Fletcher 1979

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Annu. Rev. Fluid. Mech. 1979.11:123-146. Downloaded from arjournals.annualreviews.org by UNIVERSITY OF NEW SOUTH WALES on 06/28/06. For personal use only.

Ann. Rev. Fluid Mech. 1979. 11 : 123-46 Copyright © 1979 by Annual Reviews Inc. All ri#hts reserved

AIR FLOW AND SOUND GENERATION IN MUSICAL WIND INSTRUMENTS

~8137

N. H. Fletcher Departmentof Physics, University of NewEngland,Armidale,N.S.W.2351, Australia

INTRODUCTION Over the past two decades or so, interest in musical acoustics appears to have been increasing rapidly. Wenowhave available several collections of reprinted technical articles (Hutchins 1975, 1976, Kent 1977), together with a large numberof textbooks, of which those most suitable for citation in this review are by Olson (1967), Backus(1969), Nederveen(1969), and Benade (1976). The mathematical foundations of the subject were laid primarily by Lord Rayleigh (1896) and are well treated in such standard texts as Morse (1948) and Morse & Ingard (1968). This review covers a muchmore restricted field than this preliminary bibliography might suggest. Amongall the varieties of musical instruments I concentrate on those capable of producing a steady sound that is maintained by a flow of air, and even within this family I am interested not so much in the design and behavior of the instrument as a whole but rather in the details of the air flow that are responsible for the actual tone production. Although musical instruments function as closely integrated systems, it is convenient and indeed almost essential for their analysis to consider them in terms of at least two interacting subsystems, as shownin Figure 1. The first of these is the primary resonant system, which consists of a column of air, confined by rigid walls of more or less complex shape and having one or more openings. Such a system is generally not far from linear in its behavior and it can be treated, at least in principle, by the classical methods of acoustics. The second subsystem is the airdriven generator that excites the primary resonator. This subsystem is generally 123 0066-4189/79/0115-0123501.00

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FLETCHER

Generator

air pressure

Drive tout’ling

Resonator

Acoustic output

Feedback coupling Annu. Rev. Fluid. Mech. 1979.11:123-146. Downloaded from arjournals.annualreviews.org by UNIVERSITY OF NEW SOUTH WALES on 06/28/06. For personal use only.

Fi#ure 1

Systemdiagramfor a musicalwindinstrument.

highly nonlinear, either intrinsically or through its couplings with the resonator system, and it is this nonlinearity, as we shall see below, that is responsible for the stability of the whole system as well as for muchof its acoustic character. AIR

COLUMNS

The acoustical behavior of an air column of arbitrary cross section is well understood provided the cross section is a slowly varying function of position (Eisner 1967, Benade & Jansson 1974, Jansson & Benade 1974). Columnsenclosed in tubes of exactly cylindrical or exactly conical Shape are particularly simple to analyze, as are a few other special shapes (Morse 1948, pp. 233-88, Benade 1959, Nederveen 1969, pp. 15-24), but the detailed shapes of the bores of real wind instruments usually differ significantly from these idealized models. The quantity of major importance for our discussion is the acoustical impedance Zp (defined as the ratio of acoustic pressure to acoustic volumeflow) at the input to the resonator where the driving force from the generator may be supposed to act. Various instruments have been developed to measure this impedance (Benade 1973, Backus 1974, Pratt et al 1977) following early work by Kent and his collaborators. Because of the phase shifts involved, Zp is usually written as a complexquantity, and the measuring system can be arranged to yield either the real part, the imaginary part, or simply the magnitude of either the impedance or its inverse, the admittance. For a narrow cylindrical pipe of cross-sectional area A the acousticwave propagation velocity has very nearly its free air value c, and the major losses are caused by viscous and thermal effects at the walls, comparatively little energy being lost from the end (ifopen) provided its circumference is muchsmaller than the acoustic wavelength involved. If we denote angular frequency by o~ then the propagation number is k = ~o/c, and we can generalize this to allow for wall losses by replacing k by k-jk’, where j = x/- 1 and k’ ~ k. The loss parameter k’ increases with

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3.01

lO(]

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3.1

too

O.Ol

Figure 2 Acoustic input impedance Zp, in units of pc/A, for a typical cylindrical pipe open at the far end. Zp is plotted on a logarithmic scale so that the acoustic admittance Yp = Zp 1 is obtained simply by inverting the diagram. Typically pc/A ~ 10e Pa m- 3 s ~ 1 SI acoustic megohm.

frequency like (01/2¯ If the pipe is open at the far end then the input impedanceis very nearly Zp ~ j(pc/A) tan [(k-jk’)l],

(1)

where p is the density of air and the effective pipe length l exceeds the geometrical length by an end correction equal to 0.6 times the radius. The form of this expression is shownin Figure 2, which is plotted on a logarithmic scale so that the magnitude of the input admittance Y~ = Z; 1 Can be seen by simply imagining the picture to be inverted. The impedance Zp shows peaks of magnitude (pc/A) coth k’l at frequencies O9o, 3o90, 5(00 ..... and the admittance Y~showspeaks of magnitude (A/pc) coth k’l at frequencies 2e)o, 4090, ..., where (0o is given kl = re/2 or c00 = nc/21. For a pipe of finite radius these resonances are slightly stretched in frequency, but this need not concern us here. For our present purposes we need consider only two types of excitation mechanism: the air jet of flute-type instruments, whose deflection is controlled by the velocity of the acoustic flow out of the pipe mouth as ¯ shown in Figure 3, and the reed or lip-reed generator, whose opening is flue upperlip lower lip / 7outy

foot Figure 3 features.

languid

bod/

tuningslide

Section through a typical organ flue pipe showing its main constructional

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FLETCHER

controlled by the acoustic pressure inside the pipe at the mouthpiece as shown in Figure 4. Wecan call these respectively velocity-controlled and pressure-controlled generators. A velocity-controlled generator clearly transfers maximumpower to the pipe resonator when the acoustic flow at the mouth is maximal and thus at the frequency of an admittance maximum.. A pressure-controlled generator transfers maximumpower when the acoustic pressure is maximal and thus at the frequency of an impedance maximum.I examine these statements in greater detail below and also note small modifications to allow for finite generator impedance. The consequences of this behavior are easily seen. Flutes and openended organ pipes overblow to produce the notes of a complete harmonic series 2090, 4090, 6090 .... based on the fundamental2090, while clarinets, which also have nearly cylindrical pipes, produce the odd harmonics COo, 309o, 509o, ... only (to a first approximation at any rate). Wecan also have flutelike systems in which the far end of the pipe is stopped rather than open, giving an input impedance like (1) with tan replaced by cot and a characteristic curve effectively inverted relative to Figure 2. A velocity-controlled air-jet generator leads to possible sounding frequencies Oo, 3o0, 5090, ... for such a system, while a reed generator fails to operate because of back pressure. In the case of instruments like the oboe, bassoon, or saxophone, which are based upon an approximately conical pipe, the impedance maximafor the pipe lie at frequencies 2090, 4090, 6090.... with I equal to the complete length of the cone extended to its apex (Morse 1948, pp. 286-87, Nederveen 1969, p. 2-1). Such instruments produce a complete harmonic series like the flute rather than an odd-harmonic series like the clarinet. Finally, the geometry of the brass instruments, with their mouthpiece cup, partly cylindrical, partly conical tube, and flaring bell, is adjusted mouthpiece

reed"--~

reedblades

(a)

(b)

~staple

(c) Figure 4 Reed configuration in (a) single-reed instruments like the clarinet or saxophone, (b) double-reed instruments like the oboe or bassoon. Note that in each case the blowing pressure tends to close the reed aperture (s = -1). For a lip-valve instrument like the trumpet (e), the blowing pressure tends to open the lip aperture (s = +

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by the designer so that their impedance maxima follow a progression like 0.8 ~o0, 2O~o,3090, 4~Oo.... (Backus1969, pp. 215-23, Benade1973). This progression is musically satisfactory, with the exception of the lowest mode, and the flared bell of the instrument produces a generally more brilliant sound than the straight conical horn used in some nowobsolete instruments such as the cornett, ophicleide, and serpent (Baines 1966). I do not pursue here details of the ways in which the fundamental reference frequency co0 for the air columnis varied in different instruments to produce the notes of the modern chromatic scale (Backus 1969, pp. 223-27, Benade 1960a,b, Nederveen 1969). The important thing for this analysis is that, ~or every fingering configuration of a musical wind instrument, there is an impedancecurve for the air columnthat displays a succession of pronounced maximaand minima. For musically useful fingerings the successive maximawill often have smoothly graded magnitudes and frequencies that are in nearly integer relationships, but this is by no meansuniversally true in the case of the woodwinds,particularly in the upper register. Because the acoustical behavior of the air columnis closely linear, we can consider each possible normal mode(corresponding to an impedance maximum or an admittance maximum as the case may be) quite separately and characterize it by a resonance frequency nl, a resonance width determined by ~ = k’/k at co = n~, and a displacement amplitude xi. A complete description then involves superposition of these driven modes.

SYSTEMANALYSIS A formal analysis of the system shown in Figure 1, including the nonlinearities that control its behavior, was first put forward by Benade & Gans (1968) for the case of musical instruments, though of course much of the basic work dates back to the early days of electronic circuits (Van der Pol 1934), and the general theory is of interest to mathematicians (Bogoliubov & Mitropolsky 1961). Since then the major formal developments of the nonlinear analysis of musical instruments have been in the work of Worman(1971) and Fletcher (1974, 1976a,c, 1978a) and several unpublished papers by Benade. Suppose that the pipe resonances are at angular frequencies ni when the generator is attached but not supplied with air (thus terminating the pipe with a passive impedancethat is generally either muchlarger or smaller than pc/A). Let x~ be the acoustic displacement associated with the ith mode;then, because the resonator is linear, xi satisfies an equation

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~i "-]- l~i~i -~- n~xi = 0,

(2)

where ~i is the width of the resonance. If we refer to Figure 1, the individual pipe-modeamplitudes xj influence the air-driven generator with coupling coefficients ~j, which may be directly related to either acoustic velocity or acoustic pressure, and cause it to produce a driving force F(~jxj), which depends nonlinearly upon all the influences ~z~x~. This force F then drives each individual modei through a second coupling coefficient fli according to ~i + ~i~i-b

n2i Xi = flig(~xjxj).

(3)

In general, the ~i and fli will be complex,in the sense of involving a phase shift. The careful formal development of this approach (Fletcher 1978a) involves a distinction between air-jet and reed-driven instruments in interpretation of the xi, but this need not concern us here. If the instrument is producing sound in a quasi-steady state, then it is reasonable to assume that x~ has the form Xi :

ai sin (nit +~)i),

(4)

where both the amplitude a~ and the phase ~ are slowly varying functions of time. Clearly a nonzero value of dc~i/dt implies an oscillation frequency ~oi, given by ~o~=nl +de~i/dt, (5) which is close to but not exactly equal to the free-mode frequency ni. It is now easy to show (Bogoliubov & Mitropolsky 1961, pp. 39-55, Fletcher 1976a,c) that dai/dt ~ (fll/nl) (F(~xi) cos (nit +c~i) ) - ½~iai

(6)

d~i/dt ~ - (fll/aini) (F(~jx~)sin (nit +~)

(7)

where the brackets ( ) imply that only terms varying slowly compared with n~ are to be retained. Several things are immediately clear from these equations. If the blowing pressure is zero, then F = 0, which implies that ~oi = n~ and that the amplitude a~ decays exponentially to zero. For nonzero blowing pressures, in general ~oi # n~, and because F is a nonlinear function of the x j, the terms (...) in (6) and (7) will contain slowly varying components with frequencies ~oi_+~o~_+~o~_+ .... The only situation in which a steady sound can be obtained occurs when all the blown frequencies ~oi are integer multiples of some commonfrequency ~o0. This is the normal

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playing situation for an instrument and is generally achieved after an initial transient occupying about 40 cycles of the fundamental frequency involved (Richardson 1954, Strong & Clark 1967, Fletcher 1976a). Once achieved, and this depends on the amount of nonlinearity present, the mode-locked.regime is usually stable (Fletcher 1976a, 1978a). Clearly the nonlinearity of F is also largely responsible for most if not all of the harmonic structure of the sound spectrum. Musical instruments can often be played in several different modelocked regimes for a given tube configuration and thus for a given set of pipe resonances--one has only to think of the complex fanfares that can be played on horns and trumpets without valves. In general terms we can see that this flexibility can be achieved if the generator F itself has a resonant or phase-sensitive response that can be adjusted by the player so as to concentrate F in a narrow frequency range. This is one of the aspects of generator behavior that I investigate below. The remainder of this review, in fact, is concerned with the physical nature of different generator systems and with the air flows responsible for their operation. Before leaving the general question of system behavior I should point out that there is a fundamental difference between the structure of the internal frequency spectrum of the instrument, which is what we calculate whenwe find the amplitudes x; of the internal modes, and the structure of the spectrum radiated by the instrument, which is what our ears detect. For a simple cylindrical pipe with an open end of radius r, standard acoustic theory (Olson 1967, p. 85) shows that the radiation resistance at the open end varies as ~o2 for frequencies below the cut-off tn* (which is given by eo*r/c ~ 2), while for frequencies above ~o* the radiation resistance is nearly constant. Thus the radiated spectrum below ~o* has a high-frequency emphasis of 12 dB/octave relative to the internal spectrum, while above co* the two are parallel. The situation is similar for the more complex geometry of brass and woodwindinstruments, except that for brass instruments c0* is determinedby the precise flare geometryof the horn (Morse 1948, pp. 265-88), while in woodwindinstruments with finger holes 09* is determined largely by the transmission properties of the acoustic waveguide with open side holes in the lower part of the instrument bore (Benade 1960b). AIR-JET

INSTRUMENTS

An essentially correct, though qualitatively expressed, theory of the operation of air-jet instruments was put forward as early as 1830 by Sir John Herschel (Rockstro 1890, pp. 34-35), but this was later neglected because of preoccupation with the related phenomenaof edge tones and

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FLETCHER

vortex motion in jets (Curle 1953, Powell 1961), which were made visible in fine photographicstudies like those of Brown(1935). Whileit is certainly true that edge-tone phenomenaare in some ways analogous to the action of an air jet in an organ pipe, the mechanismsinvolved differ importantly in the two cases (Coltman1976). Similarly, while vortices are undoubtedly produced by the jet in an organ pipe, tlkeir presence seems to be an incidental second-order effect rather than a basic feature of the mechanism, and a complete theory including all aspects of the aerodynamic motion will inevitably be extremely complex (Howe 1975). Our best present understanding is as set out below. The basic geometryof an air-jet generator is illustrated for the case of an organ pipe in Figure 3. A planar air jet emerges from a narrow flue slit (typically a few centimeters in length and a millimeter or so in width) and travels across the open mouth of the pipe to impinge more or less directly on the upper lip. Acoustic flow through the pipe mouth and associated with the pipe modesdeflects the jet so that it blows alternately inside and outside the lip, thus generating a fluctuating pressure that serves to drive the mode in question. The blowing pressure in the pipe foot is typically a few hundred pascals (a few centimeters, water gauge), giving a jet velocity of a few tens of meters per second and hence a transit time across the pipe mouth that is comparable with the period of the acoustical disturbance, so that phase effects are certainly important. The discussion below is in terms of the organ pipe geometry, but other instruments of the air-jet type behavesimilarly. Wave Propagation

on a Jet

The work of Rayleigh (1879, 1896, pp. 376-414) provides the foundation for understanding the behavior of a perturbed jet. He treats the case of a plane inviscid laminar jet of thickness 21 movingwith velocity V through a space filled with the same medium, and he shows that a transverse sinuous disturbance of the jet with angular frequency n = (n +jco’ and propagation number k, such that the displacement has the form y-- A exp [j(nt+_kx)],

(8)

satisfies the dispersion relation (n + k V)2 tanh kl + n2

~--

O.

(9)

Solving for n and substituting back in (8) shows that the waveon the jet propagates with velocity u = V/(1 + coth k/)

(10)

and grows exponentially with time or with the distance x traveled by

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MUSICAL WIND INSTRUMENTS

the wave as exp (#x), where

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/~ = k (coth 1/2. kl)

(11)

Equations (10)and (11) show that, at frequencies low enough that wavelength2 of the disturbance on the jet is muchgreater than l, so that kl ~ 1~ the propagation velocity u ~ kIV ~ (IVco) ~/2, while the growth parameter ~ ~ (k/l) ~/2. At the other extreme when), ~ l, we find u ~ V/2 and ~ ~ k ~ 2co/V. Rayleigh realized that these results are somewhatunrealistic since the behavior of/~ for large e) predicts catastrophic instability for the jet in this limit. He correctly identified the origin of this catastrophe in the velocity profile assumedand went on to investigate jet behavior for jets with smoother velocity profiles (Rayleigh 1896, pp. 376-414). He showed that instability (/x > 0) is associated with the existence of a point inflection in the velocity profile, and that/~ is positive in the low-frequency limit, increases with increasing frequency to a maximumwhen kb ~ 1, b being somemeasure of the jet half-width, and then decreases to become negative for kb ~ 2. Further advance did not come until Bickley (1937) investigated the velocity profile of a plane jet in a viscous fluid, showingit to have a form like V0 sech2 (y/b), and until Savic (1941) examinedthe propagation of transverse waves on such a jet in the inviscid approximation. This and more recent work has been summarized by Drazin & Howard (1966). If b is the half-width parameter defined by Bickley and J is the flow integral defined by J=

f-~o [V(Y)]2 dy,

(12)

where y is the dimension transverse to the jet, then the best numerical calculations indicate u ~0.95(bV~o) 1/2 for O 0, then Im (Z~) and from (1), using an effective length l to include the mouthcorrection, the sounding frequency must be lower than the resonance frequency of 80

_0~.1

generating

20

dissipating

Figure 5 Measured complex acoustic impedance Zj of a jet-driven acoustic generator with flutelike geometry, as a function of blowing pressure shownin pascals as a parameter (l-era water gauge = 100 Pa), other parameters being normal for a flute jet. Measurement frequency is 440 Hz and impedance is given in SI acoustic megohms(1 f~ = 1 Pam-a s). (After Coltman 1968)

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the complete pipe. If Z~ is capacitative, then the sounding frequency will be above the resonance frequency. In an elegant series of experiments, Coltman (1968) measured Z~ as function of blowing pressure for a flutelike jet system. One of his measured curves is shownin Figure 5. The spiral form of the curve is caused by the varying phase shift for wavestraveling along the jet, while the magnitude of Z~ is a rather complicated function of the amplification factor # and the interaction expression (20) as functions of jet velocity and thus of blowing pressure. Qualitatively similar curves are to be expected for organ pipe jets. The design and voicing of organ pipe ranks to produce optimum attack and sound quality is an art, the practical results of which conform fairly generally with the expectations derived from the theory (Mercer 1951, Fletcher 1976c). The muchmore complex situation of performance technique on flutelike instruments is also well accounted for (Coltman 1966, Fletcher 1975). In particular, the flute player adjusts the blowing pressure and air-jet length (increasing the first and reducing the second for high notes) in such a way that the phase relations requisite for stable oscillation are satisfied only in the vicinity of the particular resonance peak corresponding to the fundamental of the note he wishes to sound. An experienced player, for example, can easily take a simple cylindrical pipe with a side hole cut in its wall and the near end closed with a cork, thus producing a flutelike tube with an impedance curve like that in Figure 2, and blow steady notes based upon each of the first six or more impedance minima. The loudness of sound produced by the instrument is controlled almost entirely by varying the player’s lip aperture and hence the cross section of the jet, since blowing pressure must be set fairly closely to meet the phase requirements on the jet. It is still possible, however,for the player to vary the sounding frequency of a note by varying the blowing pressure, though frequency control is more usually achieved by altering the lip shape and hence the end correction at the mouthpiece. This and more subtle aspects of performance technique can also be understood on the basis of the theory (Fletcher 1975). REED

AND LIP-DRIVEN

INSTRUMENTS

Common instruments of the woodwindreed family include the clarinet, whichhas a single-reed valve, as shownin Figure 4(a), driving a basically cylindrical pipe, the saxophone, which has a similar reed driving a conical pipe, and the oboe and bassoon, which each have a double reed, as in Figure 4(b), driving a conical pipe. In all these cases the application

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FLETCHER

of blowing pressure tends to close the reed opening, and we say that the reed strikes inwards. In brass instruments like the trumpet or trombone the player’s lips form a type of reed valve as shownin Figure 4(c) but this case the blowing pressure tends to open the lip aperture and we say that the reed strikes outwards. Wehave already discussed the complex geometry of brass instrument horns. In what follows we perforce ignore the large amount of careful and detailed work that has been done on the shape of the air column and the behavior of finger holes (_Backus 1968, 1974, Benade 1959, 1960b, 1976, Nederveen 1969) and concentrate on the way in which sound is produced by the reed generator, using this term to include lip reeds. In its essentials the behavior of a reed system coupled to a pipe was first correctly described by Helmholtz (1877, pp. 390-94), and it was who clearly made the distinction between reeds striking inwards and outwards. He showed that an inward-striking reed must drive the pipe at a frequency that is lower than the resonant frequency of the reed, viewed as a mechanical oscillator, while an outward-striking reed must drive the pipe at a frequency higher than the reed resonance. Worksince that time has concentrated largely on the clarinet reed, with important advances in understanding (Backus 1961, 1963, Nederveen 1969, pp. 28-44, Worman1971, Wilson & Beavers 1974). There has been relatively little work on details of sound generation in brass instruments (Martin 1942, Benade & Gans 1968, Backus & Hundley 1971, Benade 1973). Our discussion is based largely on a recent paper by Fletcher (1978b), which incorporates this earlier work and at the same time makes possible a unified treatment of all types of reeds. Reed Generator

Admittance

Just as with the air-jet generator, it is helpful to define an impedance or in this case more conveniently an admittance Y~- Z;-a for the reed generator as viewed from inside the mouthpiece of the instrument. If p is the mouthpiece acoustic pressure and U the acoustic volume flow through the reed into the pipe at some frequency co, then Y~ = -- U/p.

(24)

Anacoustic pressure p thus leads to acoustic powergeneration p2 Re (by the reed and powerdissipation p2 Re (Yp) in the pipe, where Yp is the input admittance of the mouthpiece and pipe measured at the reed position. If sound generation is to occur then we must have

Re(Y~+ Y~)=< by analogy with (22) for the velocity-controlled

(25) air jet.

Whenstable

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WIND INSTRUMENTS

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oscillation is achieved (25) must become an equality, with nonlinear effects reducing ]Y~I at large amplitudes, while the frequency is determined by

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Im (Y, + Yp) =

(26)

Oscillation is thus always favored near frequencies for which Re (Yp) is a minimum,that is at the impedance maximaof the pipe. Wethus expect a clarinet to produce a series of odd harmonics, a saxophone, oboe, or bassoon to produce a complete harmonic series, and a brass instrument of good design to produce a harmonic series that is complete except for the "pedal" note based on the fundamental. To evaluate the admittance Y~ of the reed generator we must examine the way in which the reed opening and the flow U through it vary with blowing pressure P0 and with internal mouthpiece pressure p. Wecan formulate this in such a way that it applies to reeds striking in either direction. Because the blowing pressure Po is relatively high, the flow U through the reed opening is determined largely by Bernoulli’s law, except that we should recognize that, because of the peculiar geometry of the reed opening, there may be slight deviations from the simplest expected behavior. At very low frequencies we can therefore write U = B’b~(po- p)",

(27)

where b is the breadth of the reed opening (assumed constant) and ~ the height of the opening, which varies according to the net pressure acting on the reed; e and fl are constants which for simple Bernoulli flow would have the values e = 1, fl = 1/2; and B’ is another constant which for simple Bernoulli flow would equal (2/p) ~ where p is the density of air. In fact, from measurements on a clarinet mouthpiece and reed, Backus (1963) found e ~ ~,/3 ~ ~}, which values differ appreciably but not very significantly from those expected for simpler geometry. Wetherefore retain the general form (27). Whenthe mouthpiece pressure p fluctuates at a normal acoustic frequency we must also include the impedance of the mass of air that must movein the gap at the reed tip. If the effective length of this small channel is d, then the acoustic inertance of this massof air is pa/b~, and we can rewrite (27) in the form Po - p = B~- ~/aUI/P + (pd/b0 (dU/dt), (28) -1/~. where we have written B for (B’b) There should really be an additional term in (28) to allow for viscous losses in the reed channel, but neglect this for simplicity since it is generally small. Wemust now recognize that the reed opening ¢ will vary in response

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FLETCHER

to both the blowingpressure Po and the mouthpiecepressure p, with the reed system behavinglike a mechanicalresonator of massm, free area a, resonant frequencyco,, and coefficient of damping~, whichperhaps is providedlargely by the player’s lips. Theappropriate equation for reed motionis then

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+ co,~ (~- ~o)] sa(po -p),

(29)

where~o is the reed opening with the lips in position but no blowing pressure applied. The parameter s has the value -1 for an inwardstriking reed generator and + 1 for an outward-striking lip generator. Clearly, for the inward-striking case, if Po exceedsp~ = (m~/a)~o,the reed will be forced closed by the static blowingpressure and no sound generation can take place. No such blowing pressure limit applies to outward-strikingreeds. Further analysis of the systeminvolvessubstitution of a Fourier series for each of the acoustic variables ~, p, and U in (28) and (29) examinationof the resulting modeequations. Because(28) is quite nonlinear, there is a gooddeal of mixingbetweendifferent modes,andthis is important to the behavior of the instrument. Retaining only the linear terms, however, gives us considerable insight into the small-signal behavior. Evenin the linear case the formal result for the reed admittance~ is

0-!

-04

Figure 6 Calculated real part of the acoustic admittance Yr in SI acoustic micromhos (1 f~ 1 = 1 3 Pai s 1)for typi cal reed -valve gene rators abov e the crit ical blow ing pressure pp. Broken curves refer to woodwind-typereeds (s = - 1) and full curves to lip reeds (s = + 1). The generator resonance frequency is ~o, and its dampingcoefficient ~, given as a parameter. (After Fletcher 1978b)

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complex, and its meaningis not transparent (Fletcher 1978b). It is therefore better ~o look at the results of typical calculations. Figure 6 shows the real part of Y, plotted as a function of frequencyfor both cases s = + 1 and for a blowing pressure somewhatless than the closing pressure p~ for the s = -I case. Clearly from Figure 6 Re (Y,) is negative in the case s = - 1 only for ~o less than the reed resonance frequency ~o,, while Re (Y~)is appreciably negative in the case s = + 1 only for a small frequency range just above co,. The behavior in the two cases is thus very different. in fact it is not very hard to see whythis is so. In the s = - 1 case with co < co,, the reed behaves in a springlike manner and always tends to open whenthe mouthpiecepressure rises. In the s = + 1 case with 09 > ~o, the reed behaves like a mass load and thus moves out of phase with the mouthpiece pressure, once again opening the reed aperture as the mouthpiece pressure increases. A plot of the static behavior (27) for s = clearlyindicates a negative resistance region betweensomecritical pressure p] and the closing pressure p~, and it is in this region that the instrument operates in either case, the phase shift in reed motion referred to above effectively cancelling the sign of s. Another informative plot is given in Figure 7, which showsthe behavior of the complexreed admittance Y~for the two interesting cases s = _2" 1, co < co, and s = + 1, co > co,. The critical pressure for operation p8 is apparent in each case, as well as the closing pressure p~ when s = -1. Wealso see, remembering we are dealing with admittances now rather than impedances, that a woodwindreed with s = - 1 presents a capacitative impedance to the pipe near one of its impedancemaximaso that the sounding frequency must be slightly below the pipe resonance frequency to match the imaginary parts of the admittances as required by (26). 0"1 1

generating

-o’.2

.~ 50

-o~1

~

diSsil:)ating



0"°1i O.Ol

Figure7 Calculatedcomplex acousticadmittanceY,in SI acousticmicromhos for typical reed-valvegeneratorsas a functionof blowingpressurepo, shown in kilopascalsas a parameter(1-era water gauge= 0.1 kPa). (a) A woodwind-type reed with s = -1 co= 0.9~o,; notethat thereedclosesfor p > pb.(b) Alip-valvewiths = + 1 and~o= 1.1 09,. (AfterFletcher1978b)

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FLETCHER

Conversely, for the brass instrument case s = + i, the generator impedance is inductive and the sounding frequency must lie a little above one of the resonance impedance maximaof the horn.

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Nonlinearity

and Performance

Inspection of Figure 6 for a woodwindreed shows that two possible modesof operation are possible. If the reed damping~: is very small, as could be achieved for example with a metal reed, then the pipe will sound at a frequency close to the reed resonance ~or and associated with whatever pipe modelies in this frequency range. This is the situation with the reed pipes of pipe organs, which are tuned by adjusting the resonance frequency of the reed. If, however, the dampingis large so that K approaches unity--a condition that can be achieved by the loading effect of the soft tissue of the player’s lips--then the reed admittance has a nearly constant negative value for all ~o < ~or and the pipe will sound at the frequency that minimizes Yp and is thus at the highest of the pipe impedance maxima.This is essentially the playing situation in woodwind instruments, o~ being as much as 10 times the fundamental frequency of the note being played. Actually the nonlinearity of the reed behavior makes the situation rather more complex, as has been emphasized many times by Benade (1960a; 1976). Because all the pipe modesare coupled through the nonlinearity of the reed generator, the pipe impedancethat is important is not just that at the frequency of the fundamental but rather a weighted average over all the harmonics of that fundamental. If the instrument is well designed then its resonance peaks will be in closely harmonic relation, the weighted impedance will be large, and the instrument will be responsive and stable. If, however, some of the resonances are misplaced, not only will the weighted impedance be lower, giving a less responsive instrument, but also the frequency at which the weighted impedance is greatest will depend on the harmonic content and thus on the dynamiclevel or loudness, giving the instrument an unreliable pitch. Worman(1971) has examinedthe effects of nonlinearity in clarinetlike systems for playing levels small enough that the reed does not close, and he has shownthat, within this regime, the amplitude of the nth harmonic within the instrument tube, or indeed in the radiated sound, is proportional to the nth power of the amplitude of the fundamental. This is, in fact, a very general result that applies to nearly all weakly nonlinear systems and so to all wind instruments in their soft-playing ranges, provided adjustment of lips or other playing parameters are not made. The result no longer holds in the highly nonlinear regime in which the reed closes. Schumacher (1978b) has also applied an integral equation

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143

approach combined with computer symbolic manipulation to this problem and has been able to obtain a steady state solution essentially complete to all orders. These extended results confirm the simpler approximations in general terms while introducing modifications in detail. The onset of this highly nonlinear regime determines the maximum amplitude of the internal acoustic pressure, the peak-to-peak value of which is essentially equal to the difference between the minimum generation pressure p~ and the closing pressure p~. Each of these pressures, and so the difference between them, increases linearly with the unblown reed opening ~o, so that to produce a loud sound the player relaxes his lip pressure and allows the reed to open. Relatively small changes in blowing pressure may also be made, largely to adjust tone quality, and the playing pressure used for clarinets and oboes does not vary muchfrom 3.5 kPa and 4.5 kPa (35 and 45 cm water gauge) respectively over the whole of the dynamicand pitch range. Because a woodwindreed operates well below its resonance frequency it behaves like a simple spring, so that its deflection accurately reflects the acoustic pressure variation within the mouthpiece. A study of the clarinet reed by Backus (1961) shows this deflection to have nearly square-wave form as we should expect. Instruments such as the oboe and bassoon mayhave a pressure wave of less symmetrical form (Fletcher 1978b). Performanceon brass lip-valve instruments is quite a different matter. Figure 6 showsthat the lip valve has a negative conductance over only a small frequency range just above the lip resonance 0)r, so that playing must be based upon this regime and the lip resonance frequency adjusted so as to nearly coincide with the appropriate horn impedance maximum. In fact, skilled French-horn players can unerringly select betweenresonances lying only one semitone apart (6~ in frequency), which implies that the dampingcoefficient x for the lip vibrator must be less than about 0.1. Such a low value is probably not achieved by the lip tissue unaided by other effects, even under muscular tension, and it seems probable that the regenerative effect of the mouth cavity, fed by an airway of finite resistance, acts to decrease the effective value of the damping (Fletcher 1978b). Because the lip valve responds only close to its resonance frequency, its motion is quite closely sinusoidal, just closing during each cycle (Martin 1942). Despite this, the sound of brass instruments is rich in upper harmonics, specially in loud playing (Luce&Clark 1967), and Benade’s general principles on the alignment of resonances still apply. The primary cause of harmonic generation once again arises from the nonlinearity of

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144

FLETCHER

the flow equation (27) and particularly from the fact that, whenthe lip aperture is wide, the instantaneous generator admittance mayfall below that of the horn in magnitude, thus failing to satisfy (25) or, nearly equivalently, driving the pressure difference Po- p below the critical value p~ (Backus & Hundley 1971). Because of this one-sided limiting effect ’ and the fact that a brass instrument horn has a nearly complete harmonic resonance series, the mouthpiece pressure waveformhas a general shape approaching that of a half-wave-rectified sinusoid. However,this effect is probably not the only cause of harmonic generation at high sound levels, which may exceed 165 dB in a trumpet mouthpiece. One"must certainly suspect an additional acoustic nonlinearity in the relatively narrow constriction connecting the mouthpiece cup to the main horn of the instrument (Ingard & Ising 1967). We must also remember the transformation function between internal and radiated sound-pressure spectra, which greatly emphasizes the upper partials of the sound. Finally we should remark that, because increased blowing pressure tends to open rather than close the lip aperture in brass instruments, there is no limit (oiher than physiological) to the blowing pressure that can be used. The sound output is determined by a combination of lip opening and blowing pressure using the same general principles as set out for woodwindinstruments, except that at the highest sound levels we mayexpect additional losses and inefficiencies because of increasing turbulence and other nonlinearities in flow through the lip aperture and mouthpiece cup. CONCLUSION Our review has shown the subtle variety of air flow responsible for sound production in musical wind instruments and has indicated the extent to which the behavior of the generators involved can be said to be understood. Clearly a great deal of work remains to be done. Individual wind instruments typically have a dynamicrange of 30 to 40 dB and acoustic output powers ranging from 10-6 Wfor the softest note on a flute to 10-1 Wfor a fortissimo note on a trumpet or tuba. The maximum efficiency with which pneumatic, power is converted to acoustic power rarely exceeds 1 ~ (Bouhuys 1965, Benade & Gans 1968). Within these limits, however, the experienced player can achieve an extraordinarily sensitive control of pitch, dynamiclevel, and tone color in the steady sound of his instrument and a comparable variety of vibrato and of attack and decay transients. It is both a challenge and a useful task to try to understand howthis is done.

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145

ACKNOWLEDGMENTS

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Mostof this report is based on published material, but it is a pleasure to express my thanks to those co-workers, particularly Arthur Benade, who have sent me copies of unpublished manuscripts. I am also most grateful to Suszanne Thwaites for her help with many aspects of our own studies. This work is part of a program in Musical Acoustics supported by the Australian Research Grants Committee. Literature Cited Backus, J. 1961. Vibrations of the reed and the air columnin the clarinet. J. Acoust. Soc. Am. 33 : 806-9 Backus, J. 1963. Small-vibration theory of the clarinet. J. Acoust. Soc. Am. 35: 30513; erratum (1977) 61 : 1381-83 Backus, J. 1968. Resonance frequencies of the clarinet. J. Acoust. Soc. Am. 43: 1272-81 Backus, J. 1969. The Acoustical Foundations of Music. NewYork: Norton. 312 pp. Backus, J. 1974. Input impedance curves for the r6ed woodwindinstruments, d. Acoust. Soc. Am. 56:1266-79 Backus, J., Hundley, T. C. 1971. Harmonic generation in the trumpet, d. Acoust. Soc. Am. 49:509-19 Baines, A. 1966. European and American Musical Instruments. NewYork: Viking. 174 pp. Benade, A. H. 1959. On woodwind instrumentbores, d. 2,1coust. Soc. Am. 31 : 13746. Reprinted in Kent (1977), pp. 27d--83 Benade, A. H. 1960a. The physics of woodwinds. Sci. Am. 203 (4): 145-54. Reprinted in Kent (1977), pp. 265-73 Benade, A. H. 1960b. On the mathematical theory of woodwind finger holes. J. Acoust. Soc. Am. 32: 1591-1608. Reprinted in Kent (1977), pp. 28z~-304 Benade, A. H. 1973. The physics of brasses. Sci. Am. 229 (1):24-35. Reprinted Kent (1977), pp. 121-32 Benade, A. H. 1976. Fundamentalsof Musical Acoustics. New York: Oxf6rd Univ. Press. 596 pp. Benade, A. H., Gans, D. J. 1968. Sound production in wind instruments. Ann. N.Y. Acad. Sci. 155:247-63. Reprinted in Kent (1977), pp. 154-70 Benade, A. H., Jansson, E. V. 1974. On plane and spherical waves in horns with nonuniform flare. I. Acustica 31:80-98. Reprinted in Kent (1977), pp. 216-34 Bickley, W. G. 1937. The plane jet. Philos. Magi. 28 : 727-31 Bogoliubov, N. N., Mitropolsky, Y. A. 1961.

Asymptotic Methods in the Theory of Nonlinear Oscillations. NewYork: Gordon & Breach. 537 pp. Bouhuys, A. 1965. Sound power production in wind ingtruments. J. Acoust. Soc. Am. 37: 453-56 Brown, G. B. 1935. On vortex motion in gaseous jets and the origin of their sensitivity to sound. Proc. Phys. Soc. London 47 : 703-32 Chanaud, R. C., Powell, A. 1962. Experiments concerning the sound-sensitive jet. J. Acoust. Soc. Am. 34:907-15 Coltman, J. W. 1966. Resonance and sounding frequencies of the flute. J. Acoust. Soc, Am. 40: 99-107 Coltman, J. W. 1968. Sounding mechanism of the flute and organ pipe. J. Acoust. Soc. Am. 44:983-92. Reprinted in Kent (1977), pp. 31-4-23CoItman, J, W. 1976. Jet drive mechanisms in edge tones and organ pipes. J. Acoust. Soc. Am. 60:725-33 Cremer,U, Ising, H. 1967. Die selbsterregten Schwingungenyon Orgelpfeifen. Acustica 19: 143-53 Cu~le, N. 1953. The m~chanics of edge tones. Proc. Roy. Soc. London Ser. A 216 : 412-24 Drazin, P. G., Howard, L. N. 1966. Hydrodynamic stability of pararlelflow of invlscid fluid. Adv. Appl. Mech.9:1-89 Eisner, E. 1967. Complete solutions of the "Webster~" horn equation. J. Acoust. Soc. Am. 41:1126-46. Reprinted in Kent (1977), pp. 133-53 EldEr, S. A. 1973. On the mechanism of sound production in organ pipes. J. Acoust. Soc. Am. 54:1554-64 Fletcher, N. H. 1974. Non-linear interactions in organ flue pipes. J. Acoust. Soc. Am. 56:645-52. Reprinted in Kent (1977), pp. 328-35 Fletcher, N. H. 1975. Acoustical correlates of flute performancetechnique. J. Acoust. Soc. Am. 57:233-37. Reprinted in Kent (1977), pp. 336-40

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146 FLETCHER Fletcher, N. H. 1976a. Transients in the speech of organ flue pipes--a theoretical study. Acustica 34: 224-33 Fletcher, N. H. 1976b. Jet-drive mechanism in organ pipes. J. Acoust. Soc. Am. 60: 481-83 Fletcher, N. H. 1976c. Sound production by organ flue pipes. J. Acoust. Soc. Am. 60: 926-36 Fletcher, N. H. 1978a. Modelocking in nonlinearly excited inharmonic musical oscillators. J. Acoust. Soc. Am. In press Fletcher, N. H. 1978b. Excitation mechanisms in woodwindand brass instruments. Acustica, In press Fletcher, N. H., Thwaites, S. 1978. Wave propagation on an acoustically perturbed jet. Acustica. In press Helmholtz, It. L. F. 1877. Onthe Sensations of Tone as a Physiological Basis for the Theory of Music. Trans. A. J. Ellis and reprinted 1954. NewYork: Dover. 576 pp. Howe, M, S. 1975. Contributions to the theory of aerodynamic sound, with application to excess jet noise and the theory of the flute. J. Fluid Mech. 71: 625-73 Hutchins, C. M., ed. 1975. Musical Acoustics, Part I : Violin Family Components, Benchmark Papers in Acoustics, Vol. 5. Stroudsburg, PA: Dowden, Hutchinson & Ross. 478 pp. Hutchins, C. M., ed. 1976. Musical Acoustics, Part 2: Violin Family Components, Benchmark.Papers in Acoustics, Vol. 6. Stroudsburg, PA: Dowden, Hutchinson & Ross. 379 pp. Ingard, V., lsing, H. 1967. Acoustic nonlinearity of an orifice. J. Acoust. Soc. Am. 42:6-17 3ansson, E. V., Benade, A. H. 1974. On plane and spherical waves in horns with nonuniformflare II. Acustica 31 : 185-202. Repriuted in Kent (1977), pp. 235-51 Kent, E. L., ed. 1977. Musical Acoustics: Piano and Wind Instruments, Benchmark Papers in Aeoustics~ Vol. 9. Stroudsburg, PA : Dowden,Hutchinson & Ross. 367 pp. Luce, D., Clark, M. 1967. Physical correlates of brass-instrument tones. J. Acoust. Soc. Am. 42:1232-43 Martin, D. W. 1942. Lip vibrations in a cornet mouthpiece. J. Acoust. Soc. Am. 13:305-8. Reprinted in Kent (1977), pp. 171-74 Mercer, D. M. A. 1951. The voicing of organ flue pipes. J. Acoust. Soc. Am. 23 : 45-54

Morse, P. M. 1948. Vibration and Sound. NewYork: McGraw-Hill. 468 pp. Morse, P. M., Ingard, K. U. 1968. Theoretical Acoustics. New York: McGraw-HilL 927 pp. Nederveen, C. J. 1969. Acoustical Aspects of Woodwind Instruments. Amsterdam: Frits Knuf. 108 pp. Olson, H. F. 1967. Music, Physics and EntTineerin9. NewYork : Dover. 460 pp. Powell, A. 1961. On the edgetone. J. Acoust. Soc. Am. 33 : 395-409 Pratt, R. L., Elliott, S. J., Bowsher,J. M. 1977. The measurement of the acoustic impedance of brass instruments. Acustica 38 : 236-46 Ra.yleigh, Lord. 1879. Onthe instability of jets. Proc. London Math. Soc. 10:4-13. Reprinted 1964 in Scientific Papers by Lord Rayleigh, 1:361 71. New York: Dover Rayleigh, Lord. 1896. The Theory of Sound. Reprinted 1945. New York: Dover. 2 vols, 480 & 504 pp. Richardson, E. G. 1954. The transient tones of wind instruments. J, Acoust. Soc. Am. 26 : 960-62 Rockstro, R. S. 1890. A Treatise on the Flute. Reprinted 1967. London: Musica Rata. 664 pp. Sato, H. 1960. The stability and transition of a two-dimensional jet. J. Fluid Mech. 7 : 53-80 Savic, P. 1941. On acoustically effective vortex motion in gaseous jets. Philos. Mag. 32 : 245-52 Schumacher, R. T. 1978a. Self-sustained oscillations of organ flue pipes: An integral equation solution. Acustica. In press Schumacher, R. T. 1978b. Self-sustained oscillations of the clarinet: An integral equation approach. Aeustica. In press Strong, W., Clark, M. 1967. Synthesis of wind-instrument tones. J. Acoust. Soc. Am. 41 : 39-52 Vander Pol, B. 1934. The non-linear theory of electric oscillations. Proc. IRE (London) 22 : 1051-86 Wilson, T. A., Beavers, G. S. 1974. Operating modes of the clarinet. J. Acoust. Soc. Am. 56 : 653-58 Worman,W. E. 1971. Self sustained Nonlinear Oscillations of MediumAmplitude in Clarinet-like Systems. PhDthesis. Case Western Reserve Univ., Cleveland, Ohio. Ann Arbor: Univ. Microfilms (ref. 7122869). 154 pp.

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