First, another technical result must be pulled out of the hat: the idea, trailed in the previous chapter, of *Fourier analysis*. The idea, essentially, is that any waveform whatsoever can be built up by adding together sine waves. Fourier was a French mathematician who, among other things, travelled with Napoleon’s army and was for a while made governor of Lower Egypt. He discovered the result we are interested in while studying the flow of heat, but the mathematical result applies equally well to other waveforms. It is easiest to see what is going on if we look first at a periodic (or repeating) waveform: we will consider a *sawtooth wave*. Fourier showed that the sawtooth wave can be built up from a recipe, by taking a certain amount of sine wave at the same frequency as the sawtooth then adding, in an appropriate phase, a certain amount of the second harmonic, a sine wave at twice the frequency, plus a certain amount of the third harmonic at three times the frequency, and so on. Figure 1 shows how it works: the left-hand column shows the successive sine waves being added, while the right-hand column shows the sawtooth waveform gradually building up. By adding more and more terms, the result gets closer and closer to a perfect sawtooth wave.

The same idea can be applied to any other repetitive waveform: there is nothing special about the sawtooth wave. A little less intuitively, the approach can also be applied to non-periodic waveforms like the force pulse in the upper trace of Fig. 1 of section 2.1. The difference is that you have to add a bit of sine wave at *every* frequency, not just at separate harmonic frequencies. The mathematical procedure for calculating the “sine wave recipe” of a periodic waveform is called a *Fourier series*, and the equivalent procedure for a general non-periodic waveform is called a *Fourier transform*. This leads to a piece of jargon which may be familiar: the usual computer routine for calculating this important recipe is called the “Fast Fourier Transform” or FFT. The process of turning a recorded sound into an MP3 file, for example, makes heavy use of the FFT.

So back to the drum. If it behaves as a linear system, then we know what happens to any sine wave which is put in: nothing at all, apart from scaling by an amplitude factor and shifting in phase. We aren’t really interested in sine wave input, though: the input we want is the force pulse from the beater. But Fourier says that we can express that pulse as a combination of sine waves. We could apply each of those sine waves separately to the drum, getting a scaled and shifted sine wave out in each case. Now the second important property of linear systems comes into play, known as the “principle of superposition”. This is a grand name for a simple idea: if the *input* to a linear system is the sum of several components, then the *output* is simply the sum of the corresponding separate outputs.

We can take each separate sine wave from the Fourier recipe, find the output of the drum to that sine wave, then add them all together and we should get the actual output, the vibration waveform from the lower trace of Fig. 1 of section 2.1. It follows that we can calculate the response of the drum to *any* kind of force input provided we know what it does to sine waves. In other words, everything we need to know about the drum’s behaviour is coded (somehow) into the amplitude scale factor and the phase shift. Now remember that these might vary with the frequency of the sine wave, so we could plot a graph against frequency of the scale factor and the phase shift: this combined information is called the *frequency response function* of the drum. It is the first step in obtaining an “acoustic fingerprint” of the drum. We will see some examples soon.

For the next step we need to know a little about the theory of vibration. The essence of vibration lies in the interaction between *restoring force* and *inertia*. If a system is displaced a little from a position of stable equilibrium, a restoring force of one kind or another will make it tend to return. Inertia will guarantee that, left to its own devices, the system will overshoot the equilibrium position. A restoring force with the opposite sign will then act, and some kind of oscillatory behaviour will result, at a frequency governed by the balance between the restoring force and the inertia. Restoring forces can arise from many physical causes, for example gravity (as in a pendulum), elastic internal stresses (as in a steel spring or a wobbling jelly) or change in pressure of a fluid (as in sound waves in an organ pipe, or air compressed in a bicycle pump with your thumb blocking the outlet).

The archetypal vibrating system consists of a mass supported by a spring. The sketch in Fig. 2 shows a schematic version. The mass provides the inertia, and the spring provides the restoring force. If the spring behaves in a linear fashion, that restoring force obeys *Hooke’s law*: the force is proportional to the distance the mass has moved away from the equilibrium position. The constant of proportionality is the *stiffness* of the spring, called $k$ here. The units of $k$ are force per unit displacement, so N/m or N m$^{-1}$. Applying Newton’s law of motion to this simple system, two things are easily shown: the displacement of the mass during free vibration is a sine wave, and the frequency of that sine wave is determined by the ratio $k/m$. Specifically, the frequency in Hz is given by $\frac{\sqrt{k/m}}{2 \pi}$.

Now to delve deeper into the theory of vibration. The particular aspects we need are not new or controversial: they were all known to Lord Rayleigh in the nineteenth century. Rayleigh is the great guru of acoustics. He made other scientific contributions as well: he was the first to explain why the sky is blue, and he won the Nobel prize for the discovery of argon. He had serious health problems in his youth, and at the age of 30 he spent some months sailing on the Nile. Not only did this improve his health, but he started writing “The Theory of Sound”, published a few years later. This astonishing work is one of the few scientific books from the nineteenth century which is still in active use as a reference text, rather than merely as a historical curiosity.

The first question to answer is: how do we know that it is OK to treat the drum as a linear system? The real answer to that question is mathematical. The key conclusion is that vibration of any object will usually be linear provided the amplitude is small enough. How small is small enough? That is a matter for empirical checking by experiment, but the result is that most “musical vibration” is small enough that we can get away with assuming linearity nearly all the time. It is not always true: we will meet a striking exception later (this is a pun: think J. Arthur Rank). Also, of course, we should not forget the warning from the previous chapter that small effects cannot necessarily be ignored in musical problems. Nevertheless, on the principle of not running before we can walk, we will take advantage of the simplifying insights from linear theory for as long as we can get away with it.

Back to the drum again. After the beater has bounced off, the drum continues to produce output (i.e. to vibrate) when no input is being applied to it. The general theory of linear vibration reveals that this is only possible in a particular way. The drum has certain *natural frequencies*, which are characteristic of that particular drum. Each of these natural frequencies is associated with a particular pattern of vibration of the drum skin, called a *vibration mode*. Modes of a linear system like the drum obey the “principle of superposition”: they can vibrate simultaneously, each at its own natural frequency, uninfluenced by each other. Remarkably, each separate mode behaves exactly like the simple mass-spring oscillator (known as a “simple harmonic oscillator”) discussed above.

The first few mode shapes for a perfect “textbook” drum are illustrated in Fig. 3. The lowest mode, in the top left of the figure, has the whole skin vibrating up and down, but all the higher modes have *nodal lines*, which are lines where there is no displacement of the skin. The second mode has a single line along one diameter. Higher modes have a mixture of nodal diameters and nodal circles, and the patterns get more and more complicated as we go to higher frequencies: the wavelength of the skin motion gets shorter. Mode shapes of any vibrating structure behave in a way generally similar to this: the simplest thing you can imagine for the lowest mode, then more and more complicated patterns with shorter and shorter wavelengths as the frequency goes up.

There is one slight complication in this problem: nearly all the drum modes actually occur in *pairs*. The second mode, with a frequency 1.59 times that of the fundamental, is an example. The perfectly circular drum has no preferred direction, so how could it know where to put that single nodal diameter? The answer is that it doesn’t know: the diameter could be put anywhere. One particular choice is shown in the figure, and its natural companion would have a nodal line rotated at $90^\circ$ so that it passed through the points of maximum displacement of this one. Now, recall that modes can vibrate simultaneously, independent of each other. It turns out with a small algebraic calculation that if you have these two particular modes vibrating simultaneously, they can combine to give the same shape again, but with the nodal diameter rotated. By choosing the amplitudes of the two modes suitably, the diameter can indeed be put anywhere at all.

The sound of the drum during vibration after the input forcing has stopped can only, ever, consist of a mixture of these particular frequencies. The drummer can control the relative loudness of the different frequencies in the mixture by doing the things drummers are familiar with: choosing a hard or soft drumstick, and choosing where to hit the drum: a hit in the centre will produce a different sound than one near the rim. There is a useful formula that reveals exactly how this effect of striking position works in terms of the mode shapes of the drum. It is a good example of the kind of “deep theory” mentioned in the first chapter: one would expect this formula to apply, to a good approximation, to any structure vibrating with small amplitude. The details are given in equation (11) and the associated discussion in the next linked note. This one equation captures a great deal of important information about vibration behaviour, not only for drums but for just about every other instrument we will look at.

What about the reason a drum sounds different with different choices of drumstick? That all comes down to Fourier analysis again. We have already said that the force pulse from the bouncing drumstick can be broken down into a mixture of sine waves. When you change from a soft to a hard stick you change the shape of the force pulse, and that changes the mixture. Just as you would guess, a hard stick bounces quickly off the drum, producing a very narrow pulse of force. A soft beater stays in contact much longer, giving a broader pulse. Now there is a very simple relation between the width and the frequency mixture: wide pulses consist mainly of low frequencies only, but narrower pulses contain a lot more high frequency. As a rule of thumb, if the length of the pulse is $T$ seconds, then it generates frequencies up to about $1/T$ Hz. Rayleigh thought about this problem, and gave a simple approximate formula. A good first guess for the shape of the force pulse is a half-cycle of a sine wave, and the Fourier analysis of that waveform can be calculated mathematically. The result, as far as the drummer is concerned, is that a soft beater will only excite the first few natural frequencies of the drum, but a hard stick will excite more of them.

The actual sound of the struck drum involves one extra ingredient: it does indeed consist of a mixture of sine waves, each corresponding to a particular vibration mode of the structure, but each of these sine waves decays in amplitude as time goes on, because energy is gradually lost. To give a complete “vibration fingerprint” of the sound we thus need to know three things about each of these sine waves: the frequency, the amplitude and the *decay rate* or *damping*. The decay rate determines whether the sound goes ping (slow decay) or thud (fast decay). When a sound immediately strikes you as “metallic” or “wooden”, you are probably responding mainly to a difference in typical decay rates between those two kinds of material.

So, the set of natural frequencies, mode shapes and decay rates can be regarded as the vibration fingerprint of that particular drum. But just now, we said the same about the frequency response function of the drum, so what is the connection? The answer is simple. One way to apply force to a drum which is approximately a sine wave is to sing or hum gently, close to the drum-skin. Tympanists routinely do this when they are tweaking the tuning of an instrument. They are listening for *resonances*, frequencies where the drum, as it were, sings back at you. These resonances are nothing other than the natural frequencies we have just been talking about. Now recall that the frequency response function tells us the amplitude scaling factor, which varies with frequency. Resonances are frequencies where you get a lot of output for not very much input: in other words, they will show up as peaks in the plot against frequency.

It is time to stop talking in abstract terms, and see the response of a real drum. The small toy drum shown in Fig. 4(a) will be examined. The frequency response function has been measured, by tapping the drum at one point with a hammer equipped with a force-measuring sensor, and the motion at a nearby point on the skin has been measured. The skin of the drum is very light, so we do not want to fix any kind of heavy sensor to the surface because that would change the vibration behaviour enough to annoy the drummer. Instead, the motion has been measured using a non-contact sensor called a *laser Doppler vibrometer.* A laser beam is shone at the point to be measured: the red dot of the laser beam is visible in Fig. 4(b). A spot of reflective paint is put on the drum at this point, so some of the beam is reflected back. Now when the drum is vibrating, something happens to the reflected beam: its frequency is changed a little by the *Doppler effect*, the familiar phenomenon that makes the note of an ambulance warning horn change as the ambulance comes towards you, passes and moves away. The vibrometer detects this change, and turns it into an electrical signal proportional to the vibration velocity of the drum-skin parallel to the laser beam. That signal can be captured into a computer.

Now we follow the procedure described earlier to find the frequency response. Both the hammer signal and the vibrometer signal are turned into combinations of sine waves, using an FFT routine in the computer. For each frequency we then take the ratio of amplitudes, output divided by input, and also the phase difference between the two. For the moment we will just look at the amplitude response: it is plotted as a function of frequency in Fig. 5. In fact, two different versions are shown, found by tapping in two different places while keeping the measuring point the same. The amplitude varies over a huge range, so to make it easier to see what is going on it has been plotted on a *logarithmic* vertical scale. The left-hand axis shows the level, in terms of powers of ten. (The notation $10^2$ means $10 \times 10$, $10^{-2}$ means $1/(10 \times 10)=0.01$ and similarly for other positive and negative powers.) The right-hand axis shows the same information on a decibel scale. Decibels tell us about *ratios* of amplitudes rather than absolute amplitudes. The conventional scale for this purpose means that every 20 dB on the vertical scale corresponds to a factor of 10 in the amplitude, as can be seen by comparing the left and right scales. Decibels are familiar from noise level meters. The “loudness” of sound is usually measured in that way, because our ears work, more or less, on a logarithmic scale: more about this in Chapter ?.