What they are

Ideally, a resonant cavity is nothing but a closed space. Waves of some kind are produced inside it, and the walls bounding the cavity reflect the waves completely, so that they remain entirely trapped inside.

Examples:

• acoustic cavities (Helmholtz resonators)
• a harbour
• optical cavities: the laser
• microwave cavities: the microwave oven
• the empty part of a bottle that is not completely full

Why are they interesting?

• All closed rooms are resonant cavities for sound waves. Understanding how they work means being able to design rooms with good architectural acoustics.
• Resonant cavities are part of almost all musical instruments. The properties of the cavity contribute to the voice of the instrument, that is, to its timbre.

How they work

1. Inside the resonant cavity, waves can move freely. When they reach a wall they are reflected.
2. After a certain time (called the transient), the superposition of all the reflected waves can produce a stationary field (see standing waves). If this happens, the energy contained in the cavity stops moving and concentrates in particular fixed positions (called antinodes).
3. The stationary field does not always form, but only when the waves in the cavity have particular frequencies, called the resonance frequencies of the cavity.
4. The resonance frequencies depend on the size of the cavity.

Resonance frequencies

All we need to do to calculate the modes of the cavity is to require that certain physical quantities characterising the waves take very precise values at the wall of the cavity. Technically, we say that the resonance frequencies are determined by boundary conditions. For example, in the case of sound waves, the pressure waves in the air (or in the fluid contained in the cavity) are longitudinal.

• If we take the displacement ξ of the fluid molecules as the quantity defining the wave (and the sound field), then the boundary conditions are as follows: the displacement of the fluid must be zero at all the walls, because the fluid cannot pass through the walls (air cannot leave the ideal cavity).
• If instead we use the pressure p rather than the displacement ξ to describe the wave, the boundary condition is different: the pressure at the walls is at a maximum (antinode). Indeed, inside the box, a compression at a given point always happens "against other air", and is therefore certainly smaller than one happening "against the rigid wall".

In both cases, however, the result is the same, because what matters is not the nature of the quantity chosen to describe the field, but only the size of the box.

Finding the resonance frequencies is simple if the cavity has a very regular shape. The following table shows the conditions for a cavity shaped like a rectangular box.

For example, to amplify the sound of a tuning fork, it is mounted on a box that "resonates" exactly at the frequency produced by the tuning fork. Rewriting the boundary conditions in terms of wavelength (for this, recall that ${\displaystyle k_x=\frac{2\pi }{{\lambda }_x}}$), we obtain:

${\displaystyle {\lambda }_x=\frac{2A}{l}l=1,2,\dots }$,

${\displaystyle {\lambda }_y=\frac{2B}{m}m=1,2,\dots }$,

${\displaystyle {\lambda }_z=\frac{2C}{n}n=1,2,\dots }$.

The relations just derived show that the lower the note produced by the tuning fork (that is, the longer its wavelength), the larger the resonant cavity must be.

Musical instruments deliberately have irregularly shaped sound boxes (think of string instruments): the aim is to give the instrument a large number of resonances and to prevent the sound box from amplifying, by resonating, only certain frequencies. If you want to know more about this, visit the pages on timbre perception and on the physics of musical instruments

Number of modes

The analysis of the resonance frequencies carried out in the previous section can be summarised in a graph showing the number of possible modes for each frequency. This graph shows visually how the resonances of the system are distributed in frequency space, and gives important indications, for example, about the acoustic quality of rooms.

Let us look, for example, at the case of three rooms of equal volume (50 m³), with different ratios between the lengths of the sides.

all sides different	two equal sides	three equal sides
The sequence of resonance peaks forms a fairly continuous distribution. The peaks are all very close together and fairly evenly distributed	The distribution is still relatively even, but the peaks are higher than in the case on the left, and small gaps appear between them	The gaps between the peaks widen, creating strong resonances at some frequencies, separated by wide non-resonant frequency bands

Do not be misled by the apparent similarity of the graphs. Moving from left to right, the symmetry of the room increases (a rectangular box on the left, a box with a square base in the centre, a cube on the right). This means that the number of possible modes for each resonance frequency progressively increases (see the discussion in the previous section). In other words, instead of being spread evenly over many frequencies, the possible resonant modes tend to pile up on fewer frequencies, creating stronger resonances. From the point of view of room acoustics this is bad. A room with a nearly uniform response will return all the frequencies of a sound with roughly the same intensity, while a room with strong resonances separated by non-resonant bands will tend to filter the sound, returning mainly the resonant components and attenuating the others.

This simple example shows that the ideal geometry for good resonance in a room shaped like a rectangular box is the least symmetric one. Taking the argument further, we can say that the resonances can be made even more uniform by making the room even more irregular in shape, although in this case counting the number of states is not so simple.

Of course, examining the resonances does not exhaust the analysis of the acoustic properties of a room, which are also characterised by other parameters. For this, see the section on architectural acoustics.

Cavities for electromagnetic waves

In this case we must take into account the different, "vectorial" nature of the waves, their property of containing variations of both the electric and the magnetic field, and polarisation.

Further reading and links

• If you want to see a series of experiments on resonant cavities, you cannot miss our virtual wave laboratory. On this page you will find a series of guided tours for virtual experiments that will help you understand the formation of standing waves, acoustic resonances and the filtering action of resonant cavities.