Standing waves: from strings to rooms

A guitar string and the air between two walls look like very different things, but both support standing waves, and the same mathematics describes them. Understanding one helps with the other.

A string

A string fixed at both ends can vibrate only in patterns that fit between the ends. The longest wavelength is twice the string length; the next is equal to the length, then two thirds, and so on. The frequencies are whole multiples of the lowest:

f_n = n · v / (2L)

where v is the wave speed on the string and L its length. At the ends there are nodes (no motion), and in between antinodes (maximum motion).

A room

Between two parallel rigid walls, air behaves in a similar way. The wave speed is the speed of sound, and the frequencies are

f_n = n · c / (2L)

For a distance of 5 m, the first mode is at 34.3 Hz, the second at 68.6 Hz and so on. The room modes calculator shows them for all three dimensions, see the room mode tool.

What differs

At a rigid wall air particles cannot move into the wall, so velocity is zero and pressure is maximal. That is why a bass trap works best near walls and in corners, where pressure is strongest.

Mode shapes in a room

For the first axial mode between two walls, pressure is highest at both walls and zero at the middle. For the second mode, pressure is zero at one quarter and three quarters of the distance. So the loudness of a bass note depends on position:

• In a corner, every mode has high pressure and all bass frequencies are boosted.
• At a node of a mode, that frequency is quiet.

This is the reason why moving a loudspeaker or listener by half a metre can change the bass response drastically.

Three kinds of modes

• Axial: between one pair of walls.
• Tangential: involving two pairs.
• Oblique: involving all three.

Axial modes are the strongest; there are more tangential and oblique ones, but each is weaker.

The number of modes grows quickly

The number of modes grows roughly with the cube of the frequency. At low frequencies there are few modes and they stand apart, which makes the response uneven. At higher frequencies they overlap and merge into a smooth response, above the Schroeder frequency, see the Schroeder frequency tool.

Practical consequences

1. Avoid rooms with equal dimensions or simple ratios, which stack modes on the same frequencies.
2. Do not place the listening position at the centre of the room or at a room-wide node.
3. Treat the corners with broadband absorbers, see Room modes explained.

Draft: written by an AI assistant, awaiting technical review.