Now back to the frequency response function of the drum. Figure 1, reproduced from section 2.2, shows a sequence of very clear peaks, which correspond to the resonances or natural frequencies of the drum. Comparing the two different measurements, it is clear that they both have peaks at the same frequencies, but that the pattern of heights of those peaks is very different for the two tapping positions. This illustrates something said earlier: a drummer excites a different mixture of the vibration modes by hitting in different places. The peak heights in the measurement reflect the relative amplitudes in the mixture, and the sound of the drum will be different in the two cases.
Something else can be noticed in this plot — or rather, not noticed where it might perhaps have been expected. The peaks, showing the resonances or natural frequencies, are spaced in a rather irregular way. They are definitely not in a regular pattern in which the higher frequencies are exact multiples of the first one, called the fundamental frequency. In other words, the natural frequencies of this drum are not harmonically spaced. Now, as we will see in chapter 3, some musically-important vibrating systems do have natural frequencies which are harmonically spaced, at least approximately. But most things don’t do this: our toy drum is quite typical in this respect. For some reason, there is a widespread belief that all sequences of resonant frequencies are “harmonics”, to the extent that people sometimes use the word “harmonic” interchangeably with “resonance”. If you had this idea before you started reading this paragraph, then stop it right away! It is going to be very important in the next chapter to examine whether or not some of the natural frequencies of certain objects are harmonically spaced. If we were to call the frequencies “harmonics”, we would find ourselves saying things like “In this case the harmonics are not exactly harmonic”, and confusion would reign. Call them natural frequencies, or resonances, or overtones, or partials, but reserve the word “harmonic” for the theoretical notion of frequencies that are exact multiples of a fundamental.
We can show some of the mode shapes corresponding to the resonances of the toy drum using a time-honoured method called Chladni patterns. Ernst Chladni was an eighteenth century scientist who found he could visualise the vibration of a glass disc by holding it lightly on his fingertips, sprinkling sand on the surface, then making it vibrate using a violin bow against the edge. The sand bounces around on the vibrating disc, and tends to collect along the nodal lines. Nowadays, rather than using a violin bow it is easier and more controlled to use an electronic sine-wave generator and a small loudspeaker to “sing” at the disc or, in our case, the drum. We sprinkle powder on the surface: tea leaves are a good choice, because they are less heavy than sand and bounce more easily, and also the black colour shows up nicely on the white skin. The sine wave oscillator is gently adjusted through the frequency range to seek out resonances. The tea leaves then reveal the nodal line patterns.
Some examples for our drum are shown in Fig. 2. The first mode is not illustrated, because there is nothing to see by this method: the only nodal line is around the rim of the drum. The first strong peak in Fig. 1 shows that this lowest mode occurs at about 130 Hz. The next two, appearing as a close pair of peaks around 230 Hz, correspond to modes with a single nodal diameter: one of these is shown in the first Chladni pattern. The companion mode would have the nodal diameter rotated by $90^\circ$. For the perfect textbook drum in Fig. 3 of section 2.2, these two modes occurred at exactly the same frequency, but the real drum is not perfect, and the two modes have slightly different frequencies. This might be caused by non-uniformity in the mass distribution of the membrane, or by a slightly irregular distribution of tension: something always prevents real objects from having perfect symmetry, it only occurs in textbooks. For related reasons, the frequency of these modes relative to the lowest mode is similar to the textbook value, but not exactly equal to it.
Representative Chladni patterns for three more modes of the drum are also shown: one with one nodal circle, one with three nodal diameters, and one mode at a higher frequency with a more complicated shape not matching any of the patterns of the idealised theory. This is another universal phenomenon: as you go up the sequence of modes for any structure, the shapes become increasingly sensitive to small mechanical details, and sooner or later they cease to be recognisable in comparison to theoretical estimates. This remark does not only apply to our drum, it is equally true of large computer models of cars, buildings or aeroplanes.
Chladni patterns are not the only approach to visualising vibration patterns: more hi-tech options are also available. One way is the use a laser-Doppler vibrometer, mentioned earlier, to scan over a grid of points on the surface of the object, and then assemble animated views of the response at each peak frequency. Two examples are shown below, obtained from the stretched membrane forming the head of a banjo. The vertical scale is hugely exaggerated compared to the actual vibration when the banjo is in use. The banjo head exhibits similar shapes to the toy drum. The first one illustrated is similar to the Chladni figure with three nodal diameters, while the second one shows motion with two nodal circles.
Strictly, neither this approach nor the Chladni method really shows modes: these patterns are technically known as Operating Deflection Shapes, often abbreviated to ODS. They are the response to driving at a single frequency close to a response peak. The pattern will be dominated by the closest mode shape, but the motion will also contain some contribution from other modes. Evidence of this effect can be seen in the first of the animations: the pattern appears to rotate slightly during the motion. This is caused by a combination of the two modes which each have three nodal diameters: see the discussion in section 2.2.4. On the real banjo head these two modes do not have exactly the same frequency, so they contribute to this ODS with slightly different phases. The combination produces the effect of rotation.
There is an experimental technique which can, at least in principle, reveal true mode shapes rather than ODS patterns. It is called “experimental modal analysis”. A discussion of how this works will be deferred to section ?, but as a taster, here is an example of a mode shape of the body of a violin obtained by this method.
Returning to the toy drum, we can use it to illustrate a very different way to represent vibration or sound in graphical form. This way is known as “time-frequency analysis”, or the “spectrogram”, and it has a lot in common with the way our own ears process sound. In the description up to now, plots with time on the axis were turned into plots with frequency on the axis, using Fourier analysis and the FFT. But we don’t hear those as alternatives, we hear them somehow mixed up. Play a scale on a piano: how do you describe the sound? Surely, as a sequence of pitches, in other words as frequencies arranged in a particular pattern in time.
There are several ways to generate such a time-frequency description in the computer. The simplest is good enough to show the idea. We first get the sound or vibration waveform of interest into the computer by sampling it in the normal way for any digital recording. But instead of taking the entire chunk of sound and performing a huge FFT to turn the whole thing into a Fourier sine-wave recipe, we chop it into smaller segments and FFT each one separately. Stack the results next to each other in the order of the segments and, lo, we have a time-varying frequency spectrum. The pictures look nicer if we use a few processing tricks, such as overlapping the segments, but essentially that is all there is to it.
An example for the drum is shown in Fig. 5. The main plot shows the spectrogram, with frequency along the horizontal axis and time running vertically upwards. The colours in the plot show the distribution of energy in the sound, “hotter” for louder. To help relate this picture to the earlier ones, the original time waveform is plotted vertically on the left, synchronised with the time axis of the spectrogram. Similarly, the full FFT is plotted along the bottom, aligned with the spectrogram frequency axis. What the spectrogram shows is a set of vertical stripes, getting cooler as they go up. These are the individual modes of the drum, each vibrating at its own natural frequency, and dying away with time as the energy is dissipated into sound radiation and heat. You can listen to the sound this spectrogram is calculated from in Sound 1.
Looking closely at the bottom plot, it can be seen that the frequencies of the resonances are more sharply defined in the full FFT than in the spectrogram. Indeed at higher frequencies the separate resonant frequencies begin to overlap and blur together in the spectrogram. This is the price of trying to see time and frequency variation simultaneously. They are both aspects of the same thing, and the more you try to see details of variation in time, the more you blur the details in frequency; and vice versa. This is in fact an acoustical version of a famous result in quantum mechanics called the “Heisenberg uncertainty principle” — but it is easier to understand in acoustics!
The uncertainty effect does not only happen in the computer: your own hearing system also has to cope with it. Listen to the sound of a piece of cardboard whirring against the spokes of a bicycle wheel. If the wheel is going slowly, you hear individual little thud noises as each spoke passes. But when the wheel goes fast enough you hear a kind of note, with a pitch that goes up as the wheel speeds up. Somewhere between these two extremes you have switched from hearing “time stuff” to “frequency stuff”. Where exactly is the switchover? There is no precise answer, but it generally falls in the vicinity of 20 thuds per second. A cat’s purr happens at around about this rate: do you hear it as a low note or a fast-repeating sequence in time? This defines, a little vaguely, the lowest note you can perceive as a pitch (about 20 Hz), and at the same time the fastest temporal events you can resolve (around 1/20 second, or 50 milliseconds). Needless to say there is more to it than this: perception is endlessly subtle, but this will do us for the moment.
A few spectrogram plots of other sounds show the kind of things they can reveal. Figure 6 shows a single note on a guitar: you can hear it in Sound 2. It is broadly similar to the drum picture, with vertical stripes gradually cooling as they go up. But notice the regular pattern of the frequencies of these peaks. A stretched string has natural frequencies which fall (almost) in a harmonic pattern, and we will examine this more closely in the next chapter. Figure 7 shows a single note played on a violin, with vibrato: you can hear it in Sound 3. If you screw up your eyes a bit, or take your glasses off, this spectrogram might look quite similar to the previous picture, with a set of regularly-spaced vertical stripes. But the player’s vibrato leads to a regular ripple pattern, particular obvious at the higher frequencies. The fundamental frequency is being modulated by a few percent. Each harmonic inherits the same percentage variation, so as the frequency goes up the absolute variation gets bigger and more obvious. Notice that the FFT plot along the bottom shows only blurring of the peaks: the spectrogram analysis is far better at bringing out the structure of the “wobble”.
And now, in the famous phrase, for something completely different. Figure 8 shows a spectrogram of the sound of a Chinese orchestral gong or tamtam: you can hear it in Sound 4. The spectrogram has been smoothed to bring out the most important aspect clearly. Look at the very bottom of the spectrogram plot. This is the moment when the beater hit the tamtam. A soft beater was used, and it only excites vibration up to about 500 Hz. But as time goes on, something magical happens. There is a slanting “front” running up from about 500 Hz on the bottom line of the plot, above which hotter colours are seen. This is a striking demonstration of a nonlinear effect: over the first half second or so, energy which was initially concentrated at low frequencies makes its way, somehow, up to far higher frequencies. This is a visualisation of the characteristic “blooming” sound of the tamtam. No linear system can behave like this, and having seen this tantalising glimpse we will hurriedly return to simpler questions for the time being.