# 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.

What is the equation illustrating Williamsons ether synthesis
what is simple harmonic motion
examples: vibrating prongs of a tuning fork and a guittar string.
Salman
It is a repetitive periodic motion of a system about an equilibrium position
Felix
SHM is the repitition process of to and fro motion.
Younus
SHM is the motion in which the restoring force is directly proportional to the displacement of body from its mean position and is opposite in direction to the displacement. From Hooke's law F=-kx
Kushal
SHM is the motion in which the restoring force is directly proportional to the displacement of body from its mean position and is opposite in direction to the displacement. From Hooke's law F=-kx
Kushal
what is a wave?
show that coefficient of friction of solid block inclined at an angle is equivalent to trignometric tangent of angle
DAVID
Wave is the transfer of energy due to the periodic vibration of the particles in the medium.
Kushal
wave is the transfer of energy
Vindora
Wave is the transfer of particles in a fluid or any way.
Younus
thanks for that definition.
Hi everyone please can dere be motion without force?
Lafon
no...
Enyia
Thanks
Lafon
hi
Omomaro
whats is schrodinger equation
Omomaro
l went spiral spring
Xalat
what is position?
position is simply where you are or where you were
Shii
position is the location of an object with respect to a two or three dimensional axes or space.
Bamidele
Can dere be motion without force?
Lafon
what is the law of homogeinity?
two electric lines of force never interested each other. why?
if two electric lines of force intersect eachother then their will be two tangent at a point which represent the two forces which is impossible.
Amar
proof that for BBC lattice structure 4r\root 5 and find Apf for the BBC structure
what is physics?
physics is deine as the specific measrument of of volume, area,nd distances...
Olakojo
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
Enyia
yes
Alex
proof that for a BBC lattice structure a= 4r/ root 5 find the APF for the BBC structure
Eric
if a string of 2m is suspended an an extended 3m elasticity is been applied.... is hooks law obeyed?
tell me conceptual quetions of mechanics
I want to solve a physical question
ahmed
ok
PUBG
a displacement vector has a magnitude of 1.62km and point due north . another displacement vector B has a magnitude of 2.48 km and points due east.determine the magnitude and direction of (a) a+ b and (b) a_ b
quantum
George
a+b=2.9
SUNJO
a+b
Yekeen
use Pythogorous
Dhritwan
A student opens a 12kgs door by applying a constant force of 40N at a perpendicular distance of 0.9m from the hinges. if the door is 2.0m high and 1.0m wide determine the magnitude of the angular acceleration of the door. ( assume that the door rotates freely on its hinges.) please assist me to d
Mike
what is conditions met to produce shm
what is shm
Manzoor
shm?
Grant
Why is Maxwell saying that light is an electromagnetic wave?
Bong
1st condition; It(th e BBC's system) must have some inertia which will enable it to possess Kinetic energy 2. must be able to store potential energy
Calleb
I meant "the system" not the BBC'S....."
Calleb
Manzoor
kindly tell us the name of your university
Manzoor
GUlam Ishaq Khan INSTITUTE of engineering science
ali
Department of Environment Ionian University Zante Greece
why light wave travel faster than sounds
Why light travel faster than sounds?
ALI
Light travel faster than sound because it does not need any medium to travel through.
alhassan
when an aeroplane flies....why it does not fall on the earth?
Frazali
As an aeroplane moves, it hits a wind,we have the wind flowing at the upper and lower zone of the aeroplane, the one that is moving on the upper zone moves at a greater speed than that of the lower zone, this creates a low pressure on the upper zone and a greater pressure at the lower zone.
Kipkoech
which thing of aeroplane moves it upward?
Frazali
good question
Manzoor
Barataa
am pleased to join the group
Nesru
yea
caleb
It a privilege to be here
olajire
hi
Awode
hello
Manzoor
Light speed is more than sound speed. C=3×10*8m/s V=320-340 m/s
siva
A body of mass 2kg slides down a rough plane inclined to horizontal at 30degrees. find the energy that is wasted as a result of friction if the co-efficient of kinetic f