So far we have talked a lot about vibrating structures, but surely music is about sound, rather than vibration as such? It is time to think about sound waves — how they are made, and how they travel through the air and interact with the surrounding environment. Sound waves consist of fluctuations of pressure in the air. Usually, the fluctuation is a very small fraction of the ambient atmospheric pressure, and that means that, as with small-amplitude mechanical vibrations, most relevant aspects of musical sound can be understood using linear theory.

As explained in the next link, linearised sound waves can be modelled with a differential equation which is already familiar from earlier sections, called the *wave equation*. We have seen the one-dimensional version of this equation for the ideal stretched string (see section 3.1.1), and the two-dimensional version for a stretched membrane (see section 3.6.1). Sound waves obey the three-dimensional version: but the one-dimensional wave equation also has a simple physical interpretation for sound waves. A *plane wave* is one in which the air pressure only varies along one direction, such as the $x$ coordinate direction. The molecules of air oscillate back and forth parallel to the $x$-axis, and the wave travels in the $x$ direction. Figure 1 shows an animation of the result, for a sinusoidal waveform.

The wave equation shows that sound waves travel through air (or through water, or any other medium) at a certain speed, determined by the density, atmospheric pressure and thermodynamic properties. The speed of sound in air is somewhat dependent on the temperature and humidity, but under normal conditions it is around 340 m/s. A familiar consequence of this relatively slow speed is the delay between seeing a lightning flash and hearing the thunder: the light from the flash travels at the speed of light, so fast as to be virtually instantaneous in the context of watching a thunderstorm.

The speed of sound tells us a very useful thing straight away: it relates the *frequency* of a sound wave to its *wavelength*, the distance between successive positions where the pressure fluctuation achieves a maximum value in a travelling sinusoidal wave. The relation is very simple: the wavelength (in metres) is given by the speed of sound (in m/s) divided by the frequency (in Hz). As we saw earlier, the lowest frequency humans perceive as a musical pitch is around 20 Hz, and this corresponds to a wavelength of 17 m. At the opposite extreme, the highest frequency we can hear is around 20 kHz (and you need to be quite young to hear that high). This corresponds to a wavelength 1000 times shorter, 17 mm. Wavelength will turn out to be important in several different ways. The simplest will explain the fact, as we will see shortly, that this range of lengths encompasses the range of sizes of musical wind instruments: organ pipes, flutes, trombones and so on. Most instruments are concentrated in the middle of the range, just as most musical notes are intermediate between the extremes of pitch that we can perceive.

As well as the plane wave, there is another simple solution to the wave equation which gives a useful component for thinking about more complicated sound fields later on. If we write the equation in spherical polar coordinates and then look for a solution that only depends on the radial distance $r$, we can obtain a spherically symmetric wave field: details are given in the next link. This solution can be used to describe a wave being sent out by a pulsating sphere, as illustrated in Fig. 2. The pressure varies sinusoidally with $r$, and the amplitude dies away proportional to $1/r$.

This simple example allows a first glimpse of a characteristic feature of many acoustical problems, which often contributes to making acoustics inherently more complicated than the structural vibration we have talked about up to now. It is all to do with length scales. There are three different length scales of interest here: the radius $a$ of the pulsating sphere, the wavelength $\lambda$ of the sound, and the distance $r$ to the observer. (The Greek letter $\lambda$ is pronounced “lambda”.) We learn something important if we express both $a$ and $r$ as multiples of the wavelength, or equivalently if we introduce *non-dimensional ratios* $a/\lambda$ and $r/\lambda$.

These ratios, between them, allow the qualitative behaviour of the wave field to be classified into different regimes. The idea is similar to the use of the Reynolds number to classify qualitatively different regimes of fluid flow. The ratio $a/\lambda$ is called the *Helmholtz number*, or sometimes the *compactness ratio*. It describes in wavelength terms the size of the object making the sound: our pulsating sphere, or the body of a violin, for example. When the Helmholtz number is small, meaning that the sound source is much smaller than the wavelength, there is a useful approximate way to understand sound radiation: we will meet it in section 4.3. When the Helmholtz number is moderate, so that the size of the source is comparable with the wavelength, things get more complicated because new phenomena will come into play: we will see examples shortly. When the Helmholtz number is very large, the source is very large compared to the wavelength, and a different type of approximate analysis becomes possible, which we will also meet in section 4.3.

The second non-dimensional ratio doesn’t have a standard name, but it describes the distance of the observer in wavelength terms. For many sound fields the behaviour is different when the observer (or measuring microphone) is in the *far field*, where this ratio is large, compared to the behaviour in the *near field*, where the ratio is small. This is the case in the simple sphere example we have considered here: the details are given in the previous link. For a musical application of the idea, think about a violin. The player’s ears are very close to the vibrating body of the instrument: they are in the near field. A listener in the concert hall audience is likely to be many wavelengths distant, though: they are in the far field. When people talk about a violin sounding different “under the ear” from “the sound in the hall”, this far field/near field distinction is a key factor.

Now, the important thing about these two non-dimensional ratios is that they allow us to distinguish broad *types* of behaviour that are different. It is not the precise numerical values of these numbers that matters, so much as the distinction between “very small”, “moderate” and “very large”. So far I have defined these numbers in a particular way, based on the wavelength $\lambda$. But I have cheated a bit. When the mathematics of the approximate solutions is worked out, the natural quantities that arise are based on something called the *wavenumber*, usually called $k$. This is defined as the inverse of the wavelength, except that a factor $2 \pi$ comes in: $k=2 \pi / \lambda$. So the most familiar form of the Helmholtz number for the sphere problem is written $ka$: if you like, you can think of it as the ratio of the *circumference* of the sphere to the wavelength, because the factor $2 \pi$ cancels out. I will shortly show some particular examples, and when it comes to labelling those I will usually use values of $ka$ or $kr$ for the specific values used in the calculations.

In the remainder of this section, we will introduce some important qualitative aspects of sound waves and their interaction with mechanical objects: the ideas of *impedance*, *intensity*, *wave interference*, *diffraction* and *shadowing*. First, we look at the energy carried by a travelling sound wave. As explained in the next link, the *intensity* of a sound wave is a vector which is the product of the pressure and the particle velocity. It describes the rate and direction at which acoustic energy crosses a unit area of space. For both the plane wave and the spherical wave discussed above, the intensity is given by the expression $\frac{1}{2} \frac{p^2}{Z}$ where $p$ is the amplitude of the sinusoidal pressure, and $Z$ is a constant called the *characteristic impedance*. For the spherical wave from the pulsating sphere, we saw that pressure dies away with the inverse of distance $r$, so the intensity satisfies the *inverse square law*, dying away proportional to $1/r^2$. This is exactly what you would expect: there is no dissipation of energy, so the total power crossing a spherical surface at any distance $r$ must be the same. The area of the surface grows with $r^2$, so the intensity must indeed decay like $1/r^2$.

Perhaps the most characteristic property of waves of any kind, including sound waves, is called *interference*, or sometimes *phase cancellation*. Suppose we have two identical sound sources, sitting side by side and producing sound at the same frequency (for example, a pair of loudspeakers of a stereo system). Provided linear theory applies, the sound waves generated by the two sources each behave as if the other source was not there, and the combined sound that you would hear, or pick up with a microphone at a particular position, is simply the sum of the two.

Now, if the two sources were to produce identical, synchronised, sound waves, the result should simply be the same sound wave with twice the amplitude. However, if the sources were in *opposite phase*, the sound waves would be “equal and opposite”, and they would cancel each other out. If the match of the two fields was perfect, the combination of two sounds would be silence! It is hard to achieve perfect cancellation like this in practice, but still this effect has some very important consequences for music and musical instruments (and in many other situations involving acoustics). For example, it is how “noise-cancelling headphones” work, often used for listening to music in noisy environments such as on an aeroplane.

We can investigate this effect quantitatively by combining two of the spherical sound sources just discussed. Figures 3 and 4 show the pressure fields resulting from two pulsating spheres in close proximity, vibrating with the same amplitudes either in phase (Fig. 3) or in opposite phases (Fig. 4). The Helmholtz number based on the separation of the centres of the spheres is 1.3 for this case. (Specifically, this Helmholtz number is defined as $ka$, where $a$ is the separation of the centres and $k$ is the wavenumber of the sound wave, introduced earlier and defined in terms of the wavelength $\lambda$ by $k=2 \pi/\lambda$.) Figure 3 shows a pattern which, apart from some near-field details, is virtually indistinguishable from that of a single pulsating sphere. Figure 4 is quite different. The pressure is consistently lower because of cancellation effects, and along the horizontal mid-line the cancellation is perfect and the pressure remains zero. Sound radiation is strongest in the vertical directions, either upwards or downwards, along the line of centres of the two spheres. This pattern is called a *dipole* field, and we will come back to it in section 3.3.

Figures 5 and 6 show a similar comparison with a larger Helmholtz number of 21. The two spheres are now several wavelengths apart, and a complicated interference pattern is seen. Bright-coloured spots occur wherever the path length difference from the two spheres results in waves arriving in phase from the two places. The two figures show very similar patterns, and there is no significant difference in the pressure levels in the two cases. The interference pattern seen in these examples could be thought of as an acoustical analogue of the famous “two-slits” experiment in optics. Light waves obey the same wave equation as sound waves, although the wavelengths of visible light are much shorter than the typical wavelengths of sound so that the equivalent of the Helmholtz number is usually very large for macroscopic problems in optics.

The significance of very large Helmholtz number is revealed by the next example. This concerns the phenomena of diffraction, wave scattering and the formation of shadows. We are very familiar with the fact that “light travels in straight lines”, so that if there is an opaque object in the way then the light can’t reach the other side, and the result is a shadow. But with sound waves, things are different. You can often hear something when it is round a corner, for example a radio playing in the next room. But in truth, the distinction is not so clear. Optics and acoustics obey the same laws, and the only difference comes from the Helmholtz number. At very high Helmholtz number, sound waves start to behave much more like the familiar behaviour of light, and they can be analysed using “ray theory”.

We will not go into much detail here, but the key effects can be illustrated by a textbook example which happens to have a closed-form mathematical answer (see section 8.1 of Morse and Ingard [1] for the details). The problem concerns the interaction between a plane sound wave and an infinitely long rigid cylinder, lying perpendicular to the direction the wave is travelling. The wave *diffracts* around the cylinder, with details that depend on the Helmholtz number $ka$, where $a$ is the radius of the cylinder and $k$ is the wavenumber of the sound wave as before.

The difference between the original plane wave and the actual wavefield, including the diffraction effects, is called the *scattered field*. Figure 7 shows plots of how this scattered field behaves for a range of values of the Helmholtz number. Each case is a polar plot, to show how the intensity of the scattered field varies with angle. The red symbol marks the origin in each case, and the radial distance from there to the blue curve is proportional to the far-field intensity of the scattered field in the corresponding direction. The scale factor is the same for all the plots, so that Fig. 7 gives an accurate impression of how the strength of scattering varies with Helmholtz number.

For the lowest value of Helmholtz number shown in Fig. 7, and indeed for any value lower than this, the scattering is weak, and confined to directions on the same side of the cylinder as the incident wave. The wavelength is long compared to the size of the cylinder, and the sound wave flows round the cylinder and carries on essentially unchanged. A little of the sound energy is reflected back, with a directional pattern forming a circle in the polar plot.

As the Helmholtz number increases, the strength of scattering increases: more energy is reflected, and also we begin to see increasingly strong effects on the “downstream” side of the cylinder. By the final example plotted here, that downstream effect dominates: this is because an acoustic shadow has formed behind the cylinder. The scattered field serves to reduce the amplitude of the original plane wave; in other words, it is quieter behind the cylinder. If Helmholtz number had been increased still further, the downstream lobe describing the shadow would get longer and narrower. The scattered pattern on the “upstream” side would converge to a shape that can be analysed as a special case (see [1]). This case can be easily visualised from the optical analogy. We are shining a torch from a long way away, at a cylinder with a mirrored surface. A tight shadow forms behind the cylinder, while light that falls on the shiny surface is reflected in a spread of directions governed by the simple laws of geometric optics.

For the intermediate values of Helmholtz number, shapes are seen in the polar plots which would be hard to guess. The general pattern revealed by this example is typical of many acoustical problems, either of scattering and shadowing or of sound radiation by vibrating structures like the body of a stringed instrument. The behaviour at very low and very high Helmholtz number has a relatively simple pattern, which is sometimes amenable to analysis, at least as an approximation. But the intermediate range where the wavelength of sound is of the same order as the size of the object is more complicated. Directional patterns appear with a pattern of lobes which varies rapidly with frequency. It is rare that behaviour in this regime can be calculated mathematically: either numerical computation or direct measurement is needed.

[1] Philip M. Morse and K. Uno Ingard; “Theoretical acoustics”, McGraw-Hill (1968)