Inverse-Square Law for Sound

Sound from a small source spreads out as an ever-growing sphere, and a sphere's surface area grows as r², so the intensity thins out as 1/r². Double your distance from a firecracker and the intensity drops to a quarter; step back tenfold and it falls a hundredfold — a 20 dB drop. A 0.5 W siren heard from 10 m delivers 0.5/(4π × 100) ≈ 4 × 10⁻⁴ W/m², about 86 dB; from 100 m it is down to 66 dB, no louder than animated conversation.

The law assumes the sound spreads freely in all directions, which is why real environments bend it. A megaphone or a stage horn concentrates power into a cone instead of a sphere, staying loud farther out; a highway heard across open fields behaves more like a line source, fading as 1/r rather than 1/r². Concert engineers exploit the pure form in reverse: measure the intensity at a known distance and the formula reveals the source's total acoustic power.