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I know decibel is used as a ratio between two physical quantities with the same units on a logaritmic scale. Is also used for calculate the intensity of a sound, but I found it reported in two different ways:

$$\text {db} = 20 \log \left(\frac{p}{p_0} \right) \qquad\text{with: }p_0 = 0,0002\,\text{dyne}/\text{cm}^2$$ or $$p_0 = 10^{-12}\,\text W/\text m^2$$

are those equivalent? If so, where does that 20 come from? is that because of this (quoting from wikipeida):

"(...) commonly used in acoustics to quantify sound levels relative to a 0 dB reference which has been defined as a sound pressure level of $.0002\,$microbar, or $20\,$micropascals."

is the 20 micropascals? Then should that be equivalent to $0.0002\,\text{dyne}/\text{cm}^2$?

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Short answer: both definitions are correct, though they are off by a power of 2 from one another, as Wikipedia explains. This is complicated by the fact that "$P$" is the standard symbol for both pressure and power. – Chris White Jun 16 '13 at 3:36
up vote 3 down vote accepted

The reference pressure $p_0=20$ micropascals is equal to $2\times 10^{-5}$ pascals which is equal to $2\times 10^{-10}$ bar which is equal to $0.0002$ microbar which is also equal to $2\times 10^{-5}$ newtons per squared meter (the same unit, pascal, as watt per cubic meter) or $2\times 10^{-9}$ newtons per squared centimeter or $2\times 10^{-4}=0.0002$ dyne per square centimeters.

Note that a dyne is $10^{-5}$ newtons while a bar is $10^{5}$ pascals, a pascal is a newton per squared meter, and so on, and so on. All the unit conversions are clear from the previous paragraph.

The quantity $p_0=10^{-12} W/m^2$ is surely a wrong – dimensionally, wrong units, and wrong value – value for a pressure.

The coefficient $20$ is no coincidence. The word "decibel" is composed of "deci" and "bel". Here, "deci-" is a standard prefix for 1/10, so the number of decibels is 10 times the number of bels. That's where the coefficient of 10 comes from. However, you're defining the decibels in terms of the pressure which is an "amplitude" of a wave describing the sound.

However, one may also talk about the energy density associated with the sound which scales like $p^2$, or the squared amplitude, and $\log(p^2)=2\log(p)$. So because the energy density scales like the amplitude squared, one gets the extra factor in the formula for the decibels, thus ending up with the coefficient $2\times 10 = 20$.

The initial paragraphs were about the reference pressure (more precisely sound pressure level or SPL) for acoustics. If you look at

you will see that one may also use the sound intensity level (or SIL) or sound power level (SWL) which is given, in both cases, by $10^{-12}$ watts per squared meter, as you wrote down. It's a "different kind of decibels", dB SIL or dB SWL. You should imagine this is the power (energy per unit time) of a speaker in your hifi or PC per unit area, or the power reaching your ears per unit area.

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