Decibels

The quietest sound a young healthy ear can hear and a jet engine up close differ in pressure by a factor of millions and in intensity by a factor of a million million million. Writing numbers like that is awkward, so we squeeze them with a logarithm. The result is the decibel (dB).

For intensity I (power per square metre): L = 10 × log₁₀(I / I₀), with the reference I₀ = 10⁻¹² W/m².

For pressure p: L = 20 × log₁₀(p / p₀), with the reference p₀ = 20 micropascals (20 µPa). Compare that with the air pressure around you, about 101,325 Pa: the quietest audible sound is only about two parts in ten billion of that. A sound of 1 pascal is about 94 dB, and 20 Pa is about 120 dB.

Two rules of thumb are all you need:

• +10 dB means 10 times the intensity. +20 dB means 100 times, +30 dB means 1,000 times.
• +3 dB means double the intensity (and +6 dB means double the pressure).

0 dB is not silence. It is the reference near the threshold of hearing. Very quiet rooms can have negative levels.

Typical levels (rounded, A-weighted, from public health sources; real values vary a lot):

• Whisper at 1 m: about 30 dB.
• Normal conversation: about 60 dB.
• Busy road traffic: about 70 to 85 dB.
• Motorcycle: about 90 to 100 dB.
• Loud concert: about 100 to 115 dB.
• Jet aircraft at take-off, close by: about 120 to 140 dB.

dB(A). The ear is not equally sensitive at all frequencies, so noise meters can apply an A-weighting that reduces the low and the very high frequencies the way the ear does. A number written dBA or dB(A) is weighted that way. Health limits for noise are given in dBA.

How loud does it sound? Our sense of loudness is also roughly logarithmic. As a rule of thumb, +10 dB sounds about twice as loud, even though the intensity is 10 times bigger. A concert at 100 dB has a million times the intensity of a 40 dB room, and by the rule of thumb it sounds about 64 times as loud (six doublings).

Adding decibels. Decibels are not added like ordinary numbers, because they are logarithms. Add the intensities, then convert back. Two equal sounds of 60 dB each give 63 dB, not 120 dB: twice the intensity is only +3 dB. Ten equal sources add 10 dB. A 60 dB sound next to a 70 dB sound gives about 70.4 dB: the louder one hides the quieter one.

Hearing safety. Loud sound can damage the tiny hair cells of the inner ear, and the damage is permanent. What matters is level and time together. The US institute NIOSH recommends a limit of 85 dBA averaged over 8 hours. For every 3 dB more, the safe time is cut in half (this is called the 3 dB exchange rate; the World Health Organization gives the same 85 dB, 8-hour guidance for safe listening):

• 85 dBA: 8 hours.
• 88 dBA: 4 hours.
• 91 dBA: 2 hours.
• 94 dBA: 1 hour.
• 97 dBA: 30 minutes.
• 100 dBA: 15 minutes.

These are upper limits for a whole day, not targets: quieter is always better, and sounds you hear all through the day add up. Many headphones and phones can play above 100 dB, so volume matters. A simple check: if you cannot hear someone speaking at arm's length, or you must shout to be heard, the sound is probably above 85 dB. Earplugs made for music lower the level without muffling it.