Musicians do not normally want their instruments to exhibit chaotic behaviour. Nevertheless the ideas of chaos and sensitive dependence do have some important manifestations in musical performance, and we will meet examples later. But our final topic for this chapter is far more obviously central to the business of making music. Some nonlinear systems can produce sustained vibration “out of nowhere”: self-excited vibration. The human voice is an example, and all the musical wind instruments: woodwind, brass, organ pipes and so on. All the bowed-string instruments are also examples, as are other instruments relying on friction to cause vibration. Friction drums, for example, have a rod attached to a drum membrane of some kind, and the player rubs the rod with rosin-coated fingers to produce a sound. The familiar singing effect produced by running a wet finger round the rim of a wineglass relies on similar behaviour.
We will come to musical examples very shortly, but we will start this section with a non-musical example which allows us to make some links with the discussion in the previous sections, based around the phase plane representation. We will look briefly at something called the Van der Pol equation, originally proposed by a Dutch electrical engineer back in 1927 to describe the behaviour of certain electrical circuits based on valves (vacuum tubes). These circuits exhibited spontaneous oscillation: a phenomenon still crucial in many devices, such as the internal “clock” providing the timing signal inside every computer or mobile phone. Since the 1920s, Van der Pol’s equation has been found useful for modelling many other phenomena in physics, geophysics and biology.
You can see Van der Pol’s equation, and some extra detail about its history and behaviour, in the next link. Essentially, the equation describes a linear oscillator modified by a nonlinear damping effect. The coefficient of this nonlinear term determines the details of the behaviour. If it is zero, the system is just a linear oscillator with a single singular point at the origin in the phase plane, which is a centre as expected for an undamped oscillator. But when this nonlinear coefficient is positive, that centre turns into an unstable spiral so that any trajectory starting near the origin spirals outwards. But it doesn’t continue outwards: instead, it tends to towards a closed loop in the phase plane, in other words a periodic solution or limit cycle. Furthermore, any trajectory starting outside that limit cycle spirals inwards towards it. The limit cycle is an attractor for (almost) every trajectory.
Figure 1 shows some examples, for different values of the nonlinear coefficient. In each case four trajectories are shown: two in red starting inside the limit cycle, and two in black starting outside it. The same four starting points are used in every case. The limit cycle and its “attractor” behaviour is very clear. The shape of the limit cycle changes with the value of the coefficient. What this means for the waveform of displacement of the oscillator is shown in Fig. 2, for the same four cases. When the coefficient is small, the limit cycle is almost an ellipse and the waveform is almost sinusoidal (after a starting transient). As the coefficient grows, the waveform changes shape. It gradually acquires sharp corners, which inevitably mean that the periodic solution has a significant mixture of higher harmonics in its Fourier series. The frequency of the self-excited vibration also changes, getting lower as the coefficient increases, as is clear in the lower row of plots in Fig. 2.
We can now look at an example relating more directly to musical instruments. The discussion will be phrased in terms of a clarinet, but really the description to be given here would apply in general terms to any reed instrument: saxophone, oboe, reed organ pipe or whatever. Figure 3 shows a sketch of the mouthpiece end of a clarinet. The main tube of the instrument is an acoustic duct with internal resonances of the kind we met back in Section 4.2. The tube has tone-holes bored through it: the player can cover some or all of these using fingertips or key mechanisms, to modify the frequencies of the internal acoustic resonances. In a clarinet the tube is usually made of wood or plastic, in instruments like the saxophone it would be made of metal. These different material choices make very little difference to us here, because they all result in a tube that is essentially rigid. All the important action takes place in the air inside the tube.
At the mouthpiece, a flexible reed is attached to the tube. The player puts this part in their mouth, and blows. What happens then is a matter of common empirical experience. If the player blows very gently, the only sound is a bit of rushing wind noise associated with the turbulent air flow through the mouthpiece into the tube. But if the player gradually increases the blowing pressure, at a certain point the instrument will “light up” and start to produce a musical note. If the blowing pressure is increased further, the tone quality of the note changes (and perhaps the pitch changes a little as well). The sound tends to get brighter with higher blowing pressure. But if the blowing pressure is increased too far, the instrument “chokes up” and the sound stops.
This sequence of events can be understood in a simple way. We can suppose that the player simply provides a constant pressure in their mouth, called $p_0$ in Fig. 3. But just inside the mouthpiece there will be a time-varying pressure which we can call $p(t)$: this is the quantity we would like to understand, since it is responsible for the sound of the instrument. There is a second time-varying quantity we need to think about: this is the flow rate of air from the player’s mouth, through the mouthpiece and into the tube. We will call this $v(t)$, to describe the volume flow rate (in cubic metres per second if we wanted to put a number to it).
The two quantities $p(t)$ and $v(t)$ are related to each other in two quite different ways, illustrated schematically as a block diagram in Fig. 4. First, they are connected via the linear acoustical behaviour of the tube. We can imagine a laboratory experiment in which the instrument was supplied with a sinusoidal flow of air at its mouthpiece end by an actuator, and the pressure response inside the mouthpiece was measured by a small microphone. By varying the frequency of the sine wave, the frequency response function of the tube could be measured. This particular frequency response, with volume flow as input and pressure as output, is called the input impedance of the clarinet.
Such measurements of input impedance are indeed routinely made on wind instruments of all kinds: Fig. 5 shows one being carried out on a saxophone. The measurement is being done in an anechoic chamber, a special room with sound-absorbing walls to avoid complications from room acoustics. As with other linear response measurements we have seen earlier, the test need not necessarily be done using a sinusoidal signal. Any input can be used, provided it can be measured. The input and output signals can be converted into the frequency domain using an FFT routine in the computer, just as is done when structural measurements are made using an impulse hammer.
A typical measured example of the input impedance of a clarinet is shown in Fig. 6. The peaks in the plot correspond to the resonances of the tube. The pattern of these peaks shows them with approximately harmonic spacing, but only for the odd harmonics 1,3,5… This is as we should expect from the discussion in section 4.2: the tube of a clarinet is approximately a uniform duct (not a tapered one like an oboe or a trumpet), and it is effectively open at the bell end but closed at the mouthpiece end. As shown in Fig. 12 of section 4.2, those conditions lead naturally to odd-harmonic resonance frequencies.
The second relation between $p(t)$ and $v(t)$ involves the mouthpiece and reed, acting as a kind of nonlinear valve. For the purposes of this initial discussion, we will use a severely simplified approximation. We will take no account of the fact that the flexible reed has resonance frequencies its own, we will simply treat it as behaving like a spring. As a first step, we won’t even allow that much: we can think about how the air flow through the mouthpiece would behave if the reed were rigid. A pressure difference could be applied across this rigid mouthpiece, and the resulting air flow rate could be measured. What we would expect to see, disregarding any subtleties of fluid mechanics such as vortices, would be something like the dashed line in Fig. 7. The bigger the pressure difference, the bigger the air flow. If the air flow is dominated by viscous resistance through the small gap, the behaviour would just be linear, as sketched. The dashed line is sloping downwards rather than upwards because of the sign convention we have used: a positive value of the pressure difference $p(t)-p_0$ corresponds to sucking the mouthpiece, not blowing it, so we would expect $v(t)$ to be negative.
For very small values of the pressure difference, the actual behaviour of the mouthpiece with a flexible reed should be rather like the dashed line. But as the player tries to blow more air through, making $p(t)-p_0$ more and more negative, the reed will be pressed inwards. As a result, the flow rate $v(t)$ will be less than the dashed line would suggest. The result will be something like the solid line in Fig. 7: the further we move towards the left, the more the reed is closed and the more the air flow is restricted. Eventually, in an idealised situation, the reed will close completely against the rigid part of the mouthpiece (called the “lay”). There would be no air flow at all, indicated by the horizontal portion of the curve.
If the player applies a low mouth pressure while the clarinet is not making any sound, that would correspond to shifting to a position on the curve like the red dot in Fig. 8. Now suppose there is a little bit of pressure variation (i.e. sound) inside the tube. This will change the pressure difference a little, in the vicinity of the red dot: the air flow will then vary in a way that follows the tangent to the curve at that point, shown in the red line.
Such variations have a simple physical interpretation. The tangent line is down-sloping, similar to the dashed line in Fig. 7. But we know what that line describes: it is the response associated with a viscous resistance, and it involves energy dissipation. But now suppose the player blows a little harder, so that the operating point on the curve shifts to a position like the one shown in Fig. 9. Because we have gone past the peak in the curve, the tangent line now has an upward slope. That would correspond to a negative viscous resistance, and small fluctuations in pressure inside the mouthpiece will result in energy being gained, not lost. (Of course, energy has not been created from nowhere: this energy gain is actually supplied by the player’s lungs doing a little more work.)
We can now make an intuitive leap, and guess what might happen. The tube of the clarinet is an acoustic duct, with resonances. Each of these resonances will, of course, involve some energy dissipation. Energy can be lost in several different ways: some is “lost” because it is carried away by sound waves radiating from the instrument, while some is lost within the tube, mainly by viscous and thermal interaction with the walls. It seems plausible that if for some particular resonance these losses can be compensated by the “negative resistance” effect associated with the mouthpiece, as indicated in Fig. 9, that resonance might become unstable.
We saw just such an effect with the Van der Pol equation, in the plots of Fig. 1. The equilibrium position became unstable, leading to growing oscillations, settling down after a while into a periodic limit cycle. This is exactly what happens with our simplified model of a clarinet, and the result is the behaviour we described earlier, very familiar from real clarinets. There is a threshold of blowing pressure, at which the instrument “lights up” and starts to make sound. The phenomenon is completely dependent on the nonlinear behaviour of the mouthpiece with its flexible reed.
There are two different approaches we can use to test whether we have guessed correctly how the clarinet model will behave. The more mathematical of the two involves applying the method of harmonic balance (introduced in section 8.2.2) to the situation of Fig. 9. The details are described in the next link: the conclusion is that there is indeed a threshold of blowing pressure when a self-excited periodic oscillation becomes possible. The prediction of this analysis is that the frequency will correspond to the highest peak in the input impedance, and we can see in Fig. 6 that this is the fundamental resonance of the clarinet tube, as we might have expected.
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The alternative approach is to turn directly to numerical simulation, and use the computer to explore how the model behaves. A particularly efficient way to do this involves formulating the behaviour of the linear part of the clarinet in a slightly different way: not based on a frequency response function like the input impedance, but describing the acoustic response of the duct in terms of travelling waves. This approach was first developed in the context of the vibration of a bowed string, and we will explore that application in the next chapter. But the clarinet model gives a simple way to introduce the method. A pleasing name has been given to the approach by Julius Smith : he calls it the “digital waveguide” method.