The sensation of loudness is approximately logarithmic to human ear. The unit used to express the acoustic intensity or power, therefore, is logarithmic. Decibel (dB) is a logarithmic unit that indicates the ratio of a physical quantity (usually power or intensity) relative to a specified or implied reference level. A ratio in decibels is ten times the logarithm to base 10 of the ratio of two power quantities.
Thus, the ratio of a power value `I_1` to another power value `I_0` is represented by `L_(dB)` , that ratio expressed in decibels,which is calculated using the formula:
`L_(dB) = 10 log (I_1/I_0)`
The faintest audible sound having acoustic intensity of `1.0 *10^(-12) ` W/m^2 is the reference, `I_0` .
So, for the sound which is assigned 60 dB has actual acoustic intensity I, then
`60 = 10 log(I_1/(1.0 *10^(-12)))`
`=>`` log (I_1/(1.0 *10^(-12))) = 6`
`rArr` ` ``I_1/(1.0*10^-12)=10^6`
`rArr` `I_1= 10^6*(1.0 *10^(-12))`
= `1.0 *10^(-6)` W/m^2
Now, for the sound assigned 120 dB has actual acoustic intensity `I_2` , then
`120 = 10 log(I_2/(1.0 *10^(-12)))`
`rArr` `log (I_2/(1.0 *10^(-12))) = 12`
`I_2 = 10^12*(1.0 *10^(-12)) `
`= 1.0` W/m^2
So, `I_2/I_1 = 10^6`
Therefore a 120 dB sound, usually associated with rock concerts, is actually one million times more powerful than a 60 dB sound of a conversation.
This could damage human audible system in a matter of minutes! This is how Jacob would probably be able to convince Anderson to take the ear protection plugs in the concert.