This is a good point in our story to make a small digression to investigate the question of how a wind instrument actually makes sound. All the simulation models we have developed led to predictions of the time-varying sound pressure inside the instrument, in the mouthpiece or in the acoustical cavity in the case of our free-reed models. But that is not the sound you hear when the instrument is played: for that, we need to think about how the instrument radiates sound into the surrounding air.
First, we need to recall and extend some things we learned back in Chapter 4. As will become clear shortly, we are mainly interested in sound sources that are small compared to the wavelength of sound, and that allows us to use an important simplification. There is a rather general theory for sound radiation from this kind of “compact sound source”, which gives a hierarchy of three different patterns of sound radiation. The next link gives some details.
Whenever there is a net change in volume, for example because of air flow in and out of an open tone-hole, it behaves like a monopole sound source. An idealised example of a monopole source, which we looked at in Chapter 4, is a pulsating sphere. A monopole is the most efficient type of sound source, and if it is present it will normally dominate over the other two kinds of source that we will come to in a moment. A monopole source radiates sound equally in all directions. The strength of the source is proportional to the rate of change of the volume flow rate, which means that sound radiation tends to be stronger at higher frequencies.
If there is no change of volume, then the most likely thing to occur is a dipole sound source. The example we saw back in section 4.3 was the result of a small rigid sphere oscillating backwards and forwards. Any other small vibrating rigid body will have a similar effect. Even if there is no physical vibration driving the sound, a dipole sound source can arise if an oscillating force is applied to the air, for example the drag force associated with oscillating flow round a sharp edge. A dipole sound source has a directional pattern, in a figure-of-8 shape. Figure 1 shows two examples of the pattern: it is a copy of Fig. 1 from section 4.3. The orientation of the figure-of-8 is determined by the direction of vibration of the rigid body, or by the direction of the oscillating force applied to the air.
Finally, in the absence of volume change or rigid boundaries that might apply a force, it is still possible for sound to be generated by the air-flow itself. The most familiar example is the sound generated by jet engines. The exhaust flow from the jet is turbulent, in other words a complicated and quasi-random flow pattern. Bernoulli’s principle tells us that the variations of flow speed within this turbulence must be associated with variations in pressure. However, most of the pressure changes are confined within the area of turbulence. Only a very small fraction of the energy of the flow is converted into sound that radiates away from the jet so that it can be heard by a distant listener. Small it may be, but of course the noise from jet engines can be a major source of nuisance and noise pollution.
Because of the importance of aircraft noise, considerable research effort has been devoted to understanding and ameliorating the problem. We do not need to go into the details here, but some of the understanding resulting from that research can be applied to certain aspects of the sound of musical instruments. The cornerstone of the theory of aerodynamic sound generation is a result known as “Lighthill’s acoustic analogy”. The previous link explains a little of the background. The important upshot for our purposes is that a sound source like this has the character of a quadrupole source. We gave a simple example of a quadrupole source back in section 4.3: the sound radiated by a vibrating tuning fork. As was explained there, a quadrupole source will have a directional pattern, typically involving four lobes of strong sound radiation which can fall in a variety of detailed configurations.
To understand the importance of this monopole/dipole/quadrupole hierarchy, we need to recall another thing we learned in section 4.3. For a monopole source radiating sound at a single frequency, the sound pressure amplitude falls inversely with distance. But for dipole and quadrupole sources, the pattern is more complicated. They have a “near-field” region within which the sound pressure decays more rapidly, and only when you get into the “far-field” region does this decay rate switch over to the same inverse-with-distance pattern as the monopole. The distinction between near and far field regions relies on the ratio of distance to the wavelength of sound: when that ratio is small, you are in the near field, when it becomes large you are in the far field.
Specifically, in the near field of a dipole source the sound pressure decays like the square of distance, while for a quadrupole source it decays like the cube of distance. Now suppose you had sources of the three types which had comparable pressure amplitudes at the edge of the source region. Within the near field, the sound from the dipole source will decay more rapidly that that of the monopole source, and the sound from the quadrupole source will decay even faster. The result is that by the time you reach the far field, where all three sources have similar behaviour with distance, the monopole source is the loudest, followed by the dipole source, followed by the quadrupole source.
Now to apply these ideas to the various wind instruments. The first instrument we looked into was the clarinet, playing its lowest note with all the tone-holes closed. For that, the sound radiation behaviour is fairly simple. All the complicated fluid dynamics around the reed and mouthpiece is hidden inside the player’s mouth, and so does not contribute significantly to external sound radiation. The acoustic pressure fluctuations inside the pipe can only radiate sound at the open end of the tube, the clarinet bell. The diameter of this bell is rather small compared to the wavelength of sound except at the very highest audible frequencies, so we can expect the dominant sound radiation to have a monopole character. The strength is governed by the volume flow rate out of the bell, more specifically by the rate of change of that volume flow rate. The bell is approximately a nodal point of pressure, but it is an antinode for particle velocity, so this flow rate can be large. The net result is that the sound from this note on the clarinet can be strong, and omni-directional.
Now what happens if a more typical note is played on the clarinet, with some tone-holes open? Each individual tone-hole is small compared to the wavelength of any audible sound, so the air-flow through each hole will create a monopole acoustical source. But the spacing between the tone-holes is not so small, so when we think about the combined sound radiation from all open holes as well as from the bell, we have to take into account wave interference effects. Furthermore, we saw in section 11.1 that the first open tone-hole does not act in the same way as sawing the tube short at that point. Other open or closed tone-holes further down the tube have an influence on the pitch (for example allowing fork fingerings to used). This demonstrates that they are involved in the internal pressure distribution, so all open tone-holes may radiate some sound.
The situation is shown schematically in Fig. 2. The relative phases of these various sound sources will determine exactly how they add up to give the combined sound radiation. This will result in some directionality in the sound field. Two extreme cases are easy to explain. If all the monopole sources were exactly in phase, they would add most strongly in directions perpendicular to the tube, broadside on to the row of tone-hole sources. Not just on the side where the holes are, though: the diameter of the tube is small compared to the wavelength of sound at most audible frequencies, so each individual monopole source will send sound more or less equally in directions all around the tube.
The opposite extreme would occur if the sound inside the tube were a pure travelling wave from mouthpiece to bell, with no reflected wave travelling back. In that case, each successive tone-hole would have a short time delay relative to the previous one, determined by the hole spacing and the speed of sound. These time delays would be reflected in the phases of the monopole sources, so that they would add together most strongly in the direction of travel of the wave. The strongest sound radiation would then be in the forward-facing direction.
The real situation will fall between these extremes: beyond the first open tone-hole the internal sound field will be dominated by a travelling wave towards the bell, but there will be a component of reflected wave. As first pointed out by Benade , the result is a directional sound pattern with its maximum amplitude on a forward-facing cone, making a small angle with the axis of the tube. Some direct measurements of clarinet directivity  confirmed this pattern, with a cone angle of the order of $20^\circ$.
Turning to brass instruments, the position is in some ways simpler. Apart from unusual instruments like the cornetto, brass instruments do not have tone-holes, and essentially all the sound is radiated from the bell of the instrument, as sketched in Fig. 3. But the bell of a brass instrument like a trombone or trumpet is much bigger than a clarinet bell. This means that although it is small compared to the wavelength of sound at very low frequencies, this ceases to be the case somewhere in the mid-range of frequencies.
The result is that the sound radiation is omnidirectional at low frequency, but becomes increasingly directional at higher frequency. The behaviour will be similar to an example we saw back in section 4.3, of the sound radiation by a circular piston set in a plane baffle: the plot from that section is reproduced here as Fig. 4. The curves in different colours show the behaviour at different frequencies, indexed by the “Helmholtz number”, which for this problem we can visualise as the bell circumference divided by the wavelength of sound. The practical consequence is something quite familiar. If you stand behind a trumpeter the sound is fairly mellow, but if you walk round to the front so that you are in the direct line where the bell is pointing, the sound becomes a lot more brilliant. Instruments that are used in musical contexts that emphasise mellowness and bass sound, like the French horn or the tuba, have bells that do not point directly towards the audience.
Sound radiation from the free reeds presents a very different situation from the woodwind reeds and the brass instruments. There is no doubt that a free reed is small compared to the wavelength of sound at audible frequencies, so it will be a compact sound source of some kind. We can see three different contributions that might be relevant. First, there is a mean flow of air past the reed, and this is modulated by the nonlinear valve effect as the reed moves towards or away from the base plate. This modulated volume flow rate will constitute a monopole source of sound, rather like the sound from a siren, in which a flow of air is interrupted by a rotating disc with holes in it.
A free reed can also make sound arising directly from its own motion, interacting with the air. This produces two effects, as we noted earlier for the idealised example of an oscillating sphere. But the result is not quite the same as for the sphere example. The reed is not vibrating in empty space, because the base plate acts as a kind of baffle. So the surface motion of the reed does not generate a dipole field: the air-flow created on one side cannot cancel with the flow on the other side because the plate is in the way. As a result, the volume flow rate driven directly by the reed motion should be added to the volume flow rate from the “siren” effect to contribute to the strength of the monopole source.
But the reed (and the oscillating sphere) also exerts a force on the air, and this will create a dipole sound field. For the reasons explained above, we might expect this dipole contribution to be relatively unimportant in the far field. This is especially true at the fundamental frequency of the note, because the near field will extend a relatively long way at this lowest frequency, giving plenty of scope for the near-field decay of the dipole field. But the higher harmonics of the note will have progressively smaller near-field regions (because the size of the region is governed by the wavelength of sound), so that the dipole contribution might be more significant.
Finally, we come to the air-jet instruments like the flute. We will discuss those in detail in the next section, but we can anticipate a little in order to say a bit about their sound radiation behaviour. In some respects a flute is rather similar to a clarinet. Volume flow from the end of the tube and from any open tone-holes will produce local monopole sources in much the same way as the earlier discussion.
But there is an extra ingredient, concerning the air-jet mouthpiece. Figure 5 is a repeat of a schematic diagram from section 11.1, showing the typical geometry. An air jet emerges from a channel, crosses a hole and then impinges on a sharp edge. The hole is called the “mouth” in a recorder, and the “embouchure hole” in a transverse flute. The channel is provided by the instrument maker in a recorder or a flue organ pipe, while it is formed by the player’s lips in the case of a transverse flute.
There are two types of sound source associated with the mouth opening. The first is similar to a tone-hole: there is a net volume flow of air in and out of the hole, and this provides a monopole source. The second is more complicated. The air jet may be laminar or turbulent, depending on the instrument and the particular playing technique in use. Especially when the jet is turbulent, it provides an aerodynamic sound source. This component is responsible for a broad-band “noisy” component of the sound: it may be described as “breathy” or “wind noise”. In some musical contexts, this “breathy” noise is an important part of the characteristic sound of the instrument — a particular example is the end-blown Japanese flute called the shakuhachi.
The turbulent flow constitutes a quadrupole source, and in empty space this would be a very weak sound source for the reasons discussed above. But the jet interacts with the sharp edge, and the resulting force generates a dipole sound source, significantly more effective at radiating sound. Even so, this component of the sound can be made more prominent in a recording using a close microphone, which can be within the near field.
You may by now be wondering whether there is another way that wind instruments can make sound, which I have been ignoring. Don’t the walls of the instrument vibrate, and radiate sound rather like the body of a stringed instrument? There is certainly a widespread belief among players and makers of wind instruments that the material of which an instrument is made is somehow important for the sound. I will discuss the issue of wall vibration, and then digress slightly to discuss other ways in which the material of a wind instrument might affect its sound.
The fact is that wall vibration in wind instruments is generally not very important. It is not that the solid material of an instrument cannot vibrate — the important issue is that those vibrations usually cannot be excited by the internal pressure field created by playing the instrument. The bore diameter of all wind instruments is generally small compared to the wavelength of sound. There are perhaps exceptions for the larger sections of instruments like the euphonium, but at the mouthpiece end, all instruments have small bores. This means that the internal pressure distribution is almost entirely in the form of more-or-less plane waves, with uniform pressure across any cross-section of the bore.
Uniform pressure like this cannot easily excite vibration in the circular wall of a woodwind or brass instrument, for the reason illustrated schematically in Fig. 6. The tube would have to “inflate” in a symmetrical manner, as indicated by the dashed line in the upper sketch. That would involve stretching the wall material, which is resisted by a very high stiffness. If the tube cross-section had been square, like the lower sketch, things would be different. Now, the tube can be “inflated” by bending the side-walls as sketched. If the walls were thin, the stiffness associated with that bending action would be much lower. If you can remember back to section 10.3, we applied a somewhat similar argument to explain why softwoods like spruce are much stiffer along the grain direction than in any cross-grain direction: long-grain deformation involves stretching the cell walls, but cross-grain deformation can take place by bending them, with much lower stiffness.
Of course something like a clarinet tube will have vibration resonances, but the most obvious ones at relatively low frequencies would involve the tube behaving as a bending beam. Bending vibration like this cannot be driven by internal pressure variations in a narrow tube, for a reason that follows directly from a symmetry argument. If we ignore the holes, keywork and so on, the circular tube is symmetric in any plane through the central axis. Uniform pressure variations are symmetrical with respect to a mirror reflection in that plane, but bending motion in the plane is anti-symmetric (i.e. it reverses if you reflect it in a mirror formed by the plane). A symmetric force cannot excite an anti-symmetric motion. In the real world, with tone-holes and so on, the mirror symmetry is not perfect. That means that there might be weak but non-zero coupling between internal pressure and tube bending, but the effect is very small. Even then, bending vibration of the tube would not radiate much sound because the tube diameter is small compared to the wavelength of sound. For practical purposes, the sound of a wind instrument is not affected by wall vibration.
Admittedly, there are some exceptions under unusual circumstances. Some wooden organ pipes have a square cross-section rather like the lower sketch in Fig. 6. Way back in 1909, Dayton Miller  did an ingenious experiment to demonstrate that wall vibrations of such pipes could sometimes couple to the internal pressure, and have a detrimental effect on tone. In one experiment he made a double-walled pipe, with a space in between that could be filled with water. He demonstrated that for certain heights of the water filling, the sound was affected because a bending resonance of the walls matched a harmonic of the played note. Organ builders take some pains to avoid such coincidences of frequency, mainly by making sure that the wooden walls of pipes are quite thick.
A rather different kind of “exception” is associated with the convoluted tubing of brass instruments. Figure 7 shows a U-bend in a tube, as a simple example of the kind of thing that can be found in a trumpet or French horn. Suppose the pressure inside this U-bend is raised. That pressure exerts an equal force per unit area on all the walls of the tube. There is no net force in the vertical direction or in the direction perpendicular to the diagram, because the arrangement of pipe walls is symmetrical in those two planes and so the pressure force cancels out.
But this does not happen in the horizontal direction. The right-hand sketch in Fig. 7 shows an end-on view of the U-bend. The total projected area of pipe wall facing away from the viewer is somewhat bigger than the total projected area facing towards the viewer, because of the two holes shaded grey. This means that the raised pressure produces a net force towards the right in the left-hand sketch. So the time-varying pressure when a note is played produces an oscillating force, which may excite vibration in the rest of the tube-work. This may not radiate a lot of sound (although axial vibration of the instrument bell might behave a bit like a piston), but it can certainly be felt by the player’s hands.
However, wall vibration is by no means the only way that the sound and playing properties of a wind instrument might be influenced by the material they are made of. We can start with the broad distinctions: instruments may be made of various kinds of metal, wood or plastic. A key conclusion from what we have said previously is that the acoustical behaviour is determined almost entirely by the precise shape of the internal bore of the instrument (including details like the internal profile of closed tone-holes). There is no reason in principle why an identical bore shape could not be built from any of these materials.
But in practice things are not so simple. Different tools and working methods are used with the different types of material, and these tend to leave their trace in subtle differences of detail. And there is the most important factor of all, the skill of the instrument maker, and how long they are prepared to spend tweaking details to get things just right. Think about the extreme cases. On the one hand a plastic recorder is likely to be mass-produced, with relatively low standards of quality control. These can still be quite good instruments, depending on how much care was originally taken in making the mould template and also on how good a job the manufacturing process makes of reproducing tiny details.
But contrast that with a hand-made wooden recorder, let alone a high-end instrument like the famous golden flute of James Galway. Such instruments are, of course, far more expensive than the plastic recorder. What you are paying for is not, mainly, the cost of the material as such, but the skill of the instrument maker. The intrinsic cost and beauty of the material tends to move up in step with the level of effort devoted to making the instrument, but it is not the gold or the boxwood as such that determines the quality of the final product: it is the combination of the material characteristics with the maker’s skill and dedication.
So what are these all-important details that a skilled maker must get right? It is hard to give an exhaustive and definitive answer to that question, but we can give some examples. First, and simplest, are the mechanical things: keys, valves and trombone slides must work smoothly without leaking, key-pads must seal their holes crisply. But these are not really material-related issues. More immediately relevant is the internal surface finish of the bore. One important source of energy dissipation in an instrument tube comes from viscous effects in the boundary layer. Surface roughness of any kind will tend to increase dissipation and reduce the Q-factors of tube resonances. Roughness may come from manufacturing: tool marks from the process of boring a hole in a piece of wood, whiskers of wood or metal left over from drilling tone-holes, or surface imperfections from casting a plastic instrument.
In the case of a wooden wind instrument, the internal surface finish may also vary with exposure to the warm, moist air that the player blows down the tube. Some types of wood are relatively immune to being influenced by humidity — but these are generally tropical hardwoods, which are beginning to raise other important issues of sustainability and the restrictions imposed by the CITES rules. Historical wind instruments were (and are) often made of European hardwoods such as boxwood or pearwood, and those timbers certainly are susceptible to the effects of humidity, which can “raise the grain” on the internal surface of the bore and thus increase energy dissipation. A standard way to guard against this is to oil the wood, traditionally by applying oil to the inside of the bore with a long feather.
More subtly, the inside edges of tone-holes may be square, or chamfered, or rounded — see the sketches in Fig. 8. The same is true for other discontinuities in the internal shape, such as the exit of the air-jet channel in a recorder. Adjusting these edge details will have a small influence on the tuning of resonances, but it will also influence the shedding of vortices by the air-flow around the hole, which in turn influences energy dissipation and the Q-factors. Tweaking these things is a major part of the process of “voicing” an instrument. Naturally, the voicing process will use different tools in a wooden or a metal instrument, which may in itself have consequences for the end result. It would be unusual to carry out such voicing at all on a plastic instrument — but it is not impossible. I have a clear memory of seeing Arthur Benade demonstrating voicing adjustments on a plastic recorder, back in the 1970s.
 Arthur H. Benade, “On the mathematical theory of woodwind finger holes”, Journal of the Acoustical Society of America 32, 1591—1608 (1960)
 Fumiaki Ehara and Shigeru Yoshikawa, “Radiation directionality measurement of clarinets made of different wall material”, Forum Acusticum 2005, 525—528 (2005), available at http://www.conforg.fr/acoustics2008/cdrom/data/fa2005-budapest/paper/449-0.pdf
 Dayton C. Miller, “The influence of the material of wind instruments on the tone quality”, Science 29, 151—171 (1909).