Direct field and reverberant field

In a room, a sound source produces two kinds of sound energy at a listener: the direct field, arriving straight from the source, and the reverberant field, which has bounced off the surfaces many times. Their balance explains much about how rooms and microphones behave.

The direct field

The direct sound follows the distance law: its level falls 6 dB for each doubling of distance, see How sound falls with distance. For a source of sound power level L_W and directivity factor Q, at distance r:

direct level = L_W + 10 · log (Q / (4π r²))

The reverberant field

After many reflections the sound arrives from all directions with nearly equal strength everywhere in the room. This is a diffuse field, and its level is independent of position. For a room with total absorption area A (in square metres):

reverberant level = L_W + 10 · log (4 / A)

More absorption lowers the reverberant level. A room with twice the absorption has a reverberant field 3 dB lower.

Their sum

At any point both are present, and the energy adds:

total level = L_W + 10 · log (Q / (4π r²) + 4 / A)

• Close to the source, the first term dominates: the level falls with distance.
• Far from the source, the second term dominates: the level levels out.

The critical distance

The distance where both terms are equal is the critical distance (also room radius or reverberation radius). Setting the two terms equal and using Sabine's relation A = 0.161 · V / T gives

D_c = 0.057 · √(Q · V / RT60)

with the room volume V in m³ and RT60 in seconds. Below it the direct sound dominates; above it the room. See the critical distance tool.

For a room of 200 m³ with a reverberation time of 1.2 s and an omnidirectional source, D_c is about 0.74 m. In a small, damped studio it is shorter still; in a cathedral it can reach several metres.

What it means

• Microphones. At a distance beyond D_c, the recording is dominated by the room. A directional microphone extends the useful distance, see Microphone distance and the room.
• Loudspeakers and PA. Beyond the critical distance, adding more distance does not lower the level. Turning up the system raises the reverberant level together with the direct level, so intelligibility does not improve. Directional loudspeakers raise the critical distance.
• Speech intelligibility. It depends on the ratio of direct to reverberant sound. See the speech intelligibility tool.

Limits

The formulas assume a diffuse field and a room that is not too absorbent. In rooms with very uneven absorption or strange shapes, the reverberant field is not uniform, and the critical distance is only a rough guide.

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