Even without the added complication of room acoustics associated with a performance space, the way that sound is radiated by a musical instrument (or any other vibrating structure) is a complicated business. The details depend very strongly on the Helmholtz number: the size of the sound source compared to the wavelength. The sequence of behaviour is somewhat similar to the shadowing example we saw earlier, in section 4.1. Sources with very small Helmholtz number give simple behaviour, amenable to analysis. Intermediate Helmholtz number, around unity, gives the most complicated behaviour; but then very high Helmholtz number restores a measure of simplicity. For musical instruments, as we will see shortly, very small Helmholtz number is somewhat of a rarity. Despite this, the analysis of that case gives rise to some concepts of such importance that we will devote a disproportionate amount of space to it.
A. Small sources: monopoles, dipoles, quadrupoles
When sound is generated in a region that is very small compared to the wavelength, there will be a near-field region within which there may be vigorous motion of the air, but from the point of view of the far-field radiated sound the details of this local motion don’t matter very much. As was explained in section 4.1.3, the non-radiating part of the near-field disturbance is associated with motion of the air that is essentially incompressible. You can picture it by imagining the same source distribution vibrating very slowly, but in water rather than air. The water will flow back and forth around the vibrating object, but it will “get out of the way” rather than being compressed to form sound waves. This kind of near-field flow will carry kinetic energy, and may have a strong influence on the reaction forces experienced by the vibrating object, but it will not influence the far-field sound.
That far-field sound can be described in terms of a combination of rather simple patterns, two of which we have already met. If the vibration of the source involves a net change of volume during the cycle, then the far-field sound will have a component looking exactly like the pulsating sphere example we saw in section 4.1. This is called a monopole source. It radiates sound equally in all directions.
But if the sound source involves vibration without any net volume change, it can’t generate monopole sound. We saw an example in Fig. 4 of section 4.1, with two adjacent spheres pulsating in opposite phases so that the net volume change is zero. This is called a dipole source. The radiated sound is weaker because of cancellation between the two sources, and it is no longer the same in all directions. It has a figure-of-eight directional dependence, which we can represent in a polar plot like the ones we saw for shadowing, in Fig. 7 of section 4.1. Figure 1 shows two examples, corresponding to different orientations of the pair of sources. The two sources are indicated schematically by red and blue symbols connoting the two different phases of pulsation.
We can take the cancellation idea one stage further. If two dipoles with opposite sign are placed side by side, with all spacings between spherical sources small compared to the wavelength, the result is a quadrupole source. Two examples are shown in Fig. 2: the detailed directional pattern depends on whether the two cancelling dipoles are placed side by side, or end-to-end.
You may be wondering what these arrangements of pulsating spheres have to do with musical instruments. The first step in answering that question is to realise that there is something more familiar that generates a dipole field. If we take a single rigid object and shake it from side to side, what kind of radiated sound field will be generated? There will be no net change of volume, so it is not a monopole source. But as it vibrates in one direction, it is pushing at the air on the front face, while pulling on it with the back face. This is a little like having a positive source at the front, and a cancelling negative source at the back: exactly the recipe for a dipole source. The next link gives details, showing that this intuitive argument is in fact the right answer. A vibrating rigid object is a dipole source, provided that it is small compared to the wavelength of sound associated with the frequency of vibration.
We can go further. There is a familiar object which illustrates a quadrupole sound source: the tuning fork. When a tuning fork vibrates at the frequency marked on it, the two blades of the fork are moving in opposite directions, as sketched in Fig. 3. Each blade is a dipole source, and the two are arranged along a straight line so that they tend to cancel: just like the right-hand diagram in Fig. 2.
You can easily use a tuning fork to appreciate an important feature of a quadrupole sound field. Bang a fork to get it vibrating, then hold it at arms length. You will hardly be able to hear it: the far-field sound radiation is very weak because of the cancellation effect. But bring it very close to your ear, and you will hear it loud and clear. Now move it slowly away from your ear. The sound dies away much more quickly than would happen with the sound of the small loudspeaker in your mobile phone, for example. When you hold the fork near your ear you are in the near field, and for reasons explained in the previous link the near-field components of dipoles and quadrupoles decay faster than the $1/r$ variation of far-field sound. Furthermore, quadrupoles decay even faster than dipoles. This is what makes the sound of the tuning fork behave in that unfamiliar way.
Figure 3 also illustrates a demonstration of the influence of quadrupole cancellation on the loudness of the radiated sound, attributed to Stokes. Make a tuning fork vibrate vigorously, then bring something like a table knife very close to one corner of the vibrating fork (but without touching it). The right-hand sketch in Fig. 3 shows a top view of the tuning fork, and indicates a good position of the knife. Once the blade gets near enough, you should hear the sound get louder. The knife blade has interfered with the near-field flow of air, and the cancellation can’t work so thoroughly.
B. Musical applications of monopoles and dipoles
It is useful to summarise what we have learned so far. Any source of sound that is small compared to the wavelength will work most effectively if it produces fluctuating volume, so that it is a monopole source. Dipole and quadrupole sources are progressively less efficient at generating far-field sound. But even a monopole source has important limitations. As was shown in the previous link, the sound pressure does not depend explicitly on the size of the source, but on the product of frequency and total volume flux. So the bigger the volume flux, the louder the sound. But for any practical device there is always a limit on the volume flux that can be achieved, and then the factor of frequency tells us that the sound pressure will inevitably drop off at lower frequencies. A loud sound source at low frequency simply cannot be small.
A familiar sound generator capable of high volume flux and hence loud sound is the siren. Air is pumped under pressure through a small outlet, and the flow is periodically interrupted by a rotating disc with a ring of holes in it, or some similar system. The result is that the volume flux of the air flow from the outlet is modulated, making a strong monopole source. A somewhat similar effect is produced by our own vocal chords, modulating air flow from our lungs: a singer can indeed be quite loud. Even more striking, perhaps, is bird song. The British wren, for example, is a tiny bird with a startlingly loud and piercing song. Both factors are significant: the bird’s vocal apparatus produces a modulated volume flux, but they also sing at relatively high frequency. But even so, the size of a wren’s mouth is small compared to the wavelengths of sound they produce.
One other non-musical example is worth mentioning here. Have you ever noticed that the only time you hear sound from flowing water is when it has bubbles in it? A bubble in water can have many possible resonances, but one of them exactly matches our “pulsating sphere” example. A small spherical bubble can pulsate symmetrically, because the air inside it is not very stiff compared to the surrounding water. Unsteady forces arising from other aspects of the flowing water can excite this resonance of the bubble, and it makes a monopole sound source. Without bubbles the water flow is essentially incompressible, so however complicated the flow pattern is, little sound is made. A violent river in spate can be full of whirlpools and other exciting flow details, while being ominously silent. A trickle of water falling over a pebble and making a few bubbles can immediately be heard.
In terms of musical instruments, the closest parallel with the siren or the bird song might be an instrument like the clarinet. Sound is radiated from the bell at the end of the instrument or, more commonly, from open toneholes further up the tube. In both cases there is oscillating airflow of some kind through rather small orifices, giving a monopole sound field. (If more than one hole radiates significant sound, the spacing may violate the “small Helmholtz number” requirement, but we won’t enquire into this detail now.) For brass instruments, essentially all the radiated sound comes out of the bell. For low frequencies the Helmholtz number can be quite small and the sound field omnidirectional, but the higher overtones in the sound will violate the assumption. The sound becomes increasingly directional, beaming predominantly forwards as is only too familiar to anyone who has sat in front of the brass section in an orchestra.
The situation with stringed instruments is very different. A vibrating string cannot radiate much in the way of sound waves into the surrounding air; it is far too thin compared to the wavelengths of sound in air at audio frequencies. Most musical strings are only around 1 mm in diameter, whereas the wavelength of sound in the mid-audio range is of the order of hundreds of millimetres, and even at the very highest frequency audible to humans, around 20 kHz, it is about 17 mm. So a string is a very weak dipole source of sound. A simple illustration of this weakness is given by the (lack of) sound of an unplugged electric guitar.
The purpose of the body of an acoustic stringed instrument is to extract some of the energy from the vibrating string, and use it to excite a more efficient sound source (the details will be explored in Chapter 5). So it is no use having an instrument body that is too small, or it will still be a weak radiator of sound, especially for the lowest notes of the instrument. On the other hand, players do not want instruments to be too cumbersome to handle and carry around. The result is that instrument makers have probably settled on a compromise in which the bodies of instruments are just big enough to allow sufficient radiation of sound at the lowest frequencies of interest.
We can estimate some numbers to test this idea. Of course, there is no very precise value for the “radius” of a guitar or violin body. But a simple estimate can be made by noting that for a sphere, the conventional Helmholtz number is the radius times the wavenumber, and this is the same thing as the ratio of the circumference to the wavelength (because the factors of $2 \pi$ cancel out). Putting a tape measure round a guitar at the widest part of the body, roughly along the line of the bridge, gives a “circumference” of approximately 0.9 m. Doing the same thing for a violin gives 0.5 m. For the guitar, the lowest note in normal tuning is $E_2$ at 82.4 Hz, giving a Helmholtz number around 0.22. The frequency at which this measure of Helmholtz number would reach unity is about 380 Hz. For the violin, the lowest tuned note is $G_3$ at 196 Hz, giving a Helmholtz number about 0.29. It would reach unity around 680 Hz.
Both the guitar and the violin have “smallish” Helmholtz numbers at their lowest frequencies. The lowest useful resonances of a guitar are given by the pair of modes we looked at in section 4.2, based on coupling of the Helmholtz mode of the cavity to the first mode of the top plate. These two modes typically occur around 100 Hz and 200 Hz, and we now see that the estimated Helmholtz number remains below unity over enough of a frequency range to encompass these two modes, but only just. One would want both modes to involve volume change so that they behaved like monopole sources, and that indeed is what we saw in the animation of Fig. 8 in section 4.2. There is a similar story for the violin, although the mode shapes are rather more complicated: some images will be shown in section 5.3.
As a final example of using volume fluctuation to improve sound radiation, we can look back at the xylophone which we discussed in section 3.3. Here is a repeat of the picture shown there:
Each note of the instrument has a resonator beneath it, which is an open-closed tube tuned to the fundamental frequency of the note. We can now see what these resonators are for. A xylophone bar will vibrate with no volume change, and it is small enough to have small Helmholtz number, so on its own it would be a rather weak dipole source of sound. But the acoustic resonance of the tube does involve net volume change: air will flow in and out of the open end. By tuning the resonator to the same frequency as the bar, the two modes will couple together in a somewhat similar way as happened in the guitar body, resulting in combined motion that involves some volume change and thus creates monopole radiation.
C. Larger sound sources
As frequency goes up, so does the Helmholtz number. From a mathematical standpoint it is still possible to use monopoles, dipoles, quadrupoles and so on to describe the far field radiated sound, but it rapidly ceases to be very useful. Suppose we have a violin body, vibrating at some chosen frequency and sending out sound waves. Choose a distance that is far enough away to be in the far field, and imagine a sphere at that radius, enclosing the violin. With a bit of effort, the sound pressure can be measured at a lot of points covering that sphere, leading to a directivity pattern for the sound. Some examples for a violin are shown in Figs. 5–8, based on measurements by George Bissinger using a procedure described in .
There is a set of mathematical functions called spherical harmonics, which can be used a bit like the sines and cosines of a Fourier series (remember them from section 2.2.1?). It is always possible to express the measured directivity pattern as a combination of these functions. These functions match exactly the far-field radiation of monopoles, dipoles etc. The snag is that once you are in the far field, all these components of the sound field decay at the same rate, proportional to $1/r$. We can no longer make the argument that the monopole component will dominate over the dipole component, and the dipole will in turn dominate over the quadrupole component. That argument relied on the fact that when the Helmholtz number was very small, there was a significant region in which the near-field behaviour dominated, and it was the respective near fields of monopole, dipole and quadrupole terms that had very different decay rates, leading to the hierarchy of dominance.
Instead, the directivity patterns have complicated shapes, progressively more so as frequency goes up. We get an idea of this from the examples above. Figure 5 shows the measured pattern at the lowest resonance frequency, the Helmholtz-like resonance called “A0” in the cryptic jargon of violin acoustics (see section 5.3). The pattern is virtually spherical: this mode has net volume change, making it a monopole source, and it has a sufficiently small Helmholtz number that this monopole term dominates the behaviour. Figures 6–8 show progressively higher frequencies, giving examples of the kind of thing that happens as frequency goes up. They all have non-spherical shapes of one kind or another. Figure 6 is elongated in the vertical direction (along the length of the violin body), Fig. 7 shows more radiation towards the front (in green) than towards the back (in red). Figure 8 shows something like a 4-lobed pattern: I have chosen a different viewing angle from the other cases to show it most clearly. Finally, Fig. 9 shows the directional pattern for the same violin, evolving as a function of frequency.
The patterns continue to become more complicated as frequency goes up. They also vary rapidly with frequency, and by mid-kHz frequencies the pattern varies fast enough to show significant changes from one semitone to the next, and even within the range of a violinist’s vibrato. That means that when a note is played with strong vibrato, among the other complicated things that happen the directional pattern of each varying harmonic of the sound may have narrow lobes that are swinging around like beams from a lighthouse. This phenomenon, combined with the acoustics of a concert hall, may contribute to an effect that has been called directional tone colour . We will return to this issue later: see section ?.
D. Sources very large compared to the wavelength
When frequency gets so high that the size of the source region is very large compared to the wavelength, simplified models can often be used to give clues about how the radiated sound will behave. The reason is that once the source region is large enough, it may not make all that much difference to the local behaviour if we assume it is infinitely large. Infinite systems may seem physically implausible, but they often lead to easier mathematical models.
An important example of this is the idea of a baffle. We already mentioned the possibility of using the result for sound radiation from a small monopole source to describe more complicated sound fields. The idea is simple enough. Imagine a vibrating violin body, like the one producing the measured radiation patterns in Figs. 5–8. If we know the pattern of vibration on the surface of the instrument, we could imagine dividing that up into a lot of very tiny pistons, each one capturing the vibration in its own little area. Each of these pistons will have very small Helmholtz number, and will radiate monopole sound waves. We could imagine using a computer program to add up all these contributions to give the complete sound field.
But there is a snag: the body of the violin will produce a shadowing effect, so that the pistons on the front surface will not be able to radiate sound very effectively towards the back, and vice versa. Solving this shadowing problem is much more complicated than just adding up all the monopoles from the little pistons. But at very high frequency, we might be able to get away with a trick. There is a special case, for which the shadowing problem goes away. If the vibrating object was an infinitely-extended plane, rather than a complicated 3D shape like a violin, there is no issue of shadowing. The little piston contributions can be added up, using a formula called the Rayleigh integral which is explained in the next link.
If the top of the violin was surrounded by a rigid plane stretching off to infinity, we could use the Rayleigh integral. There is no such plane, of course, but at very high frequencies it might not matter very much to assume that there was an ” infinite baffle” like this, in order to get at least an approximate idea of how much sound is radiated by the violin top. We won’t look at a complicated shape like a violin, but there is a very simple problem which illustrates the idea, sketched in Fig. 10. We have a circular piston embedded in an infinite baffle, and the piston is made to vibrate in and out as a rigid body: this is a crude model of a loudspeaker in a rigid enclosure.
As explained in the previous link, the Rayleigh integral can be used for this problem to give a closed-form mathematical answer to the directional pattern of sound radiation in the far field. Some examples are plotted in Fig. 11, associated with different values for the Helmholtz number $ka=2 \pi f a/c$, where $a$ is the radius of the piston and $f$ is the frequency in Hz. When $ka$ is small the radiation is essentially omnidirectional, as we should by now expect. As $ka$ increases, the pattern gets more and more focussed into a beam of sound heading straight ahead from the piston, with very little being radiated in other directions. This shows immediately why your hi-fi loudspeakers need a “tweeter”, a small loudspeaker unit whose job is to radiate the higher frequencies. A smaller value of $a$ means that the beaming effect is put off until higher frequency, otherwise you would only be able to hear the high-frequency sound by sitting directly in front of your loudspeaker.
Real loudspeakers do not, of course, have this infinite baffle. But this simple model gives a good semi-quantitative impression of how a speaker will behave: once the beaming effect gets established, shadowing by a real finite-sized loudspeaker cabinet doesn’t matter very much because most of the sound is being projected forwards. The same model gives at least a qualitative idea of the sound radiation from the bell of a brass instrument like a trumpet. For this case there isn’t even a cabinet looking a little bit like the infinite plane baffle, but the trend shown in Fig. 11 still gives the right impression, of sound increasingly concentrated into a beam heading straight ahead as frequency goes up.
An interesting, and slightly counter-intuitive, example of directional radiation is illustrated in Fig. 12. In the days before GPS navigation, an important way to improve safety for ships in dangerous waters was to use foghorns to give an audible warning. Lord Rayleigh, who we have met several times before, was consulted about an observation that was puzzling the authorities: ship’s captains reported that in certain positions they were unable to hear the foghorn, even when they were quite close. Rayleigh realised the problem: the horns were too large, so that the sound radiation was strongly directional. At certain angles, the radiated sound was very weak. The cure was to use a horn with an elliptical section. The aim is to spread the sound as uniformly as possible in the horizontal plane, while concentrating it in the vertical so that sound energy was not “wasted” by being sent up into the sky. This requires a horn that is larger than a wavelength in the vertical direction, but smaller than a wavelength in the horizontal direction, exactly as seen in the picture.
To finish this chapter, we look at another important phenomenon which is most easily understood by thinking about infinite systems. This concerns the radiation of sound by the vibration of a bending plate. We are accustomed to the idea that there is a single speed of sound, the same for all frequencies. But for bending waves in plates or beams, things are more complicated. At low frequency, bending waves travel slowly, but the speed increases as frequency goes up. This has an important consequence. For sufficiently low frequencies, waves in a plate travel slower than the speed of sound. As explained in the next link, this means that they cannot generate far-field sound waves in the adjacent air. But at higher frequency, the plate waves move faster than the speed of sound, and now they can generate sound waves. Animations of the two cases are shown in Figs. 13 and 14. The crossover frequency, where the plate waves move exactly at the speed of sound, is called the critical frequency of the plate.
This has important consequences for many things, including the transmission of sound through buildings or into the passenger compartments of vehicles. The design of soundproof partitions and double glazing systems must take the critical frequency into account, because the behaviour will be different above and below that frequency.
Our interest in the idea is mainly concerned with radiation of sound by the bodies of stringed instruments. Things are made a little complicated by the fact the soundboards of stringed instruments are usually made of wood like spruce, which has very different stiffnesses in the directions along the grain and across the grain. For such plates there is not a single critical frequency: waves travelling at different angles to the grain will have different critical frequencies, so the transition is spread out over a range of frequencies. The broad behaviour is still the same, though. At high frequency, any bending plate can radiate sound efficiently, but at low frequency it is much less efficient.
Now, the argument from an infinite plate suggests that at low frequency a vibrating plate could radiate no sound at all! This would be rather embarrassing for the design of musical instruments. Fortunately, finite plates escape from this fate: a plate of finite size can always radiate some sound even at frequencies well below critical. The efficiency with which they can do so varies from mode to mode, governed by another variant of Helmholtz number. What matters here is the wavelength of plate vibration compared to the size of the instrument. Low modes involving just a few half-wavelengths can radiate relatively well, but modes involving more wavelengths behave in a way that is closer to the infinite system, and do not radiate well until the critical frequency is passed.
 George Bissinger and John Keiffer: “Radiation damping, efficiency and directivity for violin normal modes below 4 kHz”, Acoustics Research Letters Online 4 (2003), DOI 10.1121/1.1524623.
 Gabriel Weinreich: “Directional tone color”, Journal of the Acoustical Society of America, 101, 2338 (1997).