# 17.3 Sound intensity  (Page 3/14)

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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

$\beta \left(\text{dB}\right)=10\phantom{\rule{0.2em}{0ex}}{\text{log}}_{10}\left(\frac{I}{{I}_{0}}\right),$

where ${I}_{0}={10}^{-12}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ 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 ( ${10}^{\text{−12}}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ ). 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 ${10}^{-12}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}$ 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 ${10}^{-16}\phantom{\rule{0.2em}{0ex}}\text{W}$ 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 ${10}^{-9}\phantom{\rule{0.2em}{0ex}}\text{atm}\text{.}$

Sound intensity levels and intensities
Sound intensity level $\beta$ (dB) Intensity I $\left({\text{W/m}}^{2}\right)$ Example/effect
0 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-12}$ Threshold of hearing at 1000 Hz
10 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-11}$ Rustle of leaves
20 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-10}$ Whisper at 1-m distance
30 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-9}$ Quiet home
40 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-8}$ Average home
50 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-7}$ Average office, soft music
60 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-6}$ Normal conversation
70 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-5}$ Noisy office, busy traffic
80 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}$ Loud radio, classroom lecture
90 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-3}$ Inside a heavy truck; damage from prolonged exposure [1]
100 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-2}$ Noisy factory, siren at 30 m; damage from 8 h per day exposure
110 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-1}$ Damage from 30 min per day exposure
120 1 Loud rock concert; pneumatic chipper at 2 m; threshold of pain
140 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{2}$ Jet airplane at 30 m; severe pain, damage in seconds
160 $1\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{4}$ 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] ).

Ratios of intensities and corresponding differences in sound intensity levels
${I}_{2}\text{/}{I}_{1}$ ${\beta }_{2}-{\beta }_{1}$
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

## Calculating sound intensity levels

Calculate the sound intensity level in decibels for a sound wave traveling in air at $0\text{°C}$ and having a pressure amplitude of 0.656 Pa.

## Strategy

We are given $\Delta p$ , so we can calculate I using the equation $I=\frac{{\left(\text{Δ}p\right)}^{2}}{2\rho {v}_{\text{w}}}.$ Using I , we can calculate $\beta$ straight from its definition in $\beta \left(dB\right)=10\phantom{\rule{0.2em}{0ex}}{\text{log}}_{10}\left(\frac{I}{{I}_{0}}\right).$

## Solution

1. Identify knowns:
Sound travels at 331 m/s in air at $0\text{°C}\text{.}$
Air has a density of $1.29\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}$ at atmospheric pressure and $0\text{°C}\text{.}$
2. Enter these values and the pressure amplitude into $I=\frac{{\left(\text{Δ}p\right)}^{2}}{2\rho v}.$
$I=\frac{{\left(\text{Δ}p\right)}^{2}}{2\rho v}=\frac{{\left(0.656\phantom{\rule{0.2em}{0ex}}\text{Pa}\right)}^{2}}{2\left(1.29\phantom{\rule{0.2em}{0ex}}{\text{kg/m}}^{3}\right)\left(331\phantom{\rule{0.2em}{0ex}}\text{m/s}\right)}=5.04\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{-4}\phantom{\rule{0.2em}{0ex}}{\text{W/m}}^{2}.$
3. Enter the value for I and the known value for ${I}_{0}$ into $\beta \left(\text{dB}\right)=10\phantom{\rule{0.2em}{0ex}}{\text{log}}_{10}\left(I\text{/}{I}_{0}\right).$ Calculate to find the sound intensity level in decibels:
$10\phantom{\rule{0.2em}{0ex}}{\text{log}}_{10}\left(5.04\phantom{\rule{0.2em}{0ex}}×\phantom{\rule{0.2em}{0ex}}{10}^{8}\right)=10\left(8.70\right)\text{dB}=87\phantom{\rule{0.2em}{0ex}}\text{dB}\text{.}$

## Significance

This 87-dB sound has an intensity five times as great as an 80-dB sound. So a factor of five in intensity corresponds to a difference of 7 dB in sound intensity level. This value is true for any intensities differing by a factor of five.

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