Intensity and Loudness of Sound

The human ear is capable of detecting an extraordinarily wide range of sound intensities, from as low as about $10^{-2}\ \text{W/m}^2$ up to about $20\ \text{W/m}^2$ at the threshold of pain (and, more precisely, the threshold of hearing itself is usually taken as $10^{-12}\ \text{W/m}^2$) -- a range spanning many orders of magnitude, far too wide to be conveniently represented on an ordinary linear scale. To handle this, the Weber-Fechner law states that loudness $L$, as perceived by a listener, is proportional to the natural logarithm of the actual physical intensity $I$ as measured by an accurate, non-human instrument: $L=k\ln I$, for some constant $k$ that depends on the unit of measurement chosen. The difference between two loudness readings, $L_1$ and $L_0$, defines the sound intensity level between the two corresponding intensities $I_1$ and $I_0$: $\Delta L=L_1-L_0=k\ln I_1-k\ln I_0=k\ln(I_1/I_0)$. Choosing the constant $k=1$ gives the unit "bel" (in honour of Alexander Graham Bell), while choosing $k=10$ gives the more practically convenient, ten-times-smaller unit "decibel" (dB), so that $\Delta L=10\ln(I_1/I_0)$ decibel; converting the natural logarithm to the more commonly used base-10 logarithm for p …