Loudness and intensity

When a sound gets louder the increase in loudness is proportional to the fractional change in intensity or:

Loudness increase (dL) is proportional to increase in intensity (dI)/original intensity (I)

This means that equal increases in loudness are the result of equal proportional changes in intensity. So if the intensity of a sound doubles and then doubles again the loudness will increase by the same amount each time.

Writing this mathematically: dL is proportional to dI/I or dL = kdI/I where k is a constant.

If we integrate this we get the equation: L = k ln(I/I_o) or L = C log(I/I_o) where I_o is the threshold intensity, or initial intensity, and k and C are constants.

This shows the logarithmic response of the ear to the volume of a sound.

For example if the intensity of a sound goes up from say 2x10^-8 Wm^-2 to 4x10^-8 Wm^-2 and then increases again from 4x10^-8 Wm^-2 to 8x10^-8 Wm^-2 the loudness will increase by a factor of 2 each time.