Standing waves in room acoustics

A standing wave forms when sound reflecting between two surfaces overlaps with the sound travelling toward them, so that the pressure swings strongly at some fixed points and hardly at all at others. In a room, standing waves are what room modes look like in space: at each mode frequency, bass is loud near the walls and at certain spots in between, and weak at the nulls. That is why two seats a metre apart can hear very different bass.

How does a standing wave form between two walls?

Picture a single low tone travelling down the length of a room. At the far wall it reflects and travels back. The outgoing and returning waves now share the same space, and their pressures add. At most frequencies the sum shifts around as the reflections pile up. At particular frequencies, though, the returning wave lines up with the outgoing one after every round trip, and the sum settles into a pattern that does not travel: some points swing between high and low pressure as hard as they can, others stay almost still. Those still points are nodes, or nulls; the strongest points are antinodes.

For two parallel rigid walls a distance L apart, this happens when a whole number of half wavelengths fits between them:

f_n = n · c / 2L,   n = 1, 2, 3 …

With c = 343 m/s, a 5 m (16.4 ft) room gives 34.3 Hz, 68.6 Hz, 102.9 Hz and so on. These are the axial room modes along that dimension. Bass wavelengths are long: 8.6 m at 40 Hz and 3.4 m at 100 Hz, which is why the patterns span the whole room instead of repeating every few centimetres.

Where are the peaks and nulls?

A rigid wall cannot move, so the air right against it cannot move either, and pressure builds up there. Every axial mode therefore has a pressure maximum at both walls. Between them, mode n has n nulls, evenly placed:

• mode 1: one null at the middle of the room;
• mode 2: nulls at one quarter and three quarters of the length, with a maximum in the middle;
• mode 3: nulls at one sixth, one half and five sixths of the length.

A word of caution about diagrams like this one. Many illustrations draw a standing wave as a single wavy line frozen in time, which suggests the air is displaced into a fixed shape. What the diagram above shows instead is the envelope: how far the pressure swings at each point over a cycle. It also covers only one dimension. A real room holds length, width and height modes plus tangential and oblique ones, all overlapping, so the pattern you walk through is three-dimensional and changes with every frequency.

Pressure versus particle velocity

A standing wave has two sides. Where pressure swings the most, the air molecules barely move; where pressure is at a null, the air moves back and forth the fastest. At a rigid wall, velocity is zero and pressure is at its maximum. This distinction matters for acoustic treatment:

• Porous absorbers (mineral wool, foam) work by friction as air moves through them, so they need particle velocity. At bass frequencies the velocity maximum sits a quarter wavelength from the wall: about 1.7 m (5.6 ft) at 50 Hz. A thin panel mounted flat on the wall sits where velocity is near zero and does almost nothing for that frequency.
• Pressure-based traps (membrane or Helmholtz resonators) respond to pressure, so they belong at walls and especially corners, where every mode of a rectangular room has a pressure maximum.

The room acoustics guide covers when treatment helps and when placement matters more.

What standing waves mean for seats and subwoofers

The same pressure pattern governs both ends of the system. A listener sitting at a mode's null hears little of that frequency, however hard the subwoofer drives it. And a subwoofer placed at a mode's null barely excites the mode at all, because it is pushing on air that the mode does not compress there. Move either one toward an antinode and the mode comes back.

This has practical consequences:

• A seat in the exact middle of the room tends to sit on the null of every odd-numbered mode along that dimension.
• A seat against the back wall sits on the maximum of every mode along the length, which often makes bass sound heavy there.
• A subwoofer in a corner sits at a pressure maximum of every mode and excites all of them; a subwoofer centred on the front wall sits on the null of every odd-numbered width mode and leaves those unexcited.

Those are tendencies of the ideal model. Which choice sounds best depends on the seats you actually use, which is why the placement method starts with measurement rather than a rule.

Seeing a standing wave in your own room

BassMap for iPhone plays a low-frequency test signal through your system while you walk the room, and draws a heatmap of each tone's level across the area you covered. At a frequency near a mode, the heatmap shows that mode's loud and quiet regions directly, and switching frequency shows how the pattern changes. The levels are relative to the room, not calibrated SPL.

Sources

• F. Alton Everest and Ken C. Pohlmann, Master Handbook of Acoustics, McGraw-Hill.
• Heinrich Kuttruff, Room Acoustics, CRC Press.