Sound pressure, sound level and dB. Sound is usually measured with microphones and they respond (approximately) proportionally to the sound pressure, p. Now the power in a sound wave, all else equal, goes as the square of the pressure. (Similarly, electrical power goes as the square of the voltage.) The log of the square of x is just 2 log x, so this introduces a factor of 2 when we convert to decibels for pressures. The difference in sound pressure level between two sounds with p_{1} and p_{2} is therefore:
20 log (p_{2}/p_{1}) dB = 10 log (p_{2}^{2}/p_{1}^{2}) dB = 10 log (P_{2}/P_{1}) dB where again the log is to base 10.
Sound files to show the size of a decibel
What happens when you halve the sound power? The log of 2 is 0.3, so the log of 1/2 is 0.3. So, if you halve the power, you reduce the power and the sound level by 3 dB. Halve it again (down to 1/4 of the original power) and you reduce the level by another 3 dB. That is exactly what we have done in the first graphic and sound file below.

Broadband noise decreasing by 3 dB steps.
The first sample of sound is white noise (a mix of all audible frequencies, just as white light is a mix of all visible frequencies). The second sample is the same noise, with the voltage reduced by a factor of the square root of 2. 2^{0.5} is approximately 0.7, so 3 dB corresponds to reducing the voltage or the pressure to 70% of its original value. The green line shows the voltage as a function of time. The red line shows a continuous exponential decay with time. Note that the voltage falls by 50% for every second sample.
Note, too, that a doubling of the power does not make a huge difference to the loudness. We'll discuss this further below, but it's a useful thing to remember when choosing sound reproduction equipment.
Sound files and graph by John Tann and George Hatsidimitris.

How big is a decibel? In the next series, successive samples are reduced by just one decibel.

Broadband noise decreasing by 1 dB steps.
One decibel is close to the Just Noticeable Difference (JND) for sound level. As you listen to these files, you will notice that the last is quieter than the first, but it is rather less clear to the ear that the second of any pair is quieter than its predecessor. 10*log_{10}(1.26) = 1, so to increase the sound level by 1 dB, the power must be increased by 26%, or the voltage by 12%. 
What if the difference is less than a decibel? Sound levels are rarely given with decimal places. The reason is that sound levels that differ by less than 1 dB are hard to distinguish, as the next example shows.

Broadband noise decreasing by 0.3 dB steps.
You may notice that the last is quieter than the first, but it is difficult to notice the difference between successive pairs. 10*log_{10}(1.07) = 0.3, so to increase the sound level by 0.3 dB, the power must be increased by 7%, or the voltage by 3.5%. 
Standard reference levels ("absolute" sound level)
When the decibel is used to give the sound level for a single sound rather than a ratio, then a reference level must be chosen. For sound intensity, the reference level (for air) is usually chosen as 20 micropascals, or 0.02 mPa. (This is very low: it is 2 ten billionths of an atmosphere. Nevertheless, this is about the limit of sensitivity of the human ear, in its most sensitive range of frequency. Usually this sensitivity is only found in rather young people or in people who have not been exposed to loud music or other loud noises. Personal music systems with inear speakers ('walkmans') are capable of very high sound levels in the ear, and are believed by some to be responsible for much of the hearing loss in young adults in developed countries.)
So if you read of a sound intensity level of 86 dB, it means that
20 log (p_{2}/p_{1}) = 86 dB
where p_{1} is the sound pressure of the reference level, and p_{2} that of the sound in question. Divide both sides by 20:
log (p_{2}/p_{1}) = 4.3
p_{2}/p_{1} = 10^{4.3}
4 is the log of 10 thousand, 0.3 is the log of 2, so this sound has a sound pressure 20 thousand times greater than that of the reference level (p_{2}/p_{1} = 20,000). 86 dB is a loud but not dangerous level of sound, if it is not maintained for very long.
What does 0 dB mean? This level occurs when the measured intensity is equal to the reference level. i.e., it is the sound level corresponding to 0.02 mPa. In this case we have
sound level = 20 log (p_{measured}/p_{reference}) = 20 log 1 = 0 dB
So 0 dB does not mean no sound, it means a sound level where the sound pressure is equal to that of the reference level. This is a small pressure, but not zero. It is also possible to have negative sound levels:  20 dB would mean a sound with pressure 10 times smaller than the reference pressure, ie 2 micropascals.
Not all sound pressures are equally loud. This is because the human ear does not respond equally to all frequencies: we are much more sensitive to sounds in the frequency range about 1 kHz to 4 kHz (1000 to 4000 vibrations per second) than to very low or high frequency sounds. For this reason, sound meters are usually fitted with a filter whose response to frequency is a bit like that of the human ear. (More about these filters below.) If the "A weighting filter" is used, the sound pressure level is given in units of dB(A). Sound pressure level on the dB(A) scale is easy to measure and is therefore widely used. It is still different from loudness, however, because the filter does not respond in quite the same way as the ear. To determine the loudness of a sound, one needs to consult some (idealised) curves representing the frequency response of the human ear. (Alternatively, you can measure your own hearing response.)
Logarithmic response, psychophysical measures, sones and phons
Why do we use decibels? The ear is capable of hearing a very large range of sounds: the ratio of the sound pressure that causes permanent damage from short exposure to the limit that (undamaged) ears can hear is more than a million. To deal with such a range, logarithmic units are useful: the log of a million is 6, so this ratio represents a difference of 120 dB. Psychologists also say that our sense of hearing is roughly logarithmic (see under sones below). In other words, they think that you have to increase the sound intensity by the same factor to have the same increase in loudness. Whether you agree or not is up to you, because this is a rather subjective question. (Listen to the sound files linked above.)
The filters used for dBA and dBC
The most widely used sound level filter is the A scale, which roughly corresponds to the inverse of the 40 dB (at 1 kHz) equalloudness curve. Using this filter, the sound level meter is thus less sensitive to very high and very low frequencies. Measurements made on this scale are expressed as dBA. The C scale is practically linear over several octaves and is thus suitable for subjective measurements only for very high sound levels. Measurements made on this scale are expressed as dBC. There is also a (rarely used) B weighting scale, intermediate between A and C. The figure below shows the response of the A filter (left) and C filter, with gains in dB given with respect to 1 kHz. (For an introduction to filters, see RC filters, integrators and differentiators.)
On the music acoustics site, we plot the sound spectra in dB. The reason for this common practice is that the range of measured sound pressures is large. We plot acoustic impedance spectra in dB for the same reason: the input impedance of a musical instrument, such as the flute, varies over a factor of several thousand.
Loudness, phons and sones
The phon is a unit that is related to dB by the psychophysically measured frequency response of the ear. At 1 kHz, readings in phons and dB are, by definition, the same. For all other frequencies, the phon scale is determined by the results of experiments in which volunteers were asked to adjust the loudness of a signal at a given frequency until they judged its loudness to equal that of a 1 kHz signal. To convert from dB to phons, you need a graph of such results (called FletcherMunson curves), and the graph depends on sound level (it becomes flatter at high sound levels).
Curves of equal loudness determined experimentally by Fletcher, H. and Munson, W.A. (1933) J.Acoust.Soc.Am. 6:59.
The sone is derived from psychophysical measurements which involved volunteers adjusting sounds until they judge them to be twice as loud. This allows one to relate perceived loudness to phons. A sone is defined to be equal to 40 phons. Experimentally it was found that a 10 dB increase in sound level corresponds approximately to a perceived doubling of loudness. So that approximation is used in the definition of the phon: 0.5 sone = 30 phon, 1 sone = 40 phon, 2 sone = 50 phon, 4 sone = 60 phon, etc.
Wouldn't it be great to be able to convert from dB (which can be measured by an instrument) to sones (which approximate loudness as perceived by people)? This is usually done using tables that you can find in acoustics handbooks. However, if you don't mind a rather crude approximation, you can say that the A weighting curve approximates the human frequency response at low to moderate sound levels, so dBA is very roughly the same as phons. Then use the logarithmic relation between sones and phons described above.
dBi and radiation that varies with direction
Radiation that varies in direction is called anisotropic: a source that emits sound (or something else) equally in all directions is called an isotropic source. When you want to emit in (or receive from) a particular direction, you want the ratio of intensity measured in that direction, at a given distance, to be higher than that measured at the same distance from an isotropic radiator. This ratio is called the gain; express the ratio in dB and you have the gain in dBi for that radiator. This unit is mainly used for antennae, either transmitting and receiving.
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© 1998. JoeWolfe Modified 19/1/04
