How sound falls with distance: 1/r and 1/r²

How much quieter is a sound when you move away from it? The answer is one of the most useful rules in audio, and also one of the most frequently misquoted.

The inverse distance law

For a small source in free space, with nothing around to reflect sound, the energy spreads over the surface of a sphere. The sphere's area grows with the square of the radius, so:

• Intensity falls with 1/r²: double the distance, one quarter of the intensity.
• Sound pressure falls with 1/r: double the distance, half the pressure.

These are two statements about one fact, because intensity is proportional to the square of the pressure. In decibels the level drops by

ΔL = 20 · log (r₂ / r₁)

which is 6.02 dB for each doubling of distance.

A sound of 94 dB at 1 m is 74 dB at 10 m. Use the SPL and distance tool for any pair of distances.

Common mistakes

1. "It falls by 3 dB per doubling." That is true for a long line source (such as a busy road or a line array), whose sound spreads over a cylinder. Then the intensity falls with 1/r and the level with 10 · log.
2. Mixing the laws. Pressure with 1/r and intensity with 1/r² do not conflict. A formula for energy quantities uses 10 · log; one for field quantities uses 20 · log. See Sound pressure, intensity and power.
3. Applying the law indoors without thought.

When the law fails

• Near field. Very close to large or complex sources, the pressure does not follow 1/r. Interference patterns create peaks and dips.
• Rooms. At a distance the reverberant field takes over, and the level stops falling. In a venue, moving back can lower the level far less than 6 dB per doubling. Beyond the critical distance the sound level is roughly constant, see Direct and reverberant field.
• Ground and obstacles. Reflections from the ground, walls and objects change the result outdoors.
• Air absorption. The air absorbs high frequencies, adding losses on top of the geometric spread, see the air absorption tool.
• Wind and temperature gradients bend sound and can shift levels by several decibels over long distances.

What it means in practice

• A microphone twice as far from a singer gets 6 dB less of the voice, but the same room. The balance between direct and room sound changes a lot, see Microphone distance and the room.
• Sound system design: to cover a long space with the same level, speakers are distributed and delayed rather than turned up.

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