Sound levels: the decibel scale
The range of intensities the human ear can detect is extremely wide: the weakest sound we can still hear has an intensity of \(10^{-12}\,\mathrm{W} / \mathrm{m}^2,\) whilst the threshold of pain, which might then be followed by eardrum rupture occurs round intensities of \(100\,\mathrm{W} / \mathrm{m}^2\) — this means a variability of 14 orders of magnitude. For this reason, it is convenient to express sound intensities on a logarithmic scale, compared to the intensity associated with the threshold of hearing.
Sound level
The sound level associated with a given sound intensity is ten times the logarithm to the base of ten of the ratio of the given intensity \(I\) to the intensity \(I_0\) at the threshold of hearing at 1000 Hz:
\[\beta := 10 \cdot \log_{10}\frac{I}{I_0},\]
where \(I_0 = 10^{-12}\,\mathrm{W} / \mathrm{m}^2\) is the intensity at the threshold of hearing at 1000 Hz.
Values expressed on this scale are called decibel values — the ‘deci’ prefix indicates the fact that we refined the scale with a multiplier of 10 in the definition. Due to the logarithmic nature of the decibel scale, an increase of 10 dB in the sound level indicates an order of magnitude (that is, a tenfold) increase in the intensity.
