Table of Contents
The Inverse Square Law
The inverse square law describes how the intensity of energy radiating from a point source decreases with distance. As energy spreads outward in all directions, it covers an increasingly large spherical surface area. Since the total energy remains constant but the area grows as the square of the distance, intensity decreases as the inverse square of distance. This fundamental law applies to sound, light, gravity, and electromagnetic radiation.
In practical terms, doubling the distance from a source reduces the intensity to one-quarter, tripling it reduces to one-ninth, and so on. In decibels, each doubling of distance reduces the level by approximately 6 dB for a point source in free field conditions. Real-world situations include reflections, absorption, and barriers that modify this ideal behavior.
Formula
Where L is the level in dB, d is distance, and I is intensity. The 20*log10 factor applies to pressure/voltage quantities (like sound pressure); for power quantities, use 10*log10.
Attenuation vs Distance
| Distance Ratio | Attenuation | Intensity Fraction |
|---|---|---|
| 2x | -6.0 dB | 25% |
| 3x | -9.5 dB | 11.1% |
| 5x | -14.0 dB | 4% |
| 10x | -20.0 dB | 1% |
| 100x | -40.0 dB | 0.01% |
Applications
- Acoustic engineering: predicting noise levels at various distances
- Telecommunications: free-space path loss calculations
- Lighting design: illuminance at different distances from light sources
- Radio astronomy: signal strength from celestial sources
FAQ
When does the inverse square law not apply?
The law assumes a point source radiating uniformly in free space. It does not apply close to the source (near field), in enclosed spaces with reflections, when the source is directional (antennas, horns), or when the medium absorbs energy. Indoors, reverberation can maintain sound levels well beyond what the inverse square law predicts.
What is 6 dB per doubling?
Since 20*log10(2) = 6.02 dB, each doubling of distance from a point source reduces the sound pressure level by approximately 6 dB. This is a useful rule of thumb for quick calculations in acoustic engineering. For line sources (roads, railways), the reduction is only 3 dB per doubling because energy spreads cylindrically rather than spherically.
How do I account for atmospheric absorption?
At large distances, atmospheric absorption adds to the geometric spreading loss. The absorption rate depends on frequency, temperature, humidity, and altitude. High frequencies are absorbed more rapidly than low frequencies. For sound in air at 20C and 50% RH, absorption at 4 kHz is about 0.02 dB/m, which becomes significant beyond a few hundred meters.