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**Europhysics Letters 1 (6), pp. 295-302 (1986)**

Copyright © Europhysics Letters 1986

Many musical instruments are in essence nonlinear physical
systems, since their role is to
convert a (quasi) constant force, pressure, etc. applied
by the player into an oscillatory
sound wave. For a physicist, the study of wind instruments
is actually nothing but a subfield
of hydrodynamics, since it amounts to studying the motion
of the air inside and outside the
instrument; the vibrations of the walls of the acoustical
cavity are known to play a negligible
role in the direct radiation of sound in the air, at
least for woodwind instruments ^{[1]}.
Important progress has been made recently on nonlinear
hydrodynamics, instabilities,
turbulence, etc. (for a review, see for example ref ^{[2]}
or ^{[3]}) and one could wonder how
these ideas could apply to the study of musical instruments.
One specific property of wind
instruments is that the nonlinearities are indeed important,
but well localized-mostly in the
excitation system ^{[4}, ^{5]}; this contrasts with ordinary
hydrodynamics where they are present
at every point in the fluid. The aim of the present
work is to study how these localized
nonlinearities can drive instabilities and chaos. A
related work on acoustical chaos, including
sound propagation but not acoustical resonators, is
described in ref. ^{[6]}.

The physical study of musical instruments is not new
and much basic work was already
done in the last century ^{[7}, ^{8]} or 50 years ago ^{[9]}.
More recently, a nunber of advances in
our understanding of wind instruments have occurred;
for general reviews, see for example
ref. ^{[4]}, ^{[10]} or ^{[1]}. Examples of important new results
are the studies of the role of the cut-
off frequency of hole lattices ^{[4, 11]}, and of good
impedance peak co-operation ^{[4, 12]}. Here,
we take a model of the functioning of a woodwind instrument
which does not necessarily
include all these physical effects, but gives the simplest
description of nonlinear generation
of sound from a constant pressure. The model we use
draws heavily from the work published
in ref ^{[13]} and ^{[14]} (sections II-B and II-D); see in
particular the short discussion of period
doubling in appendix A of ref ^{[14]}. Among the instruments,
we choose to concentrate on the
clarinet, because it is simpler in many respects: its
acoustical cavity i approximately
cylindrical and the excitation mechanism uses a simple
reed; double reeds for the oboe or
bassoon, or flutelike excitation mechanisms are intrinsically
more complicated. A clarinet
reed is merely a thin, almost flat, flexible wedge of
cane which moves under the effect of the
pressures difference applied on both its sides (it tends
to "close" or "open"); this changes the
coupling between the applied pressure P_{0} inside the
mouth of the player and the pressure p
inside the acoustical resonator. A simple way to characterize
the reed is to use a nonlinear
function, f, which gives the flow of air entering the
clarinet as a function of the pressure
difference p = P_{0} - p; see fig. 1a). For a detailed
discussion of the shape of the curve which
gives *f* as a function of p, the reader is referred for
instance to ^{[4]} and ^{[10]}. We simply note
here that this curve includes negative (differential)
impedance parts; as is well known in
electronics for example, such negative impedances
are required to sustain permanent
oscillations.

In the model, we include only two variables, the pressure
*p* and the flow *f*, which are both
functions of time *t*; more precisely, we define *f* as
the flow divided by the cross-section of the
cylindrical resonator (*f* then gives the air velocity).
Since we ignore the dynamics of the
reed, we simply write as a first equation

(1)

The second equation is obtained by writing a boundary condition which expresses the fact that the acoustical wave is a purely outgoing wave from the instrument (energy flows out from the bell and the open holes). If f(cL) and r() are the Fourier transforrn off(t) and p(t), this gives

Fig. 1. - a) Nonlinear characteristics of the excitation system, giving the air flowfas a function of the pressure difference p = P_{0}- p across the reed. When p = p_{c}the reed closes andfvanishes. b) Geometrical construction giving the successive values of p andf, obtained by a nonlinear iteration using the function of a) after translation and a 45º rotation. O, 1, 2, 3 are successive iterates.Fis an (unstable) fixed point. The case shown corresponds to no dissipation ( = o).

where Z(w) is the acoustical impedance of the resonator measured at a point close to the reed. If G(t) is the inverse Fourier transform of Z(w), one then obtains

Equations (l) and (2) provide a closed system: (1) is nonlinear, but local in time; on the other hand, (2) is linear but includes multiple time delays, as we now discuss.

An idealized resonator ^{[1]} can be obtained by considering
a tube with constant section
ending on a small acoustical impedance Z_{0}, assumed to
be real, positive, and frequency
independent (Z_{0} = c is the acoustical impedance in
the open air). If any other source of
dissipation (viscosity, heat conduction) is ignored,
one easily obtains

where is the Dirac peak function, and T= 2L/c is the time taken by a sound wave to travel twice the length L of the tube. Clearly, this form of G(t) implies multiple delays, and complicates the solution of the problem. As discussed in

Equation (2) then becomes:

where R() is the so-called "reflexion function"; its Fourier transform R(w) is obtained from Z(w) by the homographic transformation

For the idealized resonator considered above, eq. (3) leads to

with only one time delay.

The convenient variables are, therefore, X

where the graph of F is obtained from that of F (with variables p and Z

and make use of (7), we obtain the final equation

The approximations that we have made lead to a particularly simple result: the time dependence of the variable Y = - X

The corresponding geometrical construction is shown in fig. lb). When the time dependence of X

Iterations and Feigenbaum-type scenarios ^{[15]} appear
often in the study of strongly
dissipative systems ^{[3]}. This is not the case here,
since is usually small (typically less than 0.1) and can even be put to zero.

Another difference with the usual case is that the "control parameter" P

Fig. 2. - Geometrical construction of the permanent oscillation values just above threshold, obtained by intersecting the characteristic curve with its symmetric with respect to thef-axis. In fig. a), small oscillations are obtained just above threshold, but not in fig. b)

It is well known

Fig. 3. - The second iterate ofF;Q and R give the permanent regime of normal oscillation, until they become unstable (slope less than - l); then, period doubling occurs.

To do an experiment, it was, therefore, natural to try
to obtain more flexibility on the
nonlinearity introduced by the acoustical excitation
system. The principle involved is shown
in fig. 4; an ordinary acoustical resonator was used,
in several cases a real clarinet, but the
nonlinear reed excitation was replaced by an electroacoustic
equivalent: a microphone
measuring the acoustical pressure p inside the tube,
a nonlinear system to include controlled
nonlinearities in the feedback loop, and finally a loudspeaker
to create a *p*-dependent air
velocity *f*. The nonlinear system was either an analog
electronic device, as in ref. ^{[6]}, or an
included digital processing of the data with a computer.
The latter case gives even more
flexibility on the choice of the nonlinear function,
but requires using analog-digital converters, which introduces time delays (of the order of 100 µs in our experiment with two
converters and the 4C computer of the IRCAM).

Figure 5 shows the results obtained in an experiment where the nonlinearity was obtained with an analog electronic circuit generating the function x

Fig. 4. - Sketch of the experimental set-up. The reaction loop includes a microphone M, a variable gain amplifier, a nonlinear system NL (either an analogic circuit or a digital device including AD converters and a computer), a power amplifier A and finally a loudspeaker LS. The signals are analysed through a spectrum analyser SA.

The number of observed period doublings might have been limited by the narrow domain of the experimental parameters in which they occur, but also by the poor efficiency of the electroacoustic components at low frequencies. The actual experiment is of course significantly different from an ideal experiment; for example phase rotations and band pass limitation are introduced by the loudspeaker. In practice, the time variations of the acoustical pressure have much smoother variations than steplike function. ALso, in the period doubling regime before the chaos limit, some random noise was already present, making the phenomenon slightly less clearly audible (and probably less interesting musically!).

Fig. 5. - Results of our experiments showing the spectra obtained on the SA for increasing values of the gain in the feedback loop. a) Normal oscillation; b), c) (not shown) and d) are, respectively, the first, second and third period doublings; e) chaotic regime.

Similar results were also obtained with digitally generated
nonlinearities, including
experiments with a real clarinet (with its mouthpiece
removed). In the latter case, we never
obtained more than two period doublings. This is not
extremely surprising with a real
clarinet, the bore is not cylindrical (later holes,
bell, etc.), and the reflexion function cannot
be well approximated by one delta-function. We were
never able to observe any period
tripling in any of the experimental conditions. A more
detailed report on the experimental
results obtained in various situations (*e.g* with different
sorts of nonlinearities) will be
published elsewhere. A natural prolongation of these
experiments would be to use several
acoustical resonators in parallel, to study for example
the evolution from quasi-periodic to
chaotic regimes, etc.

In conclusion, although our mathematical model and the
geometrical construction used in
fig. 1, 2 and 3 are limited by the various simplifications
on which they rely, the experiments
show that the essence of their predictions for various
regims of oscillations remains valid in
more realistic situations, including nonideal resonators
such as a real clarinet. More
generally, woodwind instruments belong to a class of
nonlinear systems which exihibit
interesting behaviour, especially if one allows for
more flexible nonlinearities. It has been
known for a long time ^{[7]} that woodwind instruments
can generate harmonic frequencies
which do not correspond to any resonance of the resonator
(e.g the even harmonics in a
clarinet tone; the effect is simply a frequency doubling
occuring in the nonlinear excitation).
The effects observed here show that even the fundamental
tone of the generated sound can
also, fall at a frequency which is much below any resonance
of the tube.

The authors are grateful to J. KERGOMARD and G. WEINREICH for many stimulating discussions and helpful advices.

- [1]
- J. BACKUS:
*J. Acoust. Soc. Am.*, 36, 1881 (1964); J. W. COLTMAN:*J. Acoust. Soc. Am.*,**49**, 520 (1971). - (2]
- J. GUCKENHEIMER and P. HOLMES:
*Appl. Math. Sci.*, Vol.**42**(Springer Verlag, Berlin, 1983). - [3]
- P.BERGE, Y. POMEAU and C. VIDAL:
*L'ordre Dans le Chaos*, (Hermann, Paris, 1985). - [4]
- A. H. BENADE:
*Foundamentals of Musical Acoustics*(Oxford University Press, 1976). - [5]
- Small nonlinear effects can indeed occur also in the acoustical resonator; see for example A. H. BENADE, preprint.
- [6]
- M. KITANO T. YABUZAKI and T. OGAWA:
*Phys. Rev. Lett.*, 50, 713 (1983). - [7]
- H. HELMHOLZ:
*On the Sensation of Tone*(Dover, New York, N. Y., 1954). - [8]
- LORD RAYLEIGH:
*Theory of Sound*(Dover, New York, N. Y., 1945). - [9]
- H. BOUASSE:
*Instrument à vent*(Delagrave, Paris, 1929). - [10]
- C. J. NEDERVEEN:
*Acoutical Apects of Woodwind Instruments*(Frits nof, Amterdam, 1969). - [11]
- H. BENADE:
*J. Acout. Soc. Am.*, 32, 1591 (1960). - [12]
- H. BENADE and D. J. GANS:
*N. Y. Acad. Sci.*, 155, 247 (1968). - [13]
- R. T. SCHUMACHER:
*Acoustica*, 48, 72 (1981). - [14]
- M. E. MCINTYRE, R. T. SCHUMACHER and J. WOODHOUSE:
*J. Acout. Soc. Am.*, 74, 1325 (1983). - [15]
- M. J. FEIGENBAUM
*Physica D*, 7, 16 (1983).

- [1]
- For a real resonator, the losses are introduced
by gas viscosity and conduction, and not
localized at the end of the tube; can be seen as a
mere (positive) convergence factor, useful to take
the
limit 0
_{+}. Also, for real resonators, there are frequency-dependent end corrections, the bore diameter is not necessarily exactly constant, etc., so that the reflexion function is not a -function; nevetheless, it has a significantly shorter memory than G(); see examples in^{[13]}.

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