2.1 Linear and nonlinear

Now to get down to business. To make sense of the acoustics of almost any musical instrument, whether a violin, a trumpet or a Caribbean steel drum, we need to understand the key concepts of resonant frequencies and vibration modes. I will give an informal but approximately correct account of the underlying theory in this chapter, together with a similar introduction to the related concept of frequency analysis. Although some of the descriptions may seem a little abstract (especially if you read the details in the additional links), everything will be illustrated by reference to our first actual musical instrument, a toy drum.

The first step in looking at how musical instruments work is to divide them into two groups. The crucial division, from the perspective of a physicist, is not at all obvious. It does not follow the traditional strings/wind/percussion groupings: the guitar lives with the xylophone, while the violin lives with the clarinet. The distinction is between instruments that can produce sustained sound, and those that can’t. If your bow is long enough, a violin note can go on as long as you like. Given enough breath, the same is true of a note on a flute. On the other hand, a guitar, a piano or a drum is set into vibration by hitting or plucking, but then left alone for the sound to die away in its own good time. Playing a violin pizzicato shifts it across the divide, as would bowing a guitar.

This distinction relates to a technical concept which is unavoidable in this discussion, so technophobic readers must grit their teeth for a bit. The instruments which die away have (usually) a property called linearity, whereas the sustained instruments must be non-linear. So the wind instruments, including the organ and the human voice, are non-linear, along with the bowed-string instruments like the violin and oddities like the musical saw, the hurdy-gurdy and the wet finger round the rim of a wine glass. Percussion instruments might be linear, along with the guitar, the piano and the humble tuning fork.

“Linear” is essentially a piece of mathematical jargon implying “easy”, while “non-linear” suggests that analysis is going to be more difficult. We can tell that something curious must be going on in sustained instruments by thinking about energy. All vibration involves energy in the vibrating system, and some of that energy is constantly being converted into other forms: for example into heat in the wood of a violin body, and into sound waves in the air which carry some of the energy away with them. If the note continues at a steady level, then something is putting just the right amount of energy back into the vibration. This energy is coming from the violinist’s bow arm, or the clarinettist’s lungs. But how does something steady and non-oscillatory, like the motion of a bow arm, get converted into vibrational energy, and furthermore do it at just the right rate to keep the note steady? That discussion must wait until later. We will look at the “linear” instruments first: they pose no dilemma about energy, since the gradual loss of energy is precisely the reason that the sound of a note on one of these instruments dies away with time.

This chapter introduces the main acoustical concepts needed. Some technical details are given in the extra links, but the key properties of a linear instrument can be explained, roughly at least, in plain words. To begin at the beginning, suppose you hit a drum with a soft beater. You could imagine a force-measuring sensor embedded in the beater, which could show the waveform of force applied to the drum-head by the beater. It will look something like the upper trace in Fig. 1: no force until the beater makes contact, then a short pulse of force rising to a peak and then reducing as the beater starts to rebound. As soon as the beater bounces off the drum there is no further force.

Figure 1: Sketch of the force applied when a drum is struck (upper trace) and the resulting motion of the drum skin (lower trace).
Figure 2: Schematic block diagram of the drum as an input-output “black box”.

In response to this force pulse, the drum skin vibrates. This vibration continues after the beater has bounced off. The waveform, for example measured by a vibration sensor on the drum or by a microphone measuring the sound produced by the vibrating drum, might look like the lower trace in Fig. 1. The whole thing could be represented schematically by Fig. 2: the input force waveform goes into the “black box” representing the mechanical behaviour of the drum, and out comes an output waveform. However, this schematic is by no means limited to drums: many things have this input-box-output pattern. A similar input force pulse could be applied to any other vibrating object, like a saucepan lid or a violin body, and some kind of vibration response generically similar to the lower trace in Fig. 1 would be produced. The input need not be force: for example the same schematic could represent a loudspeaker, in which an input waveform of electrical voltage is turned into an output waveform of sound.

If the “box” describes a linear system, then it has an unexpected and useful property when driven with a special kind of input waveform. Enter the sine wave. A sine wave is a particular shape of repeating waveform which is easy to visualise (and which has important mathematical properties, as we will see shortly). Take a glass disc and put a spot of paint on the rim. Spin the disc at a rate which you choose. Look at it edgewise on and watch the spot of paint. It goes up and down, and if you plot a graph of the height against time you get a sine wave, as illustrated in Fig. 3 The frequency of the sine wave, in cycles per second or Hertz (Hz), is the same as the rate of rotation of the disc. (Heinrich Hertz was a physicist in the nineteenth century who, among other things, discovered radio waves.)

Figure 3: Portions of two sine waves with the same frequency: the solid curve could be the input to a linear system, and the dashed curve the corresponding output. The sine wave shown in the dashed curve has a larger amplitude than the one shown solid, and its phase is shifted as indicated.

As sketched in Fig. 1, you ordinarily expect the output waveform from the vibrating drum, or whatever, to be different from the input waveform. But if the input waveform is a sine wave, then that isn’t the case. For any linear system, if you put a sine wave in you get a sine wave out. It has the same frequency as the input wave, and the only things that can be changed are the amplitude and the phase, as illustrated in Fig. 3. The change in amplitude and phase might be different for different frequencies of sine wave — an important idea to which we will return shortly.