A way of expressing a quantity in relative terms is to do the ratio with respect to a reference value. This helps to put a quantity into perspective. For example, in mechanics the acceleration is often expressed in relation to the gravitational acceleration. Instead of saying the acceleration is 22 m/s² (which is hard to relate to unless you know mechanics), we can also say the acceleration is 22 / 9.81 ≈ 2.2 times the gravitational acceleration or simply 2.2 g’s (which is much easier to comprehend).
The decibel (dB) is also a general way of expressing a quantity in relative terms, sort of a “logarithmic ratio”. And just like the ratio, it is not a physical unit or limited to any field such as mechanics, audio, etc … You can express any quantity in decibels. For example, if we take the reference value to be the gravitational acceleration, the acceleration 22 m/s² corresponds to 3.5 dB.
To calculate the decibel value L of a quantity x relative to the reference value x(0), we can use this formula:
In acoustics the decibel is used to express the sound pressure level (SPL), measured in Pascal = Pa, using the threshold of hearing (0.00002 Pa) as reference. However, in this case a factor of twenty instead of ten is used. The change in factor is a result of inputting the squares of the pressure values rather than the linear values.
The sound coming from a stun grenade peaks at a sound pressure level of around 15,000 Pa. In decibel terms this is:
which is way past the threshold of pain that is around 63.2 Pa (130 dB). Here are some typical values to keep in mind:
0 dB → Threshold of Hearing
20 dB → Whispering
60 dB → Normal Conversation
80 dB → Vacuum Cleaner
110 dB → Front Row at Rock Concert
130 dB → Threshold of Pain
160 dB → Bursting Eardrums
Why use the decibel at all? Isn’t the ratio good enough for putting a quantity into perspective? The ratio works fine as long as the quantity doesn’t go over many order of magnitudes. This is the case for the speeds or accelerations that we encounter in our daily lives. But when a quantity varies significantly and spans many orders of magnitude (which is what the SPL does), the decibel is much more handy and relatable.
Another reason for using the decibel for audio signals is provided by the Weber-Fechner law. It states that a stimulus is perceived in a logarithmic rather than linear fashion. So expressing the SPL in decibels can be regarded as a first approximation to how loud a sound is perceived by a person as opposed to how loud it is from a purely physical point of view.
Note that when combining two or more sound sources, the decibel values are not simply added. Rather, if we combine two sources that are equally loud and in phase, the volume increases by 6 dB (if they are out of phase, it will be less than that). For example, when adding two sources that are at 50 dB, the resulting sound will have a volume of 56 dB (or less).
(This was an excerpt from Audio Effects, Mixing and Mastering. Available for Kindle)