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The sound intensity level $\beta $ of a sound, measured in decibels , having an intensity I in watts per meter squared, is defined as
where ${I}_{0}={10}^{\mathrm{-12}}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ is a reference intensity, corresponding to the threshold intensity of sound that a person with normal hearing can perceive at a frequency of 1.00 kHz. It is more common to consider sound intensity levels in dB than in ${\text{W/m}}^{2}.$ How human ears perceive sound can be more accurately described by the logarithm of the intensity rather than directly by the intensity. Because $\beta $ is defined in terms of a ratio, it is a unitless quantity, telling you the level of the sound relative to a fixed standard ( ${10}^{\text{\u221212}}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ ). The units of decibels (dB) are used to indicate this ratio is multiplied by 10 in its definition. The bel, upon which the decibel is based, is named for Alexander Graham Bell , the inventor of the telephone.
The decibel level of a sound having the threshold intensity of ${10}^{\mathrm{-12}}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ is $\beta =0\phantom{\rule{0.2em}{0ex}}\text{dB,}$ because ${\text{log}}_{10}1\phantom{\rule{0.2em}{0ex}}=\phantom{\rule{0.2em}{0ex}}0.$ [link] gives levels in decibels and intensities in watts per meter squared for some familiar sounds. The ear is sensitive to as little as a trillionth of a watt per meter squared—even more impressive when you realize that the area of the eardrum is only about $1\phantom{\rule{0.2em}{0ex}}{\text{cm}}^{2},$ so that only ${10}^{\mathrm{-16}}\phantom{\rule{0.2em}{0ex}}\text{W}$ falls on it at the threshold of hearing. Air molecules in a sound wave of this intensity vibrate over a distance of less than one molecular diameter, and the gauge pressures involved are less than ${10}^{\mathrm{-9}}\phantom{\rule{0.2em}{0ex}}\text{atm}\text{.}$
Sound intensity level $\beta $ (dB) | Intensity I $\left({\text{W/m}}^{2}\right)$ | Example/effect |
---|---|---|
0 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-12}$ | Threshold of hearing at 1000 Hz |
10 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-11}$ | Rustle of leaves |
20 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-10}$ | Whisper at 1-m distance |
30 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-9}$ | Quiet home |
40 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-8}$ | Average home |
50 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-7}$ | Average office, soft music |
60 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-6}$ | Normal conversation |
70 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-5}$ | Noisy office, busy traffic |
80 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-4}$ | Loud radio, classroom lecture |
90 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-3}$ | Inside a heavy truck; damage from prolonged exposure ^{[1]} |
100 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-2}$ | Noisy factory, siren at 30 m; damage from 8 h per day exposure |
110 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{-1}$ | Damage from 30 min per day exposure |
120 | 1 | Loud rock concert; pneumatic chipper at 2 m; threshold of pain |
140 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{2}$ | Jet airplane at 30 m; severe pain, damage in seconds |
160 | $1\phantom{\rule{0.2em}{0ex}}\times \phantom{\rule{0.2em}{0ex}}{10}^{4}$ | Bursting of eardrums |
An observation readily verified by examining [link] or by using [link] is that each factor of 10 in intensity corresponds to 10 dB. For example, a 90-dB sound compared with a 60-dB sound is 30 dB greater, or three factors of 10 (that is, ${10}^{3}$ times) as intense. Another example is that if one sound is ${10}^{7}$ as intense as another, it is 70 dB higher ( [link] ).
${I}_{2}\text{/}{I}_{1}$ | ${\beta}_{2}-{\beta}_{1}$ |
---|---|
2.0 | 3.0 dB |
5.0 | 7.0 dB |
10.0 | 10.0 dB |
100.0 | 20.0 dB |
1000.0 | 30.0 dB |
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