decibel
The decibel is not a unit of quantity in the way the metre or the watt is. It is one tenth of a bel, a logarithmic expression of the ratio between two powers: a level in decibels is 10 log₁₀(P/P₀) for power quantities, or 20 log₁₀(A/A₀) for amplitude quantities such as sound pressure or voltage. Because it is a ratio, the number carries no meaning at all until the reference P₀ is stated, and there is no linear conversion factor to SI to be exact or inexact about.
Watch out: Three traps. First, a decibel figure needs its reference: dB SPL is referenced to 20 micropascals of sound pressure, dBm to one milliwatt of electrical power, dBV to one volt, and dBFS to digital full scale. These are not interchangeable. Second, dBA is a level that has been weighted by a filter approximating human hearing sensitivity, so it is a different measurement from unweighted dB SPL rather than the same measurement in different clothing. Third, decibels do not add arithmetically: two identical sources give 3 dB more than one, not double the decibel number, because you must convert to power, sum, and take the logarithm again.
| 1 dB | 1 dB |
The bel comes from Bell Telephone Laboratories in the 1920s, where engineers needed a compact way to express transmission loss along telephone lines. It was named for Alexander Graham Bell. The bel itself proved too coarse in practice, so the decibel became the working unit almost immediately.
About the decibel
The decibel exists because the quantities engineers care about span an absurd range and because perception is roughly logarithmic. Sound pressures that matter run from about 20 micropascals at the threshold of hearing to 20 pascals at the threshold of pain, a factor of a million, which the decibel scale compresses into 0 to 120 dB SPL. The defining expression for sound pressure is \(L_p = 20 \log_{10}(p / p_0)\) with \(p_0 = 20\,\mu\ ext{Pa}\), and the factor of 20 rather than 10 appears because pressure is an amplitude quantity while the underlying definition is in power, and power goes as pressure squared.
The consequences of that logarithm are worth internalising. Every 10 dB is a factor of ten in power and is heard as roughly a doubling of loudness. Every 3 dB is a factor of two in power, so adding a second identical machine to a room raises the level by 3 dB, and adding a third raises it only about 1.8 dB more. Halving the distance from a point source in the open adds 6 dB, since pressure falls as \(1/r\) and intensity as \(1/r^2\). And a level cannot be averaged by averaging the numbers: to combine \(L_1\) and \(L_2\) you compute \(10 \log_{10}(10^{L_1/10} + 10^{L_2/10})\).
Because the decibel is dimensionless, the same machinery serves acoustics, electronics and optics. A 20 dB amplifier gain, a 3 dB filter corner, a minus 40 dBm receiver sensitivity and an 85 dBA occupational exposure limit all use the identical mathematics with four different references. When a number is quoted as plain "dB" with no suffix and no stated reference, it is almost always a relative figure such as a gain or an attenuation, and treating it as an absolute level is an error.