most prominent pipe resonance foreach fingering.

In the case of the lip-driven instruments the situ-ation is quite different, as shown in Fig. 3b. Theplaying frequency a> is now greater than toT, thefrequency shift being close to ^xTcoT. There is thusa strong tendency for any mode lying about \xTtor

above the frequency of the lip resonance to be pre-ferentially excited. If xT is of order 0.1, the smallwidth of this response region gives adequate dis-crimination between pipe resonances only one semi-tone apart in pitch (6 percent different in frequency)in the case of a skilled player, while not being undulyrestrictive to the execution of rapid passages in thelow register of modern instruments using valves.

The xT value of dry cane material from whichwoodwind reeds are made is probably less than 0.1,but it is likely that the method by which two canesare tied together in instruments like the crumhornand bagpipes substantially increases this value.When the reed is wet and further loaded by the softtissue of the players lips, as in modern woodwindinstruments, it is easy to see that xT may lie in therange 0.3 to 1 as suggested by our discussion.

For the lip drive of brass instruments the situationis very different, for the xT value for the players lips,even under muscular tension, may well exceed thevalue 0.1 required for proper performance on instru-ments like the French Horn. The explanation of thisapparent difficulty seems to lie in some of the termsomitted from our simple theory, in particular theterms representing the finite volume and flow impe-dance of the airway through the player's mouth.If the mouth cavity is regarded as an acousticcapacitance fed through an air channel of finiteresistance, then the air pressure po within it willvary slightly because of flow through the lips. Thiscan be accounted for by adding a term proportionalto J U dt to the right hand side of eq. (15). SinceU is nearly n out of phase with £ for a lip reed(I < 0), J U dt is nearly in phase with d£/d£ andso reduces the effective value of xT on the left handside. This also accounts for the fact that the lipscan be made to "buzz" in the absence of any musicalinstrument resonator.

5. Non-linearitiesProvided the excitation level is not too large, so

that expansions like (12), (13) and (14) are fairlyrapidly convergent, the system behaviour can bedescribed by the set of eqs. (19), carried if necessary

to a larger number of terms. This small-signalapproach is the one usually employed and its impli-cations have been carefully explored by Benade andGans (9], Worman [7], Benade [1] and Fletcher [8].

Because the resonances of a musical horn arenever exactly harmonic, a strictly linear excitationmechanism would lead to a sound the frequencies ofwhose partials were not exact multiples of a commonfundamental frequency, giving at best a quaint andat worst a musically unpleasant sound. If the driv-ing mechanism is sufficiently non-linear, however,all the partials of the sound become locked intoexact harmonic relationship [8], [10], as is known tobe the case for virtually all non-electronic musicalinstruments producing steady sounds. The expan-sions leading to eq. (19) have already assumed thisto be so.

Investigating this situation, Benade [1], [9] hasshown that, rather than considering the regenerativebehaviour only at the fundamental frequency, aswe have done, we should really take into accountthe weighted impedance of the drive mechanism,and particularly of the musical horn, over allharmonic components of the sound being produced.We have, at present, nothing to add to Benade'sdiscussion.

Worman [7] has shown that, in the small-signalregime in which the fundamental is dominant, theamplitude of the nth harmonic varies as the nth.power of the amplitude of the fundamental. Thisconclusion is also implied by our eq. (19), as follows.From eqs. (18) and (20), if the fundamental fre-quency co is fixed, the quantities pn, Un and £n forthe nth harmonic are simply proportional to eachother. Using this conclusion to determine successive-ly the scond, third, . . . harmonic amplitudes pn

from eq. (19), on the assumption that pn > Pn+ifor all n, leads immediately to the conclusion thatPn oc pin. We must remember, however, that thisconclusion applies to the pressure amplitudes withinthe instrument mouthpiece, rather than to theradiated sound. Benade [1] has discussed, and indeedin simple cases it follows from standard acousticaltheory, that each musical instrument horn has acharacteristic cut-off frequency coc above which theradiated acoustic pressure amplitude is simplyproportional to the internal pressure amplitude pn

but below which it is proportional to pn{cojcoc)2.

Since coc is typically 104 rad s"1, corresponding toabout 1.5 kHz, there is thus a very large amountof high-frequency emphasis in the radiated soundspectrum compared with the internal spectrum.

Finally we should note that, even in this low-excitation regime, it is the non-linearity of theexcitation mechanism which determines the total

70 N. H. FLETCHER: EXCITATION IN WOODWIND AND BRASS INSTRUMENTS ACUSTICAVol. 43 (1979)

amplitude and power of the oscillation. This limita-tion depends primarily upon the relative magnitudesof the linear and cubic terms in eq. (19), as has beendiscussed elsewhere for the related case of organ pipeexcitation by an air jet [10]. Since the cubic termsvary as £~3 and the linear terms only as £-1, largeoscillation amplitude is favoured by a large value£0', and hence of £o> provided po is constant andPo > Po*-

When we come to consider the behaviour at highexcitation levels we find that non-linearity plays adominant part in determining the shape of theinternal pressure waveform. We define a high exci-tation level in this connection to be either one givingsufficient amplitude £ to the reed vibration so that£ =̂ f 0 and the reed closes, or else one giving suffi-cient amplitude p to the mouthpiece pressure sothat p ^ po — po* and the instantaneous operatingpoint for the reed generator is forced back into thedissipative region, giving a reversal in the sign ofthe real part of the flow U at the frequency consider-ed. These two non-linearities are to some extentindependent and under the control of the player butthey are also partly determined by the nature of thereed system and of the instrument horn.

The clarinet has been extensively studied [1], [6],[11] and shows this behaviour well. The optimumblowing pressure for the clarinet is midway betweenthe threshold pressure po* and the pressure poc

which closes the reed. Not only does this maximizethe dynamic range below the threshold of extremenon-linearity but also it ensures that, when non-linear behaviour begins, it does so nearly sym-metrically, giving a flow waveform which approxi-mates a square wave. This is important since theclarinet horn, being nearly cylindrical, has impe-dance maxima near coi, 3a>i, . . . and impedanceminima near 2coi, 4coi, If the pressure waveformcontains significant even-harmonic components,then the weighted impedance of the horn will belower than optimal and it will not couple well tothe reed generator.

Assuming operation in this symmetrical fashion,the horn has a linear resonant response so that themouthpiece pressure is also of square waveform.Since the reed is a compliant system of high re-sonant frequency, its displacement follows thepressure waveform quite closely and indeed this iswhat was observed in the pioneering work ofBackus [11].

Instruments such as the oboe, bassoon and saxo-phone have nearly conical horns which show impe-dance maxima at the throat at frequencies a>i, 2a>i,3a>i, . . . so that it is no longer a condition for effi-cient operation that the non-linear clipping be sym-

metrical. Indeed the relative heights of the secondand third resonance peaks suggest possible advan-tages from asymmetric clipping. In the case of theoboe, the smaller initial reed opening (<0.5)mmand higher blowing pressure (^4.5 kPa) comparedwith the clarinet (%lmm and 3.5 kPa) suggestthat the major non-linearity may be that associatedwith closing of the reed. This may also apply to thebassoon; the situation with the saxophone is lessclear. These conjectures, however, require experi-mental confirmation.

The situation with the lip-driven brass instru-ments is rather different because of the resonantbehavior of the lip reed and the fact that the dynamiclevel can be increased by a parallel increase of bothPo and |o'- The resonant behaviour of the lip reedmeans that it responds only weakly to driving pres-sures at upper-harmonic frequencies and tends to ad-just its motion so as to just close during each cycle.This behaviour was recognised long ago throughthe elegant photographic studies of Martin [12].

The horns of brass instruments are designed toproduce a harmonic series which is well tunedexcept for the horn fundamental (typically 0.8 coi,2a>i, 3a>i, . . .) and only modes based on the higherhorn resonances are used in normal playing. For eachof these modes the upper resonances provide a well-aligned complete harmonic series so that there isno obstacle to prevent, and perhaps some advantageto be gained from, asymmetric clipping. The designof a typical brass instrument mouth cup, togetherwith the high dynamic level possible, leads to ahigh mouthpiece pressure and thus to waveformclipping when the lip opening is wide. The asym-metric flow waveform contains harmonics of allorders, which may be further intensified by the non-linearity of the flow through the constriction joiningthe mouth cup to the main horn. There is, as wehave already discussed, a further emphasis of theupper harmonics in the radiated sound.

This discussion of harmonic generation in brassinstruments follows that given by Backus andHundley [13], who showed that mouthpiece pressurewaveforms calculated on this basis for a trumpet arein good agreement with measured waveforms pro-duced using a mechanically operated valve tosimulate the lip opening. In each case the pressurewaveform approached the shape of a half-wave-rectified sinusoid.

Finally we can use this discussion and the pre-vious analysis to estimate the maximum peak-to-peak value of the mouthpiece pressure waveformthat can be produced in a wind instrument. Weassume that the wall and radiation damping in thehorn is sufficiently small that the mouthpiece


pressure is driven to the clipping point in eachdirection and that the playing conditions are thosepreviously discussed: £o' > 0 and a> <̂ o)T for awoodwind instrument and £o' < 0, a> & coT(l -f- hxr)for a brass instrument.

For a woodwind instrument, from eq. (31), onecut-off point is essentially given by

(po — p)* = (mrwr2/sr) (0/«)£o (33)

where, since we are considering large signal condi-tions for the mouthpiece pressure p, eq. (16) mustbe written

£o = £o' — (sTlmTcoT2) (po — p)*. (34)

Also from eq. (16), when £o = 0 and the reed closes,we have

(po — pY = (mr eor2/sr) £„'. (35)

Combining these equations, we find for the maxi-mum peak-to-peak excursion P of the mouthpiecepressure

)'• (36)

The factor oc/(oc + /5) is about 2/3 and the value of Pis simply proportional to the product of the effectivereed stiffness rareor

2/.sr and the unblown reed open-ing io', and is independent of blowing pressureprovided po* < po < poc. Reed instrument playersgenerally control loudness by altering lip tension tochange the value of £o'-

In the case of brass instruments the resonantbehaviour of the lips makes the situation ratherdifferent. The lip motion does not respond instant-aneously to pressure variations as in a woodwindreed but executes a sinusoidal motion, the ampli-tude of which is probably equal to the equilibriumblown lip separation £o given by eq. (16) with |o ' << 0. The lip oscillation is substantially out of phasewith the mouthpiece pressure as we have alreadydiscussed. Provided the lip damping xT is smallenough that the oscillation amplitude | given byeq. (18) reaches the value £o> then the maximumpressure excursion P will approximate the blowingpressure po, the amount of clipping of the waveformdepending on the unblown lip opening £o'- Againboth £o' and po are under the control of the player.

6. Conclusion

This analysis has served to identify those featureswhich are of primary importance to the under-standing of the excitation mechanism in woodwindand brass instruments. Such an analysis shouldprovide the necessary background for more sophisti-cated studies of particular instruments or, perhapseven more interestingly, of performance techni-ques upon these instruments.

This study is part of a programme of research inmusical acoustics supported by the AustralianResearch Grants Committee. I am grateful toSuszanne Thwaites for assistance with the calcu-lations.

Appendix

Standard theory for wave propagation in finitecylindrical tubes shows that for the dimensions andoperating frequencies of most musical wind instru-ments the dominant loss mechanisms are those to thewalls. Radiation losses from the open end are notsignificant for the first few pipe modes and the openend impedance behaves, to a good approximation,like a small inductance giving an "end correction"addition to the tube length which is just 0.6 timesthe radius. If I is the length of the tube includingthis correction, then the input impedance (definedas the ratio of acoustic pressure to acoustic volumeflow) at its throat end is just

Zp = j (e c/S) tan [(jfc — j k') I] (A1)

where Q is the density and c the velocity of sound inair, S is the cross-sectional area of the tube, k =co/c is the real part of the propagation number andj A;' an imaginary component to take account of walllosses.

The expression (Al) can easily be re-expressed inthe form

(A2)Zp = j (QC/S) [tan kl — j tanh k'l] \

/ [1 + j tanh k'l tan kl]

and we see that, at the resonances corresponding toimpedance maxima, where kl = (2n -\- 1) jr/2,

Z[,res) = (QC/S) coth k'l. (A3)

If we suppose the height of these impedance peaksto be H times the reference value QC/S, where H isa slowly-varying function of frequency, since this isalso true of k' and k'l <̂ 1 in general, then we canwrite eq. (A2) as

/ (# + j tan kl).

This result will also be a reasonable approximationfor the horns of brass instruments for which theeffective length I increases with increasing frequency.

The phase 6 of Zv is given by

tan 6 =[ ^ ~H / \ l+tan2ifcZi

2H .sin2&Z.

(A5)

The real and imaginary parts of Zp can easily befound from eqs. (A4) and (A 5).

(Received February 9th, 1978.)


References

[1] Benade, A. H., Fundamentals of musical acoustics.Oxford University Press, New York 1976.

[2] Nederveen, C. J., Acoustical aspects of woodwindinstruments. Frits Knuf, Amsterdam 1969.

[3] Kent, E.L. (ed.), Musical acoustics: Piano and windinstruments. Benchmark Papers in Acoustics, Vol. 9.Dowden, Hutchinson & Ross, Stroudsburg 1977.

[4] Helmholtz, H.L.F., On the sensations of tone, 1877.Trans, by A. J. Ellis, reprinted by Dover, New York1954, pp. 390-394.

[5] Backus, J., Small vibration theory of the clarinet. J.Acoust. Soc. Amer. 35 [1963], 305, and Erratum 61[1977], 1381.

[6] Worman, W.E., Self-sustained nonlinear oscillationsof medium amplitude in clarinet-like systems. Thesis,

Cake Western Reserve University 1971, Ann Arbor,University Microfilms (ref. 71-22869).

[7] Wilson, T.A., and Beavers, G. S., Operating modes ofthe clarinet. J. Acoust. Soc. Amer. 56 [1974], 653.

[8] Fletcher, N.H., Mode locking in non-linearly excitedinharmonic musical oscillators. J. Acoust. Soc. Amer.64 [1978], 1566.

[9] Benade, A.H., and Gans, D. J., Sound production inwind instruments. Ann. N.Y. Acad. Sci. 155 [1968],247.

[10] Fletcher, N. H., Transients in the speech of organ fluepipes — A theoretical study. Acustica 34 [1976], 224.

[11] Backus, J., Vibrations of the reed and the air columnin the clarinet. J. Acoust. Soc. Amer. 33 [1961], 806.

[12] Martin, D.W., Lip vibrations in a cornet mouthpiece.J. Acoust. Soc. Amer. 13 [1942], 305.

[13] Backus, J., and Hundley, T.C., Harmonic generationin the trumpet. J. Acoust. Soc. Amer. 49 [1971], 509.

excitation mechanisms in woodwind and brass instruments

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