Here we want to examine what happens when sound waves are produced inside an enclosed space such as a box, a room or a theatre.

For simplicity, let us first consider a space shaped like a rectangular box, with walls that reflect sound.

Intuitive physical analysis

When a sound is produced inside the space, the sound waves travel through the air until they meet one of the walls. At that point a small part of the energy of the wave is absorbed by the wall, and the rest is reflected back into the box. The wave travels again until it reaches another wall, bounces again, and so on. If the walls are ideally perfectly reflecting, the wave goes on like this forever. In reality, losing a little energy at each bounce (and a little to the air while travelling from one wall to another), after some time the wave tends to die out, unless the sound source keeps sustaining it.

After many bounces two situations can arise

• For certain particular values of the frequency of the sound source, the superposition of all the waves reflected in all directions gives rise to standing waves: the sound field inside the space therefore has places where the air is always approximately at rest (nodes), and places where the air always undergoes the maximum displacement (antinodes).
• For all other frequencies no stationary field forms, and the energy of the sound waves does not concentrate at particular points.

When standing waves form, the mechanism is entirely similar to the one already seen for the vibrating string. The only difference is in the value of the resonance frequency, which is not the same as for the string (see below).

Applications

This particular resonance phenomenon is very important for several reasons (see also the section resonance for a more general discussion). Here are some:

• Everyone knows that most musical instruments have hollow parts. From what has been said above, one good reason for this is clear: for all the resonance frequencies, the standing waves formed inside the box (usually called, precisely, a "resonance box") reinforce the sound of the instrument. (There is also another, subtler reason that has nothing to do with resonance, impedance matching: see the page sound and resonance)
• The situation described above actually occurs in every space where sounds are produced, whether by voices speaking, by instruments or by industrial machinery. A classroom, a theatre, a conference hall, the interior of a car all have natural resonance frequencies at which standing sound waves form inside them. Being able to calculate the acoustic response of these spaces is important for the comfort and health of the people in them, or for good reception of music or speech. (See the section on architectural acoustics for further details.)
• An entirely analogous resonance phenomenon also occurs for all other types of waves, such as electromagnetic waves. See the section resonant cavities for further examples.

To find out more

Physicists cannot be content with the simple intuitive analysis above, but must be able to answer more quantitative questions such as the following:

• What are the values of the resonance frequencies?
• Do they depend only on the size of the box, or on other factors too?
• How does temperature affect the resonance? (In a room, the presence of people can noticeably change the temperature of the air)

The answers are in the section on resonant cavities

Virtual laboratory: standing waves in a box

Using the 2D Waves app you can build cavities of any shape and place wave sources in them to check what we have discussed on this page. Alternatively, you can follow one of the guided tours on the page Guided tours on standing waves