# 17.3 Sound intensity  (Page 3/14)

 Page 3 / 14

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.

#### Questions & Answers

how can I become confident in physics
practice more and more question ,,,,if u get trouble in any one question infact do that again and again.
jyotirmayee
is theory important or numericals
arshdeep
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jyotirmayee
which books I can follow for IIT jee prep for mcq and which for mcq
arshdeep
Can someone please tell what really happened to planet Pluto
what are the first 20 elements in periodic table.
H,O...
Arzoodan
what happened to the rest now, or have you forgotten?
Ejiba
who discover periodic table?
it wasn't discovered , it was made.and the person who made it was dmitri mendeleev. dobreinier and newland gave their laws before dmitri related periodic table but wasn't successful in their work
Ritik
Nope, numerous number of scientist had actually contributed in the making of periodic table. Dmitri Mendeleev succeeded making all the elements into the right order in accordance to their atomic number.
Dame
what is the Greek name for calcium
Greek word for calcium is asvestio while Latin name is calf meaning lime or limestone
Ejiba
different types of wave
longitudinal and transverse waves
Ravindra
a gun is kept in the state that it cannot move anywhere and the bullet is fired. Then what is the effect on the velocity of bullet and KE of gun ?
want is meant by the term solar system
it refers to the sun and all heavenly bodies revolving around it.
Danie
excatly...for sure
Arzoodan
in addition to Danie's, a solar system is a collection of planets and their moons, asteroids, and other objects bound together by the Star's gravitational force directly or indirectly.
Galiwango
what is meant by total internal reflection
what iw meant by total internal reflection
Akshay
Lorentz force?
jyotirmayee
study fibre optics. .you will get total internal reflection
Siddhansh
wha
jyotirmayee
what is Lorentz force?
jyotirmayee
a ray of light traveling at an angle of incidence greater than critical angle from denser to rarer medium is totally reflected back into the denser medium is called total internal reflaction
Manoj
motion in strat line where is this chapter
motion in straight line is kinematic's part
Ritik
yea
Manoj
this defination isn't correct
Arzoodan
motion in one dimension
Anil
what is Lorentz force?
what is maxwell electromagnetic law?
jyotirmayee
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
what answer fir this
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a dynamo is basically a dc generator while pump is usually equipped with a motor
vedanth
a dynamo converts mechanical energy to electrical while a pump is opposite to that
vedanth
nice
Piyali
okay
Friday
why sea water looks bluish?
cuz the sky is blue...
Mehdi
you see the reflexion of the "blue" sky in the water
Mehdi
somewhere sea water turns green why?
Piyali
never seen bro... are u sure ?
Mehdi
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
Piyali
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Mehdi
because of the reflection of the sky
Friday
rays coming from the sun consist of all 7 colours ie.VIBGYOR. when the ray strikes surface of water,all colors gets absorbed by it except blue which gets reflected by it.so we find the sea water appearing bluish
Ritik
how can someone identify sea water from rain water
Oniyide
i think it's not possible as because rainwater consists of water from all kind of water bodies ie.lakes,seas etc but u can predict if u have a sea nearby ur home or city
Ritik
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me too
Nirupam
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Nirupam
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john
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jyotirmayee
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Nirupam
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jyotirmayee
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jyotirmayee
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jyotirmayee
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Nirupam
its fine,,most welcome
jyotirmayee