NONLINEAR PHENOMENA IN CLARINET-LIKE SYSTEMS

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NONLINEAR PHENOMENA IN CLARINET-LIKE SYSTEMS

K. Brod, M. Lewit

To cite this version:

K. Brod, M. Lewit. NONLINEAR PHENOMENA IN CLARINET-LIKE SYSTEMS. Journal de

Physique Colloques, 1990, 51 (C2), pp.C2-889-C2-892. �10.1051/jphyscol:19902207�. �jpa-00230530�

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Colloque C2, suppl6ment au n02, Tome 51, F6vrier 1990 l e r Congrds F r a n ~ a i s dlAcOustique 1990

NONLINEAR PHENOMENA IN CLARINET-LIKE SYSTEMS

K. BROD and M. LEWIT

I n s t i t u t fiir Technische Akustik, Technische U n i v e r s i t d t Berlin, F.R.G.

Nonlinear phenomena in acoustics are often generated by a multiple acoustical interaction between global linear wave propagation and a localized nonlinearity.

Typically, these local nonlinearities can be interpreted physically as, e.g., feedback or sound generation mechanisms.

Mclntyre et at. [I] have suggested an excellent method to theoretically investigate such systems, which circumvents solving complicated nonlinear differential equations. In the case where this multiple interaction between linear and nonlinear acoustical processes is caused by the reflection of the linear wave at a given boundary, the governing equations look as follows:

For an arbitrary, but given, reflexion function r(t) the acoustical pressure qi(t) of the incoming sound wave at the point of linearlnonlinear interaction is given by a convolution integral:

qi(t) =

i

^o r(t') & (t - t') dt' := r(t)

*

^{&(t) (I]}

were qo(t) is the acoustical pressure of the outgoing wave at that point. The total pressure q(t) is the sum of these partial pressures:

The acoustical nonlinearity is now incorporated in the typical form

where F(q(t)) is a nonlinear characteristic function depecding on the total pressure q(t).

If we now eliminate qo(t) and qi(t) from eq.s (1) to (3), we obtain the final representation of a linearlnonlinear system:

This equation describes the actual state of a system (left hand side) as a function of its history (right hand side).

We now apply these considerations to a one-dimensional clarinet-like system, where the nonlinearity is caused by the openinglclosing mechanism of the reed.

According to Benade [2] or Stewart and Strong [3] the flow rate, f(q), in the clarinet mouthpiece as a function of the difference of the pressures inside and outside the mouthpiece, p

-

q, is given in Fig. 1.

Article published online by EDP Sciences and available at http://dx.doi.org/10.1051/jphyscol:19902207

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Fig. 1: Relation between flow rate and pressure difference in the mouthpiece

Here, p is the pressure in the player's mouth (assumed to be constant), and q, is the pressure below of which the reed remains closed.

As a first approximation f(q) can be assumed to be quadratic in q:

This characteristic has a symmetric maximum, whereas experiments show that the maximum is shifted away from the center towards q = p

-

q, see [3], which suggests a cubic function f(q):

( 5 ) or (6) combined with (4) leads to

where Z is the wave impedance in the clarinet tube

Z = (with R being the cross-sectional radius of the tube)

.

(8) R~

The reflection function, r(t), is obtained from the consideration that the amplitude of the sound waves reflected at the clarinet bell decreases with increasing frequency; this leads in the time domain to a Gauss-like form of r(t):

- b ( t - T ?

r(t) = a e (T: round-trip time; a, b: constants)

.

₍₉₎

Introducing a, the half width of r(t), and A, defined by

which is a measure for the energy content of the reflected wave, (7) reduces to

The appearance of what looks like subharmonics in the time behavior of q(t) for very narrow reflection functions suggests to investigate the system with a delta

-

reflexion function

(which is in agreement with (10)).

With this new refexion function the convolution integral (7) is reduced to

Here still, as in equation (4), the right hand side describes the history of the system and the left hand side its actual state, which leads to the conclusion that (13) can be discretized in the form

with a prescribed qo.

Equations of the form of (14) are called 'iterated maps'; a diagram of the fixed points q* of the iteration of equation (14) as a function of the energy parameter A (Feigenbaum diagram) can be found in fig. 2.

Here, the physically relevant domain is the A region between -1 and 0 (passive, with open clarinet tube).

We see that the curve bifurcates at A n 0.6, with two coexisting pressures below that point (period-doubling, subharmonics). Outside the. passive range, where

-

15 A I 0, we see for decreasing A a continuous cascade of period-doublings, chaos, and intermittent periodic windows. The same is true for A > 0 (which corresponds to a closed-tube situation).

Iterations for A values to the left and right of fig. 2 do not lead to finite fixed points and are therefore omitted. The corresponding diagram of the Lyapunov exponents X(A) can be found in fig. 3.

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Fig. 2: Feigenbaum diagram of the iterated clarinet model

Fig. 3: Lyapunov coefficients for the iterated clarinet model

References

[I] Mclntyre,

M.E.;

Schumacher, R.T.; Woodhouse, J.: On the oscillations of musical instruments. J. Acoust. Soc. Am. 74 (1983) 1325-1345

[2] Benade, A.H.: Fundamentals of musical acoustics. New York: Oxford University Press, 1976

[3] Stewart, S.E.; Strong, W.J.: Functional model of. a simplified clarinet. J. Acoust.

Soc. Am. 68 (1 980) 109-120