17.3 Sound intensity and sound level  (Page 2/7)

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$\beta \phantom{\rule{0.25em}{0ex}}\left(\text{dB}\right)=\text{10}\phantom{\rule{0.25em}{0ex}}{\text{log}}_{\text{10}}\left(\frac{I}{{I}_{0}}\right),$

where ${I}_{0}={\text{10}}^{\text{–12}}\phantom{\rule{0.25em}{0ex}}{\text{W/m}}^{2}$ is a reference intensity. In particular, ${I}_{0}$ is the lowest or threshold intensity of sound a person with normal hearing can perceive at a frequency of 1000 Hz. Sound intensity level is not the same as 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 ( ${\text{10}}^{\text{–12}}\phantom{\rule{0.25em}{0ex}}{\text{W/m}}^{2}$ , in this case). 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.

Sound intensity levels and intensities
Sound intensity level β (dB) Intensity I (W/m 2 ) Example/effect
$0$ $1×{10}^{–12}$ Threshold of hearing at 1000 Hz
$10$ $1×{10}^{–11}$ Rustle of leaves
$20$ $1×{10}^{–10}$ Whisper at 1 m distance
$30$ $1×{10}^{–9}$ Quiet home
$40$ $1×{10}^{–8}$ Average home
$50$ $1×{10}^{–7}$ Average office, soft music
$60$ $1×{10}^{–6}$ Normal conversation
$70$ $1×{10}^{–5}$ Noisy office, busy traffic
$80$ $1×{10}^{–4}$ Loud radio, classroom lecture
$90$ $1×{10}^{–3}$ Inside a heavy truck; damage from prolonged exposure Several government agencies and health-related professional associations recommend that 85 dB not be exceeded for 8-hour daily exposures in the absence of hearing protection.
$100$ $1×{10}^{–2}$ Noisy factory, siren at 30 m; damage from 8 h per day exposure
$110$ $1×{10}^{–1}$ Damage from 30 min per day exposure
$120$ $1$ Loud rock concert, pneumatic chipper at 2 m; threshold of pain
$140$ $1×{10}^{2}$ Jet airplane at 30 m; severe pain, damage in seconds
$160$ $1×{10}^{4}$ Bursting of eardrums

The decibel level of a sound having the threshold intensity of ${\text{10}}^{–\text{12}}\phantom{\rule{0.25em}{0ex}}{\text{W/m}}^{2}$ is $\beta =0\phantom{\rule{0.25em}{0ex}}\text{dB}$ , because ${\text{log}}_{\text{10}}1=0$ . That is, the threshold of hearing is 0 decibels. [link] gives levels in decibels and intensities in watts per meter squared for some familiar sounds.

One of the more striking things about the intensities in [link] is that the intensity in watts per meter squared is quite small for most 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 ${1 cm}^{2}$ , so that only ${\text{10}}^{–\text{16}}$ 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 ${\text{10}}^{–9}$ atm.

Another impressive feature of the sounds in [link] is their numerical range. Sound intensity varies by a factor of ${\text{10}}^{\text{12}}$ from threshold to a sound that causes damage in seconds. You are unaware of this tremendous range in sound intensity because how your ears respond can be described approximately as the logarithm of intensity. Thus, sound intensity levels in decibels fit your experience better than intensities in watts per meter squared. The decibel scale is also easier to relate to because most people are more accustomed to dealing with numbers such as 0, 53, or 120 than numbers such as $1\text{.}\text{00}×{\text{10}}^{–\text{11}}$ .

One more observation readily verified by examining [link] or using $I={\frac{\left(\Delta p\right)}{2{\text{ρv}}_{\text{w}}}}^{2}$ 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, ${\text{10}}^{3}$ times) as intense. Another example is that if one sound is ${\text{10}}^{7}$ as intense as another, it is 70 dB higher. See [link] .

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