Reed vibration in lingual organ pipes without the resonators

Reed vibration in lingual organ pipes without the resonators Andra´s Miklo´s,a) Judit Angster,b) and Stephan Pitsch Fraunhofer-Institut fu¨r Bauphysik, Nobelstrasse 12, D-70569 Stuttgart, Germany

Thomas D. Rossingc) Physics Department, Northern Illinois University, DeKalb, Illinois 60115

共Received 12 January 2002; revised 31 October 2002; accepted 5 November 2002兲 Vibrations of plucked and blown reeds of lingual organ pipes without the resonators have been investigated. Three rather surprising phenomena are observed: the frequency of the reed plucked by hand is shifted upwards for large-amplitude plucking, the blown frequency is significantly higher than the plucked one, and peaks halfway between the harmonics of the fundamental frequency appear in the spectrum of the reed velocity. The dependence of the plucked frequency on the length of the reed reveals that the vibrating length at small vibrations is 3 mm shorter than the apparent free length. The frequency shift for large-amplitude plucking is explained by the periodic change of the vibrating length during the oscillation. Reed vibrations of the blown pipe can be described by a physical model based on the assumption of air flow between the reed and the shallot. Aerodynamic effects may generate and sustain the oscillation of the reed without acoustic feedback. The appearance of subharmonics is explained by taking into account the periodic modulation of the stress in the reed material by the sound field. Therefore, a parametric instability appears in the differential equation of vibration, leading to the appearance of subharmonics. © 2003 Acoustical Society of America. 关DOI: 10.1121/1.1534101兴 PACS numbers: 43.75.Np, 43.75.Pq, 43.25.Ts, 43.28.Ra 关NHF兴

I. INTRODUCTION

The pipe organ has been called the ‘‘king of musical instruments.’’ No other instrument can match it in size, range of tone, loudness, or complexity. No two pipe organs in the world are exactly alike. There are two basic types of organ pipes: flue 共labial兲 pipes and reed 共lingual兲 pipes. Flue pipes produce sound by means of a vibrating air jet, in a manner similar to the flute or recorder, while reed pipes use a vibrating metal reed to modulate the air stream. Although sound production in labial pipes has been studied fairly extensively,1–10 relatively little research has been reported on sound production in lingual pipes.11–14 In these pipes, there is a strong and rather complicated interaction between the vibrating reed and the pipe resonator. In most lingual pipes the reed 共tongue兲 vibrates against the fixed shallot. The foot 共boot兲 of a typical reed pipe is shown in Fig. 1. Air under pressure from the windchest flows through the bore 共foot hole兲 into the boot and through the thin opening into the shallot. The wind sets the thin, flexible tongue into vibration; this in turn modulates the flow of air passing through the shallot into the resonator. The reed is pressed against the open face of the shallot by a tuning wire that can be adjusted up and down to tune the vibrating reed. The pipe voicer generally adjusts the reed and the resonator to produce the best sound, and tunes the pipe by adjusting the tuning wire in several subsequent steps. According to the experience of organ building, the pitch and timbre of the pipe can be affected by many factors.15,16 Although reed pipes speak without their resonator, certain a兲

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b兲 c兲

J. Acoust. Soc. Am. 113 (2), February 2003

partials or partial groups can be reinforced and the loudness can be increased by applying a proper resonator. The sound of the reed pipe is strikingly affected by the relationship between tongue length and resonator length. Natural ‘‘full length’’ cylindrical resonators correspond roughly in length to stopped flue pipes of the same pitch, while ‘‘full length’’ conical resonators are somewhat longer; the ‘‘resonance length’’ is about three-quarters of the length of a corresponding open flue pipe. Reed pipes with cylindrical resonators have mainly odd-numbered harmonics, while the sound of a reed pipe with a conical resonator contains all harmonics. The timbre of reed pipes can also be affected by the tongue 共reed兲 and shallot. The thinner the reed the richer is the sound in harmonics; the thicker the reed, the smoother and more fundamental the sound. Wider cylindrical resonators produce stronger tone, while the sound of pipes with narrow resonator tubes is weaker and has a character similar to that of reed woodwind instruments. Helmholtz resonators are used in some families of lingual organ pipes. A broad variation in shape and size of the resonator can be found in contemporary organ building.15 In order to facilitate the attack 共speech兲, the reeds of lingual organ pipes are curved. During the prevoicing procedure the voicer, holding a metal rod, applies a compression force on the reed, which lies on a flat metal surface. He moves the rod slowly towards the end of the reed and gradually increases the force. By repeating this procedure 共called ‘‘polishing’’ by organ builders兲 several times, the reed will be curved up. The resulting curvature is quite even, although it may increase towards the end of the reed. Because of this curvature there is a gap between the reed and the shallot, which cannot be closed completely when the reed is pressed against the shallot by the wind pressure in the foot. There-

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FIG. 2. Frequency versus reed length for a plucked reed with and without a resonator attached. FIG. 1. The foot 共boot兲 of a reed pipe.

fore, the air flow between foot and shallot is never interrupted entirely. Lingual organ pipes have been recently investigated17–19 and several features were observed. In order to understand better how sound is produced in reed organ pipes, more detailed experiments were carried out. The results of the measurements of reed vibrations without resonators and theoretical considerations are presented in this paper. Reed vibrations of pipes with resonators and sound generation in reed pipes will be discussed in a second paper.

long with diameters 9.2 and 65 mm at the ends 共conical angle: 4.7 deg兲. The shallot was also slightly conical 共internal diameters: 6.5 mm at upper end and 8 mm at lower end兲, with a length of 56.5 mm and a wall thickness of 1.5 mm. The opening on the face of the shallot has a triangular shape with 4.8-mm maximal width and 40.4-mm height. The bronze tongue 共reed兲 has a trapezoidal shape with length of 55.8 mm and widths of 7.2 and 5.5 mm at the ends. The thickness of the reed is 0.31⫾0.005 mm. The smaller end was clamped by means of a wedge into the block 共head, nut兲 of the pipe. The total length of the reed from the wedge to the free end was 45 mm. The vibrating length could be adjusted from 37.5 to 15 mm.

II. EXPERIMENTAL METHOD

In the experiments reported in this paper, reed pipes were tested on a slider chest, with the wind supplied by an organ blower and pressure regulator 共bellows兲. Wind pressure was measured on a water manometer. A window was installed on the regular pipe boot, so that the reed could be observed while it vibrated. The reed velocity was recorded by means of a laser vibrometer 共Polytec, OFV 3000兲. A 41-in. microphone 共B&K 4135兲 was inserted into the shallot wall to record the sound pressure inside the shallot, and another microphone 共B&K 4165兲 recorded the sound pressure near the open end of the pipe. The fundamental frequency of the free vibration of the reed was measured by plucking the reed and measuring the velocity with the laser vibrometer. The reed did not strike against the shallot in this experiment; it was vibrating freely. The frequency and damping were determined from the decaying velocity signal. The reed velocity and the sound pressure inside the shallot, as well as the sound pressure near the open end of the pipe, were recorded as the vibrating reed length was varied by means of the tuning wire. The wind pressure was set to the optimal value 共80-mm water兲 for sounding the pipe, as recommended by an experienced organ voicer. Sound pressure and reed velocity waveforms were recorded both with and without the pipe resonator in place. Sound spectra, velocity spectra, and frequency were obtained with the help of a dual-channel FFT analyzer 共HP 35670A兲. A G4 trumpet pipe was used for the study presented in this paper. The conical resonator of the pipe was 687 mm 1082

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III. EXPERIMENTAL RESULTS A. Plucked reed

In the first experiment the reed was plucked by hand and the resulting reed velocity waveforms were measured by the laser vibrometer. In case of plucking with a small amplitude, the velocity waveform, disregarding the first two-three periods, is an exponentially decaying sine signal. Frequencies and Q factors are determined from the exponentially decaying waveforms. Attaching the resonator does not influence the plucked frequency of the reed, but the Q factor decreases about 20% 共for example, from Q⫽16 to Q⫽12.4 at 200.8 Hz兲, probably due to air mass loading. The frequency vs reed length curves for plucked reed with and without a resonator in Fig. 2 are practically the same. If the reed is plucked with large initial amplitude, the velocity waveform is more complicated. The frequency increases as the amplitude of the vibration decreases. This effect was observed in an earlier experiment,17 where the frequency determined from the period of the second cycle was 247 Hz, whereas it has increased to 272 Hz by the 16th cycle of the decaying velocity waveform. B. Blown reed pipe without resonator

In this case the resonator is not attached to the boot of the pipe. The investigated system, as shown in Fig. 1, consists of the boot, the shallot inserted into the block 共head, nut兲, and the reed, attached to the shallot. This system, when blown, produces quite a strong sound, whose frequency can Miklos et al.: Reed vibration in lingual pipes

FIG. 5. rms value of reed velocity versus reed length for a blown reed without a resonator. FIG. 3. Frequency versus reed length for plucked and blown reeds without a resonator.

be tuned continuously by the tuning wire. Reed vibration waveforms and spectra, as well as sound pressure in the shallot, are measured for several different positions of the tuning wire. The effect of the wind pressure on the frequency of the blown reed is also investigated. Plucked and blown frequencies for different reed lengths are shown in Fig. 3. Blown frequencies are somewhat higher than that of the plucked reed for the longer lengths. For a vibrating length of 30.5 mm, for example, a frequency of 202 Hz is measured for the plucked reed, whereas the frequency of the blown reed is 278 Hz, almost 40% higher. Plucked and blown frequencies are equal at 18-mm length, while for shorter reeds the plucked frequency is larger than the blown one. Waveforms of reed velocity and acceleration are shown in Fig. 4 for a reed length of 37.5 mm and a wind pressure of

80-mm water without a resonator. The velocity waveform has a sawtooth-like profile. Positive velocity values correspond to a movement towards the laser vibrometer, i.e., outwards from the shallot. Velocity zeros show the turning points of the vibration. Zero crossing of the steep slope shows the inner turning point close to the shallot, while the other zero crossing corresponds to the outer turning point of the vibration, i.e., to the maximal displacement of the reed. There is no indication of complete closure of the shallot opening, which would appear as a short horizontal part 共zero velocity兲 in the waveform. The acceleration of the reed is determined by calculating the time derivative of the velocity waveform. The acceleration 共see Fig. 4兲, which is proportional to the force acting on the tongue, shows a sharp positive peak, corresponding to a strong restoring force. The smaller but wider negative acceleration 共force兲 corresponds to a slowly accelerating movement of the reed towards the shallot. The velocity waveform approximates a sine wave as the frequency of the vibration increases. The magnitude of the vibration is estimated by calculating the root-mean-square 共rms兲 velocity. The rms values as functions of the reed length can be seen in Fig. 5. The curve has a broad maximum around 25-mm reed length. For shorter reeds the rms value decreases quickly with decreasing reed length. The blown frequency increases with increasing wind pressure. The frequency of the reed with L⫽29.5 mm length is determined at different pressures as measured at the foot of the pipe. The frequency versus wind pressure plot is shown in Fig. 6. The data reported in this paper are from a single G4 trumpet pipe, but similar effects have been observed recently in other reed pipes as well.17–19

IV. THEORETICAL CONSIDERATIONS AND COMPARISON WITH RESULTS A. Plucked reed

FIG. 4. Velocity and acceleration waveforms for a blown reed (l ⫽30.5 mm) without a resonator. J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

Although the investigated reed has a slightly trapezoidal shape, it will be regarded as a cantilever with clamped and free ends in the following discussion. The differential equation of a lossless cantilever is given as20 Miklos et al.: Reed vibration in lingual pipes

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a way that its value at the free end equals unity: ␸ (l)⫽1. This function for the fundamental vibration can be written as

冋 冉 冊 冉 冊册 冋 冉 冊 冉 冊册

␸ 共 z 兲 ⫽0.5 cosh ␣ 1

z z ⫺cos ␣ 1 l l

⫺0.37 sinh ␣ 1

z z ⫺sin ␣ 1 l l

.

The function ␺ (t) can be regarded, then, as the timedependent displacement of the free end of the reed. By substituting Eq. 共4兲 into Eq. 共1兲, a second-order ordinary differential equation can be derived for ␺ (t) FIG. 6. Frequency versus pressure 共water column height兲 for a blown reed (l⫽29.5 mm) without a resonator.

⳵ 2 ␰ 共 z,t 兲 Eblh 3 ⳵ 4 ␰ 共 z,t 兲 ␳ blh ⫹ ⫽F 共 z,t 兲 , ⳵t2 12 ⳵z4

共1兲

where ␳, E, b, l, h, ␰, z, and F are the density, Young’s modulus, width, length, thickness, and displacement of the reed, the spatial variable, and the force acting on the reed, respectively. The resonance frequency of the plucked reed can be determined by solving Eq. 共1兲 with F⫽0. Assuming exp(⫾knz⫹i␻t) or exp(⫾iknz⫹i␻t) dependence, the following equation can be derived for the angular frequency ␻ n : Eh 2 4 ␣ 4n h 2 E 2 ␻ n⫽ k ⫽ 4 , 12␳ n l 12␳

共2兲

where ␣ n is a numerical constant. Its value can be determined from the boundary conditions.20 For the fundamental vibration ␣ 1 ⫽1.8748. The frequency of the plucked reed vs reed length is shown in Fig. 2. Since the fundamental resonance frequency of the reed is inversely proportional to its length squared 关see Eq. 共2兲兴, a function of the form of f⫽

A , 共 l⫺ ␬ 兲 2

共3兲

was fitted to the measured frequencies. The fit with ␬ ⫽3 mm is excellent 共see the solid line in Fig. 2兲, showing that the effective vibrating length of the reed is about 3 mm shorter than the distance between the tuning wire and the free end of the reed. That is, a 3-mm length of the reed near the wire is pressed so hard against the shallot that it does not vibrate. However, this ‘‘end point’’ is not fastened mechanically. Therefore, it may be expected that the vibrating length decreases when the reed moves towards the shallot and increases when the reed moves outwards. This effect may be the reason for the observed shift of the frequency at largeamplitude plucking. This assumption can be investigated as follows. Assume that the displacement can be written as a product of a time-dependent and a z-dependent function

␰ 共 z,t 兲 ⫽ ␺ 共 t 兲 ␸ 共 z 兲 .

共4兲

The function ␸共z兲 must fulfill the boundary conditions of a clamped-free cantilever,20 i.e., ␸ (0)⫽0, ␸ I(0)⫽0, ␸ II(l) ⫽0, and ␸ III(l)⫽0. Moreover, it will be normalized in such 1084

J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

d 2 ␺ 共 t 兲 Eh 2 ␣ 41 ⫹ ␺ 共 t 兲 ⫽0. dt 2 12␳ l 4

共5兲

Assume that the length of the reed depends on the displacement in the following way:

冉

l⫽l 0 ⫹␧ ␺ 共 t 兲 ⫽l 0 1⫹

冊

␧ ␺共 t 兲 , l0

共6兲

where the second term in the bracket is much smaller than unity. Then, 1/l 4 can be approximated to the second order as

冉

冊

4␧ 6␧ 2 1 1 ␺ 共 t 兲 ⫺ 2 ␺ 2共 t 兲 . 4 ⫽ 4 1⫺ l l0 l0 l0

共7兲

By substituting Eq. 共7兲 to Eq. 共5兲, a nonlinear differential equation can be derived

冉

冊

␧ 2 ␧2 d 2␺共 t 兲 2 ␺ 共 t 兲 ⫺6 2 ␺ 3 共 t 兲 ⫽0. 2 ⫹ ␻ 0 ␺ 共 t 兲 ⫺4 dt l0 l0

共8兲

Equation 共8兲 can be solved by the method of successive approximations.21 The first approximation is the solution of the linear equation ␺ 1 (t)⫽a cos(␻0t). The solution of Eq. 共8兲 to second order in the small variable ␧a/l 0 can be written as21

冋

␺ 共 t 兲 ⫽a 2 ⫹ where

冋

␧a 2 ␧a ⫹cos ␻ t⫺ cos 2 ␻ t l0 3 l0

冉 冊

1 ␧a 6 l0

␻ ⫽ ␻ 0 1⫺

册

2

cos 3 ␻ t ,

冉 冊册

107 ␧a 12 l 0

共9兲

2

.

共10兲

The small variable ␧a/l 0 is essentially the relative change of the length ⌬l/l 0 . The nonlinear terms in Eq. 共8兲 cause a decrease of the frequency and the appearance of higher harmonics. The frequency shift is proportional to the square of the amplitude. Equation 共10兲 can explain the observed behavior of the plucked frequency at large-amplitude plucking. Since the amplitude a decreases during the slow decay of the reed vibration, the frequency, in accordance with Eq. 共10兲, shifts upwards. From the observed frequency shift of 25 Hz at 272-Hz fundamental frequency, the value of 0.1 can be calculated for ␧a/l 0 from Eq. 共10兲. This value corresponds to ⬃10% length change during the decay process. Miklos et al.: Reed vibration in lingual pipes

B. Blown pipe without a resonator

At a first glance it is quite surprising that the plucked reed frequency is much lower than the blown frequency. Moreover, the measurements in Fig. 6 show that the vibration frequency of the reed increases with increasing pressure. Both phenomena can be explained by assuming that the pressure in the boot reduces the free vibrating length of the reed. It was shown in the previous section that the free vibrating length of the plucked reed is already smaller than the distance between the tuning wire and the free end of the reed. The equilibrium position of the plucked reed without the boot pressure is determined by the curvature given by the curving process mentioned in the Introduction. The boot pressure pushes the reed towards the shallot to a new equilibrium position. In this position the part of the reed which cannot vibrate will be somewhat bigger than the value of the fit parameter ␬ for a plucked reed. Therefore, the vibrating length will be smaller, and the blown frequency higher than that of the plucked reed. Since the frequency is inversely proportional to the square of the vibrating length, quite small length changes could be expected. The relative change of the vibrating length calculated from the measured frequencies between pressures of 63- and 22-mm water column would be less than 5%. This would be the simplest possible explanation of the observed increase of the blown frequency compared to the plucked one. However, another physical effect, the stiffening of the reed through aerodynamic forces, could also raise the blown frequency. This effect will be discussed later. The reed generator in the lingual organ pipes is composed of the reed and the shallot. The basic operation of the system can be described by the model given by Fletcher and Rossing 共Ref. 2, Chap. 13兲. The pressure in the boot tends to blow the reed closed,11 but, due to the special shape of the reed, the pressure in the boot cannot close the opening between the reed and the shallot completely, and a thin gap always remains. Since the boot pressure is usually quite small, the displacement of the reed is also small between the equilibrium positions without and with boot pressure. The displacement of the free end of a clamped-free cantilever due to a distributed force can be written as20

冉冊

␰ 0 3p 0 l ⫽ h 2E h

4

,

共11兲

where p 0 is the force per unit area of the cantilever. In the case of a blown reed p 0 is the difference between wind pressures in the boot of the pipe and in the shallot. Since p 0 ⬇800 Pa, E⬇1011 Pa, h⬇0.3 mm, and l⬇30 mm in our experiments the shift of the equilibrium position is about ␰ 0 ⬇1.2 h⬇0.36 mm, smaller than the gap between the reed and shallot without boot pressure, i.e., the boot pressure cannot close the reed. The gap between the new equilibrium position and the shallot is very small, less than 1 mm. The vibration of the reed around its fundamental frequency can be modeled by a mass–spring system. The differential equation of motion can be derived from Eq. 共5兲 by adding a loss term to the equation. The resulting differential equation can be written as J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

FIG. 7. A typical calculated flow velocity versus gap-width curve. Gap width y is dimensionless.

m

d 2␺共 t 兲 d␺共 t 兲 ⫹K ␺ 共 t 兲 ⫽F, 2 ⫹2 ␤ m dt dt

共12兲

where ␺, m, ␤, K, and F are the displacement of the free end of the reed around the equilibrium position, reed mass, attenuation constant, spring constant 共stiffness兲, and driving force, respectively. The mass m and spring constant K can be given as m⫽ ␳ bhl

and

K⫽1.03Eb

冉冊 h l

3

,

共13兲

where ␳, b, h, l, and E are the density, width, thickness, and length of the reed and the Young’s modulus, respectively. The driving force F on the right side of Eq. 共12兲 has two components: the aerodynamic force due to the air flow through the slit between the reed and shallot, and the acoustic force produced by the sound fields in the boot and shallot. It will be shown in the following discussion that the aerodynamic force alone is sufficient for producing a self-sustained oscillation of the reed. C. Reed oscillation due to the aerodynamic force

The slit between the reed and the shallot is very narrow. For such a slit the flow could be laminar up to about 30– 40m/s flow velocity. Since the flow velocity through the slit would probably be smaller, it can be assumed that the flow between the reed and shallot is laminar. However, the flow through the foothole and the outflow from the shallot should be described by the Bernoulli equation. For stationary flow the volume velocity at the foothole, slit, and shallot opening is equal. Under this condition the flow velocity w in the slit can be determined 关see the Appendix, Eq. 共A13兲兴. The details of the calculation can be found in the Appendix. The dependence of the flow velocity on the dimensionless gap variable y, defined in Eq. 共A14兲, can be seen in Fig. 7. As mentioned in the Appendix, the velocity curve has a maximum at y⫽ 冑 2. For small y values the velocity scales approximately with y 2 , as expected for a laminar flow. As y increases, the second term under the square root of Eq. 共A13兲 will dominate; therefore, the velocity curve should approach a 1/y dependence. The physical reason for the decrease of the velocity is the decreasing pressure difference between the boot and shallot at larger gaps. Miklos et al.: Reed vibration in lingual pipes

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FIG. 8. Calculated aerodynamic and spring forces acting on the reed. Gap width y is dimensionless.

FIG. 9. The resultants of the aerodynamic and spring forces for weak, medium, and strong reeds. Gap width y is dimensionless.

The velocity shown in Fig. 7 was calculated for a constant gap. In the case of an oscillating reed the gap changes during the travel time of a fluid particle through the slit. This effect modifies the velocity 关see Eq. 共A16兲兴. The modified velocity was used for calculating the f (y) and g(y) components of the aerodynamic force 关see Eqs. 共A19兲–共A21兲兴. Both components are strongly nonlinear. The differential equation of the blown reed can be derived by substituting the displacement ␺ for the change of the gap width x – x P (x P is the equilibrium position of the plucked reed兲 and the driving force F from Eq. 共A19兲

ferent for weak, medium, and stiff reeds. In all cases a strong restoring force appears as the gap approaches zero. This force impedes closing the gap entirely. The force curve crosses the y axis at least in one point, but three zero crossings 共or one zero crossing and one tangential point兲 are also possible for weak reeds. Positions where the resultant force is zero are the new possible equilibrium positions of the reed when wind pressure is applied. In the cases of medium and stiff reeds the zero crossings can be regarded as the new equilibrium positions of the reed. As the force is negative for bigger y values, the applied pressure will drive the reed from the original equilibrium position (y P ) to the new position. Since this position is stable, the reed may oscillate around the new equilibrium. However, the force curve is nonlinear, and quite asymmetric to this position. For example, the reed with medium stiffness in Fig. 9 would provide a small negative force in the entire domain between the old and new equilibrium positions; therefore, the reed would approach the new equilibrium quite slowly. That is, the onset of the oscillation would be delayed after applying the pressure. Such behavior is very common in organ reed pipes. The curve of the weak reed with three zero crossings in Fig. 9 can describe the response of certain reed pipes for slowly increasing pressure. In this case the reed will move from the original equilibrium position to the first zero from the right in Fig. 9. This position is stable; therefore, the reed remains in this position. It may respond with small oscillations for small disturbances. However, these oscillations may drive the reed over the local maximum of the curve, bringing it into the domain of negative force, which will drive the reed to the equilibrium position closest to the shallot. The investigated reed pipe must have this type of force curve, because the observed behavior corresponds to that described above. In the case of slow valve opening the pipe did not sound, but after quite a long time 共⬃1–2 min兲 a slow and quite uncertain onset of the sound could be observed. The pipe responded normally, however, for a sudden onset of the pressure, because the reed was then driven over the first and second zero crossings by the first pressure stroke. Stiff reeds cannot be easily influenced by the aerodynamic forces. It can be observed in Fig. 9 that the shape of the resultant force is quite close to that of a linear force. The

m

d 2x dy dx ⫹K 共 x⫺x P 兲 ⫽ f 共 y 兲 ⫺g 共 y 兲 . ⫹2 ␤ m dt 2 dt dt

共14兲

Equation 共12兲 can be written in a more convenient form by substituting the dimensionless gap variable y in the left side of the equation and by rearranging the loss and force terms

冋

册

2 ␤ ma dy ma d 2 y ⫹g 共 y 兲 2 ⫹ dt dt & &

冋

⫽ ⫺

Ka &

册

共 y⫺y P 兲 ⫹ f 共 y 兲 ,

共15兲

where a is the gap at maximum flow velocity. It can be seen from Eq. 共15兲 that f (y) corresponds to a force, while g(y) corresponds to a loss or gain. The expression on the right side of the equation is the sum of the elastic restoring force and the aerodynamic force. The calculated aerodynamic force f (y) per unit area of the reed vs gap size is shown in Fig. 8. The sign of this force is negative, i.e., it would drive the reed towards the shallot. The spring force of a reed of medium strength is also shown in the figure, assuming an original equilibrium position of the plucked reed at y P ⫽2.65. This force is positive for y ⬍2.65 and proportional to the dimensionless displacement y⫺y P . Depending on the elasticity of the reed material, the reed geometry, and the wind pressure, a broad variation of force curve profiles is possible. The resultants of the aerodynamic and spring forces are shown in Fig. 9 for reeds having different stiffness (K W /K M ⫽0.39, K S /K M ⫽1.56, y P ⫽5.15, y P ⫽2.04 for weak and strong reeds, respectively兲. It can be seen that the shape of the resultant force is very dif1086

J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

Miklos et al.: Reed vibration in lingual pipes

FIG. 10. Aerodynamic force 共dotted line兲 and loss/gain 共solid line兲 versus gap width. Gap width y is dimensionless.

blown frequency will then be close to the plucked one. For a given pipe geometry and wind pressure the gap y P in equilibrium position without pressure in the foot should be bigger for weaker reeds than for stronger ones. That is, weaker tongues have to be curved up more than stiffer ones. Since y P is adjusted by the polishing process, this voicing step is extremely important. Investigate first the expression in the square bracket on the left side of Eq. 共15兲. Since g(y) corresponds to a mechanical resistance, this component provides a negative resistance for small y values. For larger y values the resistance becomes positive; thus, the aerodynamic effect increases the loss of the reed in this domain. Functions g(y) and f (y) are shown in Fig. 10. The expressions in the left and right square brackets, i.e., the total loss/gain and the total force of Eq. 共15兲, are shown in Fig. 11. Assume a symmetric small-amplitude oscillation around the equilibrium position given by the zero crossing of the force curve. In a small domain around this point the resistance is negative; thus, the amplitude of an initial small oscillation will increase. The loss/gain during one single period of oscillation can be represented by the integral of the loss curve over the shaded domain of oscillation 共see Fig. 11兲. Therefore, the amplitude will increase until the value of the integral equals zero. At this amplitude the oscillation becomes self-sustained. It is clear from the figure that selfsustained oscillation can be achieved without closing the gap entirely.

FIG. 11. Total loss/gain curve 共solid line兲 and total force curve 共dotted line兲 versus gap width. Shaded area: domain of self-sustained oscillation. Gap width y is dimensionless. J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

FIG. 12. Subharmonics in reed vibration spectra without and with resonator attached.

The frequency of the self-sustained oscillation can be estimated in the following way: The derivative of the spring force with respect to the displacement is proportional to the negative of the squared angular frequency for a simple harmonic oscillator. Since a small part of the force curve around the equilibrium position 共zero crossing兲 can always be regarded as linear, the slope of the force curve at the zero crossing will be proportional to ⫺ ␻ 2 . The slope, as shown in Fig. 9, is always negative, and it is always steeper than the slope of the pure elastic restoring force 共see Fig. 8兲. That is, the blown frequency of the reed is always higher than the plucked frequency. The relative frequency shift, however, will be the biggest for weak reeds, somewhat smaller for medium reeds, and the smallest for strong reeds. That is, the presented model of reed oscillation can explain the increase of the blown frequency by taking into account the stiffening of the reed due to aerodynamic effects. D. Effects of the acoustic field on reed vibrations

Two observed phenomena can be explained by the effect of the acoustic field on the vibration of the reed. It is obvious that the air mass load on the reed should decrease the blowing frequency compared to the plucked one. Therefore, the observed upshift of the blown frequency was very surprising. However, the decrease of the blown frequency can be seen also in the presented experiments. The difference between the curves of blown and plucked frequency in Fig. 3 will be smaller for shorter reeds 共higher frequencies兲. For very short reeds the blown frequency is smaller than the plucked one. Since the acoustic load 共inertance兲 due to the sound field in the shallot increases linearly with the frequency in the range of the presented measurements, the observed shape of the blown frequency curve can be explained by the combination of the frequency-increasing effects discussed before and of the frequency-lowering effect of the acoustic load. Another interesting property of reed pipes is the generation of subharmonics. Measured spectra of reed vibration of the blown pipe with and without a resonator attached are shown in Fig. 12. Sharp peaks can be observed halfway between the harmonics. Since subharmonic peaks appear without a resonator, too, they are not produced by the interaction between the reed generator and the acoustic resonator. It was also found that the subharmonic is not reproducible; the Miklos et al.: Reed vibration in lingual pipes

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subhar- monic content in the spectrum may be significantly different in subsequent soundings of the pipe. It seems that some kind of instability may be responsible for this effect. A physical explanation of such instability can result from taking into account the effect of the curving 共polishing兲 process. The reed will be curved, because a mechanical stress is produced in the reed material by the curving process. Since the shape of the curved reed is very complicated, it cannot be described unambiguously by a mathematical function. For a qualitative description, however, it can be assumed that the deformation of the reed corresponds to the deformation of a clamped-free cantilever under a uniform stress ␴20

␰共 z 兲⫽

␴ 2 2 z 共 z ⫺4lz⫹6l 2 兲 . 2Eh 3

共16兲

d 2␺共 t 兲 ⫹ ␻ 20 共 1⫹ ␩ cos共 ␻ t 兲兲 ␺ 共 t 兲 ⫽0, dt 2

共24兲

where

␻ 20 ⫽

E ␣ 41 h 2 12␳ l 4

and

␩⫽

3l 4 2 ␣ 21 h 3 ␰ 共 l 兲

P . E

共25兲

Equation 共24兲 is the well-known Mathieu equation of parametric instability.21,22 The amplitude of the pump function can be estimated as ␩ ⬇0.09. Because of the parametric instability an oscillation around half of the pump frequency ␻ may be developed.22 That is, Eq. 共25兲 can explain the appearance of subharmonic peaks observed in the measured sound spectra of blown pipes. The physical reason of this effect is the periodic modulation of the mechanical stress in curved reeds due to the sound pressure in the boot and shallot.

The displacement of the free end is given by

␰共 t 兲⫽

3␴l4 . 2Eh 3

共17兲

In turn, the stress produced by the curving process can be estimated from the displacement of the free end

␴⫽

2Eh 3 ␰共 l 兲. 3l 4

共18兲

The radius of curvature at z⫽0 can be given as l2 , R共 0 兲⫽ 4␰共 l 兲

共19兲

and the mean radius of curvature (R(l)⫽0) R⫽

l2 . 8␰共 l 兲

共20兲

Since the reed is not stress-free, the differential equation of a stressed plate22 has to be applied in the form of

␳

⳵ 2 ␰ 共 z,t 兲 ⳵ 2 ␰ 共 z,t 兲 Eh 2 ⳵ 4 ␰ 共 z,t 兲 ⫹ ⫺T ⫽0, ⳵t2 12 ⳵z4 ⳵z2

共21兲

where T is the stress due to the curving of the reed. This stress will change during the vibration of the reed, because the curvature of the reed changes. Therefore, T can be written as the sum of a constant part and a variable part, i.e., T ⫽T 0 ⫹T 1 . The variable part T 1 of the stress due to a pressure difference p between the upper and lower surface of the reed can be given as22 T 1⫽

l2 Rp ⫽ p. h 8h ␰ 共 l 兲

共22兲

The solution of Eq. 共22兲 can be separated into a timedependent part ␺ (t) and a z-dependent part ␸ (z), as described above. The constant part T 0 contributes only to the displacement distribution function ␸ (z); therefore, the following equation can be derived for ␺ (t): R ␣ 21 d 2 ␺ 共 t 兲 Eh 2 ␣ 41 ⫹ ␺共 t 兲⫺ p ␺ 共 t 兲 ⫽0. dt 2 12␳ l 4 ␳ hl 2

共23兲

Assuming a periodic pressure in the form of p⫽ P cos(␻t), Eq. 共24兲 can be written in the following form: 1088

J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

V. DISCUSSION

Three rather surprising phenomena 共the frequency shift by large-amplitude plucking, higher blown frequency than plucked one, and the appearance of subharmonics in the vibration spectrum兲 can be explained by qualitative physical models of reed vibration. The vibration of the plucked reed corresponds to the expected exponentially decaying sine function, but the effective vibrating length is somewhat shorter than the distance between the tuning wire and the free end of the tongue. The observed frequency difference between small- and large-amplitude plucking can be explained by the change of the vibrating length with the amplitude. Two different effects contribute to the increase of the blown frequency over the plucked one: the shortening of the vibrating length due to the boot pressure and the increase of the stiffness of the reed due to aerodynamic forces. These effects overcompensate the frequency-lowering effect of air mass loading on the reed; thus, the frequency of the blown reed will be higher than that of the plucked reed. However, air mass loading scales with the frequency; therefore, the blown frequency becomes lower than the plucked one for short reeds, as can be seen in Fig. 3. The vibrating reed did not strike the surface of the shallot in our experiments. Due to the curved shape of the reed, it can vibrate freely around the equilibrium position. The gap between reed and shallot is very small at the lower turning point of the oscillation, but no mechanical contact of the metal surfaces occurred. Air flow through the thin slit between shallot and reed has been regarded as laminar in this paper. This is the main difference between the present and earlier treatments2,11,12,14 of the reed generator of lingual organ pipes. While the velocity of the laminar flow has a shape shown in Fig. 7 as a function of the gap, the velocity of a Bernoulli flow would be maximal at the limit y⫽0 and it would decrease monotonically with increasing gap 关 w⫽a/(1⫹by 2 ) dependence兴. The blown frequency then would always be lower than the plucked one. Thus, the observed behavior of the blown frequency contradicts the assumption of Bernoulli flow. Moreover, the Reynolds number calculated from the velocity Miklos et al.: Reed vibration in lingual pipes

shown in Fig. 7 approximates a constant value 共⬃1600兲 as the gap increases, and this limit value is smaller than the laminar limit (Re⬍2320) for the usual lingual pipe geometry. The laminar theory presented can explain several properties of lingual organ pipes known from the tradition of organ building, and supports the opinion of organ builders concerning the ultimate importance of reed curving for the proper speech and sound quality of reed pipes. The theory given in this paper creates only a qualitative model, because several effects cannot be quantified properly 共shape of the curved tongue, flow resistance of shallot opening and of the bore in the foot of the pipe, etc.兲. The main problem is the modeling of the flow in the slit. The length of the slit 共essentially the thickness of the shallot wall兲 is not long enough to allow the full development of a laminar flow. Thus, Eq. 共A1兲 in the Appendix probably overestimates the laminar flow velocity. Therefore, the maximal flow velocity U and the gap of the maximal velocity a calculated by Eq. 共A14兲 may deviate significantly from the real values. The calculated value of a⫽0.23 mm may be only about half of the real value, because the amplitude of vibration determined by integrating the velocity profile shown in Fig. 4 is ⬃0.45 mm. The real shape of the side opening between the shallot and the reed is far from a triangle, as is assumed in the Appendix. Since each tongue has slightly different shape, it is impossible to take into account the real shape of the side opening at the model calculations. However, this shape can have quite a large influence on the average flow velocity and on the aerodynamical forces. The flow resistances of the bore and the open end of the shallot are also not known. For both resistance 共drag兲 coefficients, a value of 1.5 was used in the calculation. For better results, the flow resistances should be determined by measurements. Only one dynamic effect, the change of the gap due to reed vibration, was taken into account. Flow in the boot and the shallot, however, should be described by the timedependent Bernoulli equation, and compressible 共acoustic兲 flow should be taken into account. In this case the volume of the boot and the acoustic properties of the shallot would enter into the theory. According to the opinion of the authors, however, these effects play only a secondary role in the mechanism of reed oscillation. The main features can be described by the simple model presented in this paper. This treatment also shows that the reed oscillation is essentially an aerodynamic phenomenon. Although the vibrating reed produces sound, feedback from the acoustic field is not necessary for sustaining the oscillation. However, the acoustic field may modify the vibration of the reed. Such an effect can be observed, in the case of weak reeds, in the vibration spectrum of the reed.17,18 Much more influence can be expected in reed pipes with resonators. Sound generation by reed pipes with resonator, and the effect of acoustic feedback on the vibration of the reed will be discussed in a second paper. J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

ACKNOWLEDGMENTS

The authors express their thanks to Neville Fletcher for his helpful comments and to the organ builder B. Welde for providing the lingual pipes for the measurements. T.D.R. expresses thanks to Karl Gertis for supporting his research visit to the Fraunhofer-Institut fu¨r Bauphysik, Stuttgart. APPENDIX: AIR FLOW BETWEEN THE REED AND THE SHALLOT

It is assumed that the side opening between the shallot and the reed has a triangular form so that the height of the opening equals zero at z⫽0 and the value x at z⫽l. Thus, x(z)⫽xz/l. The front opening is a rectangle with height x and width b. The length of the gap in the direction of the flow equals to the thickness ␦ of the wall of the shallot. The flow through the slit is regarded as laminar. The velocity through the side opening can be calculated as ⌬ px 2 z 2 , w共 z 兲⫽ 12␳␯ ␦ l 2

共A1兲

where ⌬p, ␳, and ␯ are the pressure difference between boot and shallot, the density, and kinematic viscosity of the air, respectively. The velocity of the laminar flow through the side opening depends on coordinate z, also. Therefore, the velocity has to be averaged over the length l of the side opening. The average flow velocity can be expressed as the ratio of the volume flow and the area of the opening ¯⫽ w

2 xl

冕

l

0

w共 z 兲

xz 2⌬ px 2 dz⫽ l 12␳␯ ␦ l 4

冕

l

z 3 dz⫽

0

1 ⌬px 2 . 2 12␳␯ ␦ 共A2兲

The total flux q through the slit can be given as the sum of the fluxes through the front slit and the side ones q⫽bx

冉 冊

xl 1 ⌬ px 2 l ⌬ px 2 ⌬ px 2 ⫹2 ⫽x b⫹ . 共A3兲 12␳␯ ␦ 2 2 12␳␯ ␦ 2 12␳␯ ␦

The average velocity of the laminar flow through the slit is defined as the ratio of the total flux q and the total open area of (b⫹l)x w⫽

⌬px 2 b⫹ l/2 12␳␯ ␦ b⫹l

⫽

⌬px 2 12␳␯˜␦

,

共A4兲

where an effective gap length was introduced by the definition ˜␦ ⫽ ␦

2l⫹2b . l⫹2b

共A5兲

The stationary flow through the pipe can be determined from the following four equations: 共 1⫹␵ F 兲 21 ␳ w F2 ⫽ p 0 ⫺ p B ,

共A6兲

12␳␯˜␦ w⫽p B ⫺ p S , x2

共A7兲

␵ S 12 ␳ w 2S ⫽ p S ,

共A8兲

and Miklos et al.: Reed vibration in lingual pipes

1089

A F w F ⫽ 共 b⫹l 兲 xw⫽A S w S ,

共A9兲

where w F , w S , ␨ F , ␨ S , A F , A S , p 0 , p B , p S are flow velocities in the foot hole and at the open end of the shallot, the flow resistance 共drag兲 coefficients of the foothole and shallot opening, the cross-sectional areas of the foothole and the shallot, and the pressures in the windchest, pipe boot, and shallot, respectively. Equations 共A6兲 and 共A8兲 describe the free flow through the foot hole and the open end of the shallot. Equation 共A9兲 is the mass conservation equation for incompressible flow. Equation 共A7兲 is the same as Eq. 共A4兲, and it describes the laminar flow through the slit between reed and shallot. The velocities w F and w S can be substituted by w using Eq. 共A9兲. By adding Eqs. 共A6兲–共A8兲 the pressures p B and p S fall out and an equation quadratic in the variable w remains

冉

1⫹␵ F A F2

冊

␵S 24␯˜␦ 2p0 ⫹ ⫽0. 共A10兲 共 l⫹b 兲 2 x 2 w 2 ⫹ 2 w⫺ AS x ␳

where ␯ ⫽dx/dt is the velocity of the reed. The flow velocity w can be calculated then from Eq. 共A11兲. The first-order approximation can be written as w⫽2U

2B C w⫺ 2 ⫽0, x4 x

冊

冊

1⫹

冑

3

C2 , 8B

冊

冑1⫹y ⫺1 C 6 , 2 x ⫺1 ⫽2U B y4

x y⫽& , and a

共A16兲

共A17兲

共A18兲

dy dy , ⫽ f 共 y 兲 ⫺g 共 y 兲 dt dt

⫺2 ␳ U ␥

冋

g共 y 兲⫽ ⫺

6

共A13兲

where U and the dimensionless variable y were introduced by the definitions U⫽

␦ dy . 2y dt

共 2l⫹b 兲 ␦ . lb

2

The solution of Eq. 共A11兲 can be written as

冉冑

冊

dy 1 ⫽⫺ 共 p B ⫺ p S 兲 bl 共 l⫺ ␥ 兲 ⫺ ␳ w 2 bl ␥ , dt 2

f 共 y 兲 ⫽⫺2 p 0 共 1⫺ ␥ 兲

2 p 0/ ␳ . 1⫹␵ F ␵ S 2 ⫹ 2 共 l⫹b 兲 A F2 AS

B x4

⫺1

The force can be calculated by substituting Eq. 共A16兲 into Eq. 共A17兲. After some algebra, the force can be written in the following form:

共A12兲

w⫽

冑1⫹y

6

共A19兲

where

and

冉

3

where ␥ denotes the ratio of the area over the gap to the total area of the vibrating reed

F y,

12␯˜␦ B⫽ 1⫹␵ F ␵ S ⫹ 2 共 l⫹b 兲 2 A F2 AS

C⫽

冉

冉 冊

where B and C are the following constants:

冉

⫹

冉 冊

F y,

␥⫽ 共A11兲

y4

The force acting on the reed can be calculated by assuming that a pressure difference of p B ⫺ p S pushes the middle part of the reed towards the shallot, while a Bernoulli force tries to close the gap. Both components must have a negative sign, because they are directed to the shallot. The force can be given as the sum of the two components

This equation can be written in a simpler form w 2⫹

冑1⫹y 6 ⫺1

冑

6

a⫽

8B 2 . C

⫻

冉

冉

冑1⫹y 6 ⫺1 y6

冑1⫹y 6 ⫺1 y4

冊

2

共A20兲

,

冉

3⫺ 冑1⫹y 6 3 p 0 共 1⫺ ␥ 兲 ␦ ⫹␳U␦␥ 2U y2

冑1⫹y 6 ⫺1 y 3 冑1⫹y 6

冊

.

冊册 共A21兲

Calculated curves of f (y) and g(y) vs y are shown in Fig. 10.

共A14兲 1

The quantities U and a depend only on the geometry of the pipe foot and the wind pressure p 0 . The flow velocity w has a maximum at x⫽a where w⫽U. Therefore, parameters U and a can be regarded as the maximum flow velocity through the slit between reed and shallot and the value of the gap width at maximum velocity. For the pipe studied U ⫽37.5 m/s and a⫽0.23 mm can be calculated from the geometrical data and assuming 800-Pa wind pressure. The calculated velocity w vs y curve can be seen in Fig. 7. In case of vibration, the gap changes during the time needed for a fluid particle to cross the slit. Therefore, x should be replaced by the following expression: x⫹ 1090

␯␦ , 2w

J. Acoust. Soc. Am., Vol. 113, No. 2, February 2003

共A15兲

N. H. Fletcher, ‘‘Sound production by organ flue pipes,’’ J. Acoust. Soc. Am. 60, 926 –936 共1976兲. 2 N. H. Fletcher and T. D. Rossing, The Physics of Musical Instruments, 2nd ed. 共Springer, New York, 1998兲, Chap. 16. 3 M. P. Verge, B. Fabre, W. E. Mahu, and A. Hirschberg, ‘‘Feedback excitation mechanism in organ pipes,’’ J. Acoust. Soc. Am. 95, 1119–1132 共1994兲. 4 M. Castellengo, ‘‘The role of mouth tones in the constitution of attack transients of mouth pipes,’’ Acust. Acta Acust. 85, 387– 400 共1999兲. 5 A. Miklo´s and J. Angster, ‘‘Properties of the sound of labial organ pipes,’’ Acust. Acta Acust. 86, 611– 622 共2000兲. 6 J. W. Coltman, ‘‘Jet drive mechanisms in edge tones and organ pipes,’’ J. Acoust. Soc. Am. 60, 725–733 共1976兲. 7 S. A. Elder, ‘‘On the mechanism of sound production in organ pipes,’’ J. Acoust. Soc. Am. 54, 1554 –1564 共1973兲. 8 T. L. Finch and A. W. Nolle, ‘‘Pressure wave reflections in an organ note channel,’’ J. Acoust. Soc. Am. 79, 1584 –1590 共1986兲. 9 S. Yoshikawa and J. Saneyoshi, ‘‘Feedback excitation mechanism in organ pipes,’’ J. Acoust. Soc. Jpn. 共E兲 1, 175–191 共1980兲. Miklos et al.: Reed vibration in lingual pipes

W. Lottermoser and J. Meyer, Orgelakustik in Einzeldarstellungen 共Verlag Das Musikinstrument, Frankfurt am Main, 1966兲. 11 N. H. Fletcher, ‘‘Autonomous vibration of simple pressure-controlled valves in gas flows,’’ J. Acoust. Soc. Am. 93, 2172–2180 共1993兲. 12 A. Hirschberg, R. W. A. van de Laar, J. P. Marrou-Maurie´res, A. P. J. Wijnands, H. J. Dane, S. G. Kruijswijk, A. J. M. Houtsma, ‘‘A quasistationary model of air flow in the reed channel of single-reed woodwind instruments,’’ Acust. Acta Acust. 70, 146 –153 共1990兲. 13 G. R. Plitnik, ‘‘Vibration characteristics of pipe organ reed tongues and the effect of the shallot, resonator, and reed curvature,’’ J. Acoust. Soc. Am. 107, 3460–3473 共2000兲. 14 A. Z. Tarnopolsky, N. H. Fletcher, and J. C. S. Lai, ‘‘Oscillating reed valves—An experimental study,’’ J. Acoust. Soc. Am. 108, 400– 406 共2000兲. 15 P. Williams and B. Owen, The Organ 共Norton, New York, 1988兲. 16 R. Janke 共organ builder兲, http://www.orgel-info.de 10

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T. D. Rossing, J. Angster, and A. Miklo´s, ‘‘Reed vibration and sound generation in lingual organ pipes,’’ J. Acoust. Soc. Am. 104, 1767–1768 共1998兲. 18 T. D. Rossing, J. Angster, and A. Miklo´s, ‘‘Reed vibration and sound generation in lingual organ pipes,’’ in Proc. of Int. Symp. on Mus. Acoust., Perugia, Italy, Vol. 1, 313–316 共2001兲. 19 J. Braasch, J. Angster, and A. Miklo´s, ‘‘The influence of the shallot leather facing on the sound of lingual organ pipes,’’ in Proc. of Int. Symp. on Mus. Acoust., Perugia, Italy, Vol. 1, 325–328 共2001兲. 20 L. D. Landau and E. M. Lifshitz, Theoretical Physics, VII. Elasticity 共Pergamon, Oxford, 1960兲, Chap. 3. 21 L. D. Landau and E. M. Lifshitz, Theoretical Physics, I. Mechanics 共Pergamon, Oxford, 1960兲, Chap. 5. 22 M. A. Mironov, ‘‘Parametric instability of a circular cylindrical shell propagating a Korteweg wave,’’ Acoust. Phys. 41, 707–711 共1995兲.

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