NON-LINEAR REGENERATION MECHANISMS IN WIND INSTRUMENTS

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NON-LINEAR REGENERATION MECHANISMS IN WIND INSTRUMENTS

S. Elliott

To cite this version:

S. Elliott. NON-LINEAR REGENERATION MECHANISMS IN WIND INSTRUMENTS. Journal

de Physique Colloques, 1979, 40 (C8), pp.C8-341-C8-345. �10.1051/jphyscol:1979861�. �jpa-00219567�

NON-LINEAR REGENERATION MECHANISMS IN WIND INSTRUMENTS

S.J. E l l i o t t

Department o f Physics, University of Surrey, GuiZdford, Surrey, GUZ 5 X H .

Resume.- On s a i t depuis p l u s i e u r s annees que l a colonne d ' a i r d'un instrument I vent se comporte de facon l i n e a i r e I des niveaux de s o u f f l e nomaux ; on s a i t egalement que l e mecanisme n o n - l i n e a i r e q u i p r o d u i t l e s harmoniques de l a p r e s s i o n sonore a i n s i degagee se s i t u e dans l'anche, l a q u e l l e , r e g i e par l a pression, m a i n t i e n t l ' o s c i l l a t i o n .

Des progres importants o n t & t e accomplis recemment dans l a f o r m u l a t i o n mathematique de ce mecanisme regenerateur, mais l a r e s o l u t i o n exacte de ces f o r m u l a t i o n s e x i g e une connaissance q u a n t i t a t i v e de l a forme que revCt l a n o n - l i n e a r i t s . C e l l e - c i e s t encore ma1 comprise e t j u s q u l I present l e s travaux q u i o n t e t & f a i t s dans ce domaine se sont bornes l e p l u s souvent I l ' e t u d e de l a n o n - l i n e a r i t 6 du f l u x q u a s i - s t a t i q u e au t r a v e r s de l'anche.

Cependant il d e v i e n t de p l u s en p l u s e v i d e n t que l a dynamique de l ' a n c h e c o n t r i b u e e l l e aussi pour une grande p a r t au comportement n o n - l i n e a i r e ; aussi une d e s c r i p t i o n complete d o i t - e l l e pouvoir t e n i r compte

a

l a f o i s de l a dynamique de l ' a n c h e e t du f l u x , v a r i a b l e dans l e temps, q u i l a traverse.

L ' o b s e r v a t i o n d i r e c t e t a n t de l a pression que du f l u x de 1 ' a i r q u i passe par l'embouchure d'un i n s t r u ment 2 vent en c u i v r e l o r s de son emploi a permis de mesurer directement l a n o n - l i n e a r i t 6

a

diverses frequences e t 6 diverses puissances de s o u f f l e . L ' a u t e u r presente dans c e t a r t i c l e quelques r e s u l t a t s obtenus de ces etudes p r e l i m i n a i r e s ; ces r e s u l t a t s sont egalement compares avec l e s travaux a n t e r i e u r s s u r l ' a c o u s t i q u e musicafe e t l a synthese du discours.

Abstract.- I t has been known f o r some years t h a t t h e a i r column o f a wind instrument behaves l i n e a r l y a t normal p l a y i n g l e v e l s , the n o n - l i n e a r mechanism which g i v e s r i s e t o harmonics o f t h e r a d i a t e d sound pressure i s contained i n t h e pressure c o n t r o l l e d reed which maintains t h e o s c i l l a t i o n .

Considerable advances have been made r e c e n t l y i n the mathematical f o r m u l a t i o n o f t h e regenerative mechanism, b u t these depend f o r t h e i r accurate s o l u t i o n on a q u a n t i t a t i v e knowledge o f the form of the n o n - l i n e a r i t y . This i s n o t w e l l understood and previous work i n t h i s area has been l a r g e l y confined t o the n o n - l i n e a r i t y o f t h e q u a s i - s t a t i c f l o w through the reed.

I t has become i n c r e a s i n g l y c l e a r however t h a t t h e dynamics o f t h e reed a l s o c o n t r i b u t e s u b s t a n t i a l l y t o t h e n o n - l i n e a r behaviour, and any complete d e s c r i p t i o n must be a b l e t o take account o f both t h e dynamics o f t h e reed and t h e t i m e v a r y i n g f l o w through i t .

By d i r e c t observation o f b o t h t h e pressure and f l o w i n t h e mouthpiece o f a brass wind instrument d u r i n g p l a y i n g , i t has been p o s s i b l e d i r e c t l y t o measure t h e n o n - l i n e a r i t y a t various frequencies and p l a y i n g l e v e l s . Some r e s u l t s o f t h i s p r e l i m i n a r y i n v e s t i g a t i o n are presented and compared t o previous work i n musical acoustics and speech synthesis.

I n t r o d u c t i o n . - Musical acoustics i s an o l d and A t t h e moment t h i s problem i s more than acade- d i v e r s e f i e l d o f study, and h i s t o r i c a l l y t h e m o t i - mic since, w i t h t h e i n t r o d u c t i o n o f mass p r o d u c t i o l l v a t i o n behind i t has been mostly academic. The pro- techniques, t h e crafsmen's s k i 1 1 becomes a1 i e n a t e d blems a r e so complicated t h a t q u a n t i t a t i v e s c i e n t i - and a t e c h n o l o g i c a l e x p e r t i s e must be sought.

f i c t h e o r i e s have been of l i n i t e d use t o n u s i c a l instrument makers. I n many cases t h e b a s i c mecha- nism o f o s c i l l a t i o n i n an instrument may be w e l l understood, b u t musical l y important phenomena i n t h e o s c i l l a t i o n may be second o r t h i r d o r d e r e f f e c t s from a s c i e n t i f i c p o i n t o f view. So t h e design and manufacture o f instruments has been a m a t t e r f o r craftsmen w i t h a wealth o f e m p i r i c a l and a r t i s t i c knowledge, b u i l t up over generations.

Nevetheless t h e challenge has been such t h a t many eminent s c i e n t i s t s have spent time conside- r i n g t h e fundamenta4 problems. O s c i l l a t i o n s i n wind instruments f o r example were s t u d i e d by Helmholtz / I / , A.G. Webster /2/ who f i r s t drew t h e analogy between e l e c t r i c a l and a c o u s t i c impedance, and Bouasse /3/ whose c o n t r i b u t i o n i s o n l y now being r e a l ised.

Previous Theories.- Hemhol t z ' s theory o f a 1 in e a r reed ( a reed i n t h i s sense meaning any form o f dynamic valve, e.g. a cane reed, t h e l i p s o r even t h e g l o t t i s ) , i n t e r a c t i n g w i t h a s i n g l e resonance i n a p i p e was published i n 1877 / I / . He d e r i v e d t h e amplitude and phase o f t h e response o f t h e p i p e necessary t o cause o s c i l l a t i o n , assuming t h a t t h e reed was a simple, damped mass-spring system d r i v e n by t h e pressure i n t h e tube, and t h a t t h e flow through t h e reed was p r o p o r t i o n a l t o i t s opening.

Taken i n broader c o n t e x t t h i s i s e s s e n t i a l l y t h e c o n d i t i o n o f s t a b i l i t y i n a feedback l o o p now known as t h e Barkhausen o r N y q u i s t c r i t e r i o n . So t h e i n t e r p r e t a t i o n was s i q n i f i c a n t , b u t he d i d n o t t a k e i n t o account two important e f f e c t s i n reed instruments:

Article published online by EDP Sciences and available at

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:1979861

c8-342 JOURNAL DE PHYSIQUE

F i r s t l y , the f l o w through t h e reed i s a non-tinear expansions (two i n t h e double T a y l o r s e r i e s and one f u n c t i o n o f b o t h t h e opening and t h e pressure d i f - f o r the F o u r i e r c o e f f i c i e n t s ) make t h e f i n a l equa-

ference across it. t i o n very unwieldy.

Secondly, t h e f a c t t h a t as w e l l as being d r i v e n For t h e case o f a reed w i t h opening area pro- by t h e pressure i n t h e pipe, the reed a l s o has a p o r t i o n a l t o reed displacement, w i t h a t u r b u l e n t B e r n o u l l i f o r c e a c t i n g on i t because o f t h e f l o w . flow, t h e pressure d i f f e r e n c e i s r e l a t e d i n a

The more complete i n t e r a c t i o n i s i n d i c a t e d i n r e l a t i v e l y simple way t o t h e v e l o c i t y /6/. If t h i s F i g . 1 where i t can be seen t h a t Helmholtz's i n t e r - pressure i s compared t o t h e pressure response from a c t i o n i s a s i m p l i f i e d v e r s i o n o f t h i s , c o n s i s t i n g t h e instrument as a F o u r i e r s e r i e s then more t r a c -

s i n g l e loop. t a b l e s o l u t i o n s should be obtained, as here t h e r e

pressure in mouth- piece

FIGURE 1 : Diagram i l l u s t r a t i n g regeneration processes.

The theory o f a n o n - l i n e a r source e x c i t i n g i s e s s e n t i a l l y o n l y one expansion.

m u l t i p l e resonances i s f a r more complicated than Any such p e r i o d i c s o l u t i o n must assume an t h e l i n e a r case. The flow of energy i n t o harmonics initial oscillation frequency which prevents solu- of t h e m o u t h ~ i e c e Pressure means t h a t t h e resPO"se

tion of t h e m u s i c a l l y i m p o r t a n t t r a n s i e n t s . schu- o f an instrument a t harmonic frequencies must be

macher /7/ has suggested a f o r m u l a t i o n u s i n g non- considered.

l i n e a r i n t e g r a l equations and t h i s may be a b l e t o By assuming t h a t t h e o s c i l l a t i o n i s steady i . e .

give transient solutions and has the possibility p e r i o d i c , a F o u r i e r expansion of t h e pressure and

of takinq

-

the Bernoulli force into account.

v e l o c i t y i n time may be s u b s t i t u t e d i n t o t h e cou-

The aim o f t h e present work i s t o i n v e s t i g a t e p l e d equations f o r t h e reed and instrument. T h i s e x p e r i m e n t a l l y some o f t h e assumptions made i n was achieved i n a general way by Benade and Gans these t h e o r i e s f o r the p a r t i c u l a r case o f a brass /4/ who were a b l e t o e x p l a i n q u a l i t a t i v e l y many instrument ( a trombone).

o f t h e e f f e c t s found i n reed instuments. The work was extended by Worman /5/, who used a T a y l o r

Previous Experimental Work.- The apparatus used i n expansion f o r t h e f l o w as a f u n c t i o n o f reed ope-

t h i s i n v e s t i g a t i o n was o r i g i n a l l y used t o measure n i n g and pressure d i f f e r e n c e , which was equated

t h e i n ~ u t im~edance o f brass instruments w h i l e t o t h e flow as a f u n c t i o n o f t h e response of t h e

being d r i v e n from a loudspeaker /8/.

instrument. By u s i n g a simple c l a r i e n t - l i k e

The a c o u s t i c Dressure and v e l o c i t v a r e measu- i n s t r u m e n t W0rma.n was a b l e t o o b t a i n numerical r e d w i t h a horn-coupled probe microphone and a s o l u t i o n s f o r t h e o s c i l l a t i o n , however t h e t h r e e constant temperature, h o t w i r e anemometry system.

The v o l t a g e outputs o f these devices a r e d i g i t i s e d and processed by a mini-computer which ' l i n e a r i s e s ' t h e anemometer output, e x t r a c t s t h e fundamental components o f pressure and v e l o c i t y and then compa- r e s t h e magnitudes and phases t o o b t a i n t h e complex impedance ( d e f i n e d as Acoustic Pressure/vol ume

-4 -1 v e l o c i t y and measured i n S . I . 'ohms', Kg m s ) .

An example o f t h e i n p u t impedance o f a trombo- ne i s given i n Fig. 2. The computer may a l s o be used t o compute t h e e f f e c t o f a mouthpiece on t h i s impedance, modelling t h e mouthpiece as lumped e l e - ments. T h i s i s t h e impedance i n t h e plane o f the l i p s and i s i m p o r t a n t because i t i s t h e 'response' o f the instrument w h i l e being played. The same impedance as i n F i g . 2 'transformed back' t o i n c l u - de t h e e f f e c t o f a mouthpiece i s shown i n F i g . 3.

Ti/2

+

p/

RAO

-

FIGURE 2 : Measured, i n p u t impedance as the t h r o a t o f a trombone.

Ti /?

+

pl

RRD

-

ii /?

FIGURE 3 : C a l c u l a t e d i n p u t impedance i n t h e plane o f the l i p s f o r a trombone.

.Measurement o f t h e Pressures.- The f i r s t s e t of experiments conducted were t o determine t h e magni- tudes o f t h e s t a t i c and a c o u s t i c pressures d u r i n g a blown note i n t h e mouthpiece o f t h e i n s t r u m e n t and i n t h e mouth. Two probe microphones were used f o r t h i s , and t h e average pressure d i f f e r e n c e between t h e mouth and cup o f t h e i n s t r u m e n t measu- r e d w i t h a water manometer. The r e s u l t s f o r a s o f t n o t e a t 174 Hz (note F3 played p ) a r e presen- t e d i n Fig. 4 where t h e mouth pressure w i t h t h e steady pressure d i f f e r e n c e i s superimposed on t h e pressure i n t h e t h r o a t .

FIGURE 4 : Measured pressure i n t h e mouth (above) and pressure i n t h e t h r o a t (below) d u r i n g t h e note F3 (174 Hz) played s o f t l y . (The zero o f pressure i s appro- x i m a t e l y atmospheric).

The a l t e r n a t i n g pressure i n t h e mouth i s o f s m a l l e r magnitude than t h a t i n t h e mouthpiece.

Because t h e f l o w i n t o t h e i n s t r u m e n t i s t h e same as t h a t o u t o f t h e mouth, t h e impedance o f t h e mouth c a v i t y may be determined from t h e r a t i o of t h e two pressures and t h e impedance o f t h e i n s t r u - ment. The impedance l o o k i n g i n t o t h e mouth may be as much as 10% o f t h e i n s t r u m e n t impedance. A l t h - ough i t i s s t i l l a good f i r s t approximation t o consider t h e mouth pressure as constant (as has been done i n a l l previous t h e o r i e s ) , m u s i c a l l y i m p o r t a n t e f f e c t s may w e l l be achieved by changing t h e shape o f t h e mouth c a v i t i e s .

The general form o f t h e pressure i n t h e i n s t r u m e n t has been e x p l a i n e d by comparing t h e r e l a t i v e r e i s t a n c e s o f t h e reed opening and t h e i n s t r u m e n t / 9 / .

c8-344 JOURNAL DE PHYSIQUE

The reed opening i s assumed approximately s i n u s o i d a l (as measured f o r a p a r t i c u l a r case by M a r t i n

/lo/).

Over t h e p a r t o f t h e c y c l e where t h e reed opening i s s u f f i c i e n t f o r i t s r e s i s t a n c e t o be small compared t o t h a t o f the instrument, t h e pressure i n t h e mouthpiece i s about t h e same as t h a t 4n t h e mouth. T h i s c o n d i t i o n may a p p l y o v e r a l a r g e p a r t o f t h e cycle, e s p e c i a l l y 'when t h e n o t e i s o f low p i t c h , so t h e amplitude of reed v i b r a t i o n i s l a r g e

/lo/.

When t h e reed i s n e a r l y closed i t s r e s i s t a n c e r i s e s considerably and c u t s off t h e f l o w t o t h e instrument where t h e pressure f a 1 1s t o atmospheric..

The i n t e r e s t i n g t h i n g about t h e observed i n s t r u m e n t pressure i s t h a t f o r some p a r t s of t h e c y c l e i t r i s e s above t h a t i n t h e mouth and f o r o t h e r s i t f a l l s considerably below atmospheric.

This i n d i c a t e s an energy storage mechanism, presu- mably i n t h e reactances o f t h e impedances i n v o l v e d .

Measurement o f Pressure and V e l o c i t y . - A second s e t o f measurements i n v e s t i g a t e d t h i s by measuring t h e pressure and v e l o c i t y i n t h e t h r o a t , t o g e t h e r w i t h t h e average pressure d i f f e r e n c e as before.

Figures 5, 6 and 7 show t h e r e s u l t s a t s o f t p l a y i n g l e v e l s and frequencies o f 174 Hz (F3), 233 HzX(B3b) and 350 Hz (F4) r e s p e c t i v e l y . Again t h e mouthpiece pressure i s g r e a t e r than t h a t * i n t h e mouth f o r c e r t a i n p a r t s o f t h e c y c l e . The h o t w i r e anemometer can o n l y measure t h e magnitude o f t h e v e l o c i t y , n o t i t s d i r e c t i o n , a l s o i t i s r a t h e r i n a c c u r a t e a t low v e l o c i t i e s under t h e c o n d i t i o n s used here. Despite t h i s , evidence o f r e c t i f i c a t i o n i n t h e v e l o c i t y waveform i s observed i n a l l t h e notes studied, i n d i c a t i n g t h a t t h e v e l o c i t y passes through zero and a c t u a l l y reverses a t some p o i n t s i n t h e cycle, r o u g h l y corresponding t o when t h e pressure i n t h e i n s t r u m e n t i s g r e a t e r than t h a t i n t h e mouth.

For comparison t h e same n o t e as i n Fig. 5, played l o u d l y ( f f ) i s shown i n f i g u r e 8, t h e gene- r a l form o f ' the waveforms i s s i m i l a r , b u t t h e magnitudes are a p p r e c i a b l y hjgher. The o t h e r impor- t a n t 'piece o f i n f o r m a t i o n from t h e v e l o c i t y wave- forms i s t h a t t h e f l o w i s t u r b u l e n t , which i s t o be expected from s t u d i e s o f the a c o u s t i c n o n - l i n e a r i t y of, o r i f i c e s /6/. The i m p l i c a t i o n i s t h a t t h e turbu- l e n t equations should b e used t o d e s c r i b e t h e f l o w c o n d i t i o n s o f t h e reed opening i n F i g . 1.

I

TIME l m S l

FIGURE 5 : Measured pressure i n t h e t h r o a t , w i t h s t a t i c pressure d i f f e r e n c e superimposed, and v e l o c i t y i n t h e t h r o a t d u r i n g t h e note F3 played s o f t l y .

PRESSURE . 2 - A n n n A

I K P o 1

I

^{TIME 1 m S l}

FIGURE 6 : As F i g . 5 f o r n o t e B3b (233 Hz) played s o f t l y .

PRESSURE

TIME l m S l

FIGURE 7 : As F i g . 5 f o r note F4 (350 Hz) played s o f t l y .

Conclusions.- The study has been u s e f u l i n esta- b l i s h i n g some new methods o f observing t h e i n t e r - a c t i o n between t h e p l a y e r and an instrument. The p r e l i m i n a r y r e s u l t s have shown t h a t ( w i t h r e f e r e n - ce t o Fig. 1) :

VELOCITY i ~ s " 1 ²⁰ -

0 PRESSURE *to-

lK Po l

S I0 IS 2 0

T I M E 1mSI

/9/ Backu- J . and Hundley

T.C.,

J. Acoust. Soc.

FIGURE 8

:

As Fig. 5 f o r note F3 (174 Hz) played 1 oudl y .

n n A A Am.,

1971, 49, p. 509.

0 -

/lo/ Martin D.W., J . Acoust. SOC.

Am.,

1942, - 13,

-10-

p. 309.

1) the pressure source, from the lungs, can be considered constant a t the mouth.

2 ) t h a t the flow through the reed i s turbulent, indicating the b e s t equations t o use f o r the f 1 ow conditions .

3) t h a t the flow through the reed actually changes direction a t some parts of the cycle, indicating t h a t the reactive parts of t h e impedances a r e important.

4) t h a t t h i s part of the cycle seems t o be where the reed i s nearly closed suggesting t h a t the 1 inear velocity i s large through the reed, which would give r i s e t o an appreciable Bernoul- l i force a t t h a t i n s t a n t .

References

/1/ Helmo7tz H , 'Sensations of tone' 3rd e d i t i o n of English t r a n s l a t i o n , 1695, Appendix VI I .

/2/

Webster

A.G.,

Physical Rev. 1919,g,

p.164 /3/

Bouasse H., 'Instruments v e n t ' ,

2

vofs.,

L i b r a i r i e Delagrave, P a r i s , 1929.

/4/

Benade A.H. and Gans D.J., Ann. N.Y. Acad.

S c i . , 1968, 155, p. 247.

/5/ Woman W.E., Ph. D. Thesis, 1971, Case Western Reserve University.

/6/ Ingard

U.

and Ising

M.,

J . Acoust. Soc. Am., 1967, 42, p. 6.

171 Schumacher R.T., Acustica, 1978, 2, p. 225.

/8/ P r a t t R.L., E l l i o t t S.J. and Bowher J.M.,

Acustica, 1977, 38, p. 236.