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I am trying to decipher what decibels are:

http://en.wikipedia.org/wiki/Decibel

It seems to be a log ratio of audio amplitude multiplied by a constant. I am confused by what this means though.

If my original volume is X, what does say increasing the volume by Y decibels mean?

Does it mean New Volume = 10 log ( Y / X )?

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You almost had it right but you applied the formula backwards. The power of the sound waves (at any given point) is formalized in Wikipedia as the following using the bel unit:

$$P_1 = 10^{L_B} P_0$$

A value in bel is 10 times the value in decibels

$$ L_{dB} = L_B / 10$$

You are interested in the power at a power level $X+Y$ decibels versus the power at a previous power level, $X$ decibles. To get this ratio we will divide the power values.

$$ \frac{P_2}{P_1} = \frac{10^{X/10+Y/10} P_0}{10^{X/10} P_0} = 10^{Y/10}$$

That is to say, increasing power by $10 Y$ decibels will multiply power by 10 to the $Y$. If you look at loudness charts, this sounds right. Truck traffic (90 dB) seems about 10 times as loud as a telephone dial tone (80 dB).

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$Y$ in the second equation is in bels, while in the third equation it is decibels. This is clear because $\frac{P_2}{P_1}$ has two inconsistent expressions. – Ross Millikan Aug 27 '12 at 2:39
@RossMillikan I was using two different definitions of Y, this is confusing and unnecessary so I edited, put all variables in decibels now. Thanks. – AlanSE Aug 27 '12 at 12:24

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