How To Calculate Decibels

A decibel is not a unit of sound in the way a metre is a unit of length. It is a ratio expressed on a logarithmic scale, which means it always describes one quantity relative to another rather than an absolute amount. The reference has to be stated or implied for the number to mean anything.

For sound in air, the reference pressure is twenty micropascals, written as 0.00002 pascals. That is roughly the quietest sound a healthy young person can hear at the most sensitive frequency, so the scale conveniently starts near zero at the threshold of hearing.

The formula multiplies by twenty rather than ten because sound pressure and sound intensity are related by a square. Intensity is proportional to pressure squared, and the logarithm of a square is twice the logarithm of the original, so the factor of twenty for pressure is equivalent to a factor of ten for intensity.

The logarithmic scale exists because the range is vast. The pressure at the threshold of hearing is about twenty micropascals, while the pressure at a rock concert can be twenty pascals, a million times larger. A linear scale would need seven digits to cover that range, and the useful detail would be lost at the quiet end.

Doubling the sound pressure adds about six decibels, because twenty times the logarithm of two is 6.0206. That is a handy rule: a doubling of pressure is a six decibel increase, and a tenfold increase is twenty decibels.

The intensity ratio is the square of the pressure ratio, so the numbers grow even faster. A pressure ratio of ten thousand corresponds to an intensity ratio of a hundred million. This is why decibels are described as compressing a huge range into a small one.

Because the scale is logarithmic, decibels cannot simply be added when sources combine. Two identical eighty decibel sources together produce about eighty three decibels, not one hundred and sixty, because the intensities add while the logarithms do not.

The three decibel rule follows from the same arithmetic. Doubling the intensity adds three decibels, which is a barely perceptible change to most listeners. Ten decibels is usually described as twice as loud, even though the intensity has increased tenfold.

Everyday reference points make the scale concrete. A whisper is around thirty decibels, normal conversation about sixty, a busy street eighty, a chainsaw around one hundred and ten, and a jet engine at close range about one hundred and forty. Sustained exposure above eighty five decibels risks hearing damage.

The same mathematics describes other logarithmic quantities, though the reference and the multiplier change. Power ratios use ten times the logarithm, voltage ratios in electronics use twenty, and the decibel is used in telecommunications and audio engineering for exactly this reason.

It is worth noting that decibels can be negative. A sound pressure below the reference gives a negative level, which simply means it is quieter than the nominal threshold. Negative decibels are perfectly meaningful, just as a negative temperature is.