Distance and the inverse square

Stand next to a loudspeaker and walk away. The sound gets quieter, and quickly at first. The reason is that the sound energy spreads out.

Imagine a tiny source in the open air, sending the same power in every direction. At distance r the energy is spread over a sphere of area 4πr². Double the distance and the sphere has four times the area, so each square metre gets a quarter of the intensity. This is the inverse square law: I ∝ 1/r².

In decibels, the change between distance r₁ and r₂ is ΔL = −20 × log₁₀(r₂ / r₁). In practice:

• Doubling the distance lowers the level by 6 dB.
• Ten times the distance lowers it by 20 dB.

Example: a source of 90 dB at 1 m. At 2 m it is 84 dB, at 4 m 78 dB and at 8 m 72 dB (the formula gives 71.9 dB, which we round).

It is not a perfect rule. The law is exact only for a small source in free space, with nothing to reflect the sound. In real life:

• Indoors, the walls send sound back. Close to the source you hear mostly the direct sound and the level falls 6 dB per doubling. Farther away you are in the reverberant field, and the level levels off. The distance where the two are equal is the critical distance. In a 1,000 m³ room with a reverberation time of 1.6 s a small omnidirectional source has a critical distance of about 1.4 m, which is why you cannot hear a speaker clearly from the back of a big, echoey room.
• Direction matters. A person talking sends more sound forward. High frequencies are the most directional. Low frequencies, with long wavelengths, bend around objects and spread in all directions, so you hear the thump of your neighbour's music through the wall and not the voices.
• Obstacles and the ground reflect, absorb and block sound. Air itself absorbs high frequencies, a little more as the distance grows. Wind and temperature layers bend the sound.

Perceived loudness. A 6 dB drop is a big change in the measurement but only a modest change in how loud it sounds. Because about −10 dB sounds half as loud, you need to go about three times as far from a source for it to sound half as loud, which is why distance is the cheapest hearing protection there is.

More than one source. Two identical speakers that are not related to each other (say two different people talking) give +3 dB, not twice as loud. Four give +6 dB. If the two speakers play the same signal, the waves combine in a different way, and the level can change a lot from place to place, which you will meet in World 4.