Nonlinear propagation in woodwinds

Joël Gilbert, Jean-Pierre Dalmont, Thomas GuimezanesLaboratoire d’acoustique de l’université du Maine – UMR CNRS 6613

Avenue Olivier Messiaen – 72085 Le Mans cedex 9 – France

joel.gilbert@univ-lemans.fr, jean-pierre.dalmont@univ-lemans.fr, thomas.guimezanes@univ-lemans.fr

The objective of this study is to investigate if nonlinear propagation in reed instruments may result into aperceptible effect. This question is first investigated by comparing theoretical results obtained by usingboth linear and nonlinear propagation theories. Measurements of pressures in the mouthpiece of a clarinetand outside the instrument in different situations are then used to reveale the evidence of non linearpropagation effect. The comparison between the linear and nonlinear cases exhibits differences in thecalculated radiated pressures, but these differences would not imply audible effects.

1 Introduction

Linear acoustics is a fruitfull first approach to modelacoustic propagation in wind instruments. Despiterewarding musical sounds have been obtained withthese models, it has become obvious that someessential phenomena escape such a description. Inparticular, the brightness of the sound generated by abrass instrument is due to the nonlinear propagation ofthe wave in the pipe [1, 2, 3, 4, 5]. Such “brassysounds” are perceptually associated to sounds producedby brass instruments at high sound level. If reedinstruments are not known for generating “brassysounds” , one can imagine that nonlinear propagationmay have some importance in such instruments. Theobjective of this study is to investigate if nonlinearpropagation in a clarinet may result into a perceptibleeffect.This question is first investigated by comparingtheoretical results obtained by using both linear andnonlinear propagation approaches. Input data being themeasured acoustic pressure spectrum in the clarinetunder playing conditions, the pressure and velocityacoustical field are calculated inside the bore andoutside. The calculation is carried out by using linearpropagation theory first, then by using nonlinearpropagation theory. The two sets of results arecompared in order to see how much they are different.The present paper is divided into five sections. Afterthis brief introduction, some insights of woodwinds arepresented in section 2. The theoretical approach takinginto account non-linear or linear propagationphenomenum is presented in section 3. In section 4, theresults are discussed. Finall y the perspectives of thiswork are discussed in the concluding section 5.

2 Woodwinds, internal andradiated acoustical pressures

Sound production in reed instruments is the result ofself-sustained oscill ations. A mechanical oscill ator, thereed, acts as a valve which modulates the air flowentering into the mouthpiece. The oscill ations of thereed are strongly influenced by the oscillations withinthe air column, and the resulting fundamentalfrequency is very close to one of the resonancefrequencies of the instrument. When the instrument isoperating, a periodic oscill ation inside the bore isreached. At the open end of the instrument the pressurewave is almost totally reflected with a change of sign(boundary condition close to a node of pressure). Onlya small part of pressure is radiated outside.In our study, the reed instrument considered underplaying condition is a “simplified clarinet” – i.e. aclarinet mouthpiece and a barrel extended by a straightcylindrical pipe. A pipe termination with roundededges is attached on the end of the pipe in order tominimise the nonlinear losses localised at the end ofthe pipe [6]. The “simplified clarinet” is blown with anartificial mouth (see Figure 1) which has been used inprevious studies [7].

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Figure 1 : “simplified clarinet” artificially blown.

The experimental procedure is the following. Startingfrom zero, the blowing mouthpressure is increaseduntil oscill ations start, grow up, decrease quickly andstop. The mouthpressure increase is carried out step bystep to get a set of representative acoustic periodicregimes. Pressure sensors are used to record theacoustic signal in several positions - three inside thebore located in the mouthpiece, at the entrance andnear the end of the pipe, and one outside (see Figure 2).Some typical periodic regimes which seem to be themost “brassy” are selected by informal auditary tests.The corresponding recorded pressure at the entrance ofthe pipe are the input data of the theoretical workdescribed in the following section 3 and discussedsection 4.

3 Non-linear and linearpropagation

Our work is based on the comparison of twocalculations by using linear and nonlinear propagationapproaches respectively. In each calculation, theinternal pressure field and then a radiated pressure arecalculated. For each approach, the acoustical velocityand pressure along the pipe are obtained from inputdata which are the acoustical pressure in two points :the measured acoustical pressure at the entrance of thepipe (see Figure 2 for example) and the pressure at theopen end which is supposed to be zero. Then theradiated pressure is calculated by assuming a monopolesource located at the open end and having a volumestrength equal to the volume velocity at the open end ofthe pipe.Knowing the pressure in two points – the two boundaryconditions – and assuming a linear plane wavepropagation everywhere between these two points, it isstraightforward to deduce the acoustical pressure andvelocity everywhere along the pipe by assuming asuperposition of one inward traveling wave and one

outward travell ing wave. Otherwise, if a weaklynonlinear plane wave propagation is assumed insidethe pipe, the acoustical pressure and velocity field inthe pipe can be deduced too by using a more elaboratemethod which is described in detail i n [8], andsummarized in the following. The method is based onthe superposition of periodic inward (q+) and outwardtraveling (q-) wave with space dependent ampli tudes σ(Equation 1) :

( ) ( ) ( ) ( )( )∑+∞

=

±±±±±± +=1

cossin),(n

nn nbnaq θσθσθσ . (1)

These traveling waves are solutions of the followinggeneralised Burgers equations (Equation 2) :

−−

−

−

−−

−

++

+

+

++

+

∂∂+

∂∂−=

∂∂

∂∂−

∂∂=

∂∂

πθθεθσ

πθθεθσ

1*

1*

qTqq

q

qTqq

q

. (2)

The variables and parameters used in the Equations (1)and (2) are carrefully defined in [8],The Burgers equations are solved with a numericalfinite-difference method from one boundary conditionto the other. Then the harmonic ampli tudes an and bn ofthe traveling waves are obtained by an iterative methodusing a harmonic balance method on these ampli tudes.The acoustical velocity and pressure along the pipe areobtained. Then the volume velocity Qout at the openoutput of the pipe is estimated by multyplying theoutput area by the output acoustic velocity. By usingthe low-frequency approximation from a monopolehaving a volume strength equal to Qout (Equation 3), aradiated pressure is estimated at the distance d from theopen end of the pipe as follows :

outrad Qjd

p ωπρ

4= . (3)

4 Results and discussion

1 2 3 4 5 6 7 8 9 10110

115

120

125

130

135

140

145

150

155

160

harmonic

dBsp

l

Figure 2 : Ampli tude of the first ten harmonics of theentrance internal acoustic pressure in [dBspl] as a

function of the harmonic’s rank varying from 1 to 10.

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A periodic regime associated to a loud tone obtainedfrom the artificial mouth (Figure 1) is selected. Thisone corresponds to the fundamental regime of the pipe,that is a playing frequency equal to 143 Hz. Theinternal acoustic pressure pin at the intrance ismeasured and its frequency spectrum (see Figure 2) isused as input data of the linear and nonlinearcalculations detailed in section 3. The acoustic velocity(or volume flow) and the pressure are calculated alongthe pipe in the linear and nonlinear cases. Theampli tudes of the first five harmonics along the pipeare displayed Figure 3. The boundary conditions p=0 atthe open end and p=pin at the entrance of the pipe areverified by the two calculations. Having a first look onFigure 3, no great differences between the twocalculations are visible.

0 0.1 0.2 0.3 0.4 0.5 0.6

0

1000

2000

3000

4000

5000

[m]

[Pa]

harm 1harm 2harm 3harm 4harm 5

0 0.1 0.2 0.3 0.4 0.5 0.6

0

5

10

15

20

25

[m]

[m3/

s]

harm. 1harm. 2harm. 3harm. 4harm. 5

Figure 3 : Acoustic pressure (up) in [Pa] and volumeflow (down) in [m3/s] amplitudes along the pipe in [m].

The first five harmonics are displayed for the linearcase (straight line) and the nonlinear case (stars).

The nonlinear calculation case is based on an iterativemethod, the iteration being operative until an inequali tytest is reached. The calculation is stopped when theharmonic’s amplitudes of the entrance acousticpressure are less than 0.1% close at the beginning andat the end of the loop. Then, in one hand the entrancepressure time signal calculated in the nonlinear case is“equal” to the original one (see top of Figure 4). In theother hand, some differences between the twocalculation cases are slightly visible on the time signalsclose to the end of the pipe (see middle of Figure 4)and on the radiated pressure (see bottom of Figure 4).

0 0.002 0.004 0.006 0.008 0.01 0.012

-5000

0

5000

[s]

[Pa]

nonlinearlinear

0 0.002 0.004 0.006 0.008 0.01 0.012

-1000

0

1000

[s]

[Pa]

nonlinearlinear

0 0.002 0.004 0.006 0.008 0.01 0.012

-0.5

0

0.5

[s]

[Pa]

nonlinearlinear

Figure 4 : Acoustic internal pressure signal at theentrance of the pipe (top), close to the end of the pipe(middle), and a radiated pressure (bottom). The two

curves displayed in each sub-plot are corresponding tothe linear case and the nonlinear case respectively.They are slightly shifted each other in order to see

small differences.

In order to emphasize the differences, the ampli tude ofthe first ten harmonics of the radiated pressure aredisplayed for the two calculations in Figure 5. Themost highly powerful harmonics do not showsignificant differences, even if differences on theampli tude of some harmonics of the order ofmagnitude of 5 dB can be observed. But this do notresult in a perceptible effect as verified by listening thetwo simulated “clarinet tones” . These radiated soundshave been simulated by adding the first 20 harmonicscoming from the linear and nonlinear calculationsdescribed in section 3.

1 2 3 4 5 6 7 8 9 1040

45

50

55

60

65

70

harmonic

dBsp

l

nonlinearlinear

Figure 5 : Ampli tude of the first ten harmonics of theradiated pressure in [dBspl] as a function of the

harmonic’s rank varying from 1 to 10. The two resultsassociated to each harmonic are corresponding to the

linear case and the nonlinear case respectively.

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Forum Acusticum 2005 Budapest Gilbert, Dalmont, Guimezanes

5 Conclusion

In this paper, we have investigated if nonlinearpropagation may result into a perceptible effect forreed instruments. It has been first investigated bycomparing theoretical results obtained by using bothlinear and nonlinear propagation theories. Thesetheoretical results have been calculated from measuredinput data : the measured acoustical pressure at theentrance of a “simplified clarinet” artificially blown.The comparison between the linear and nonlinear casesexhibits differences in the calculated radiatedpressures, but these differences do not imply audibleeffects. As a conclusion of this preliminary study, wecan assert that it is propably not necessary to take intoaccount nonlinear propagation in the pipes of reedinstruments.

References

[1] Beauchamp J.W. (1980). Analysis of simultaneousmouthpiece and output waveforms. AudioEngineering Society, preprint No. 1626 (1980) 1-11.

[2] Hirschberg A., Gilbert J., Msallam R. et WijnandsA.P.J. (1996). Shock waves in trombones. Journalof the Acoustical Society of America 99, 1754-1758.

[3] Gilbert J. and Petiot J-F. (1997). Brassinstruments. 4th International Symposium ofMusical Acoustics, Edimbourg, Grande -Bretagne.

[4] R. Msallam, S. Dequidt, R. Caussé, S. Tassart :Physical model of the trombone includingnonlinear effects, application to the soundsynthesis of loud tones. Acustica 86 (2000), pp.725-736.

[5] Thompson M.W. and Strong W.J. (2001).Inclusion of wave steepening in a frequency-domain model of trombone sound production. J.Acoust. Soc. Am. 110, 556-562.

[6] Atig M., Dalmont J-P. and Gilbert J. (2004).Saturation mechanism in clarinet-like instruments,the effect of the localised nonlinear losses. AppliedAcoustics 65, 1133-1154.

[7] Dalmont J-P., Gazengel B., Gilbert J. andKergomard J. (1995). Some aspects of tuning andclean intonation in reed instruments. AppliedAcoustics 46, 19-60.

[8] Menguy L. and Gilbert J. (2000). Weakly non-linear gas oscill ations in air-fill ed tubes ; solutionsand experiments. Acustica 86, 798-810.

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