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d V V = lim Δ x 0 A [ s ( x + Δ x , t ) s ( x , t ) ] A Δ x = s ( x , t ) x .

The fractional change in volume is related to the pressure fluctuation by the bulk modulus     β = Δ p ( x , t ) d V / V . Recall that the minus sign is required because the volume is inversely related to the pressure. (We use lowercase p for pressure to distinguish it from power, denoted by P .) The change in pressure is therefore Δ p ( x , t ) = β d V V = β s ( x , t ) x . If the sound wave is sinusoidal, then the displacement as shown in [link] is s ( x , t ) = s max cos ( k x ω t + ϕ ) and the pressure is found to be

Δ p ( x , t ) = β d V V = β s ( x , t ) x = β k s max sin ( k x ω t + ϕ ) = Δ p max sin ( k x ω t + ϕ ) .

The intensity of the sound wave is the power per unit area, and the power is the force times the velocity, I = P A = F v A = p v . Here, the velocity is the velocity of the oscillations of the medium, and not the velocity of the sound wave. The velocity of the medium is the time rate of change in the displacement:

v ( x , t ) = y s ( x , t ) = y ( s max cos ( k x ω t + ϕ ) ) = s max ω sin ( k x ω t + ϕ ) .

Thus, the intensity becomes

I = Δ p ( x , t ) v ( x , t ) = β k s max sin ( k x ω t + ϕ ) [ s max ω sin ( k x ω t + ϕ ) ] = β k ω s max 2 sin 2 ( k x ω t + ϕ ) .

To find the time-averaged intensity over one period T = 2 π ω for a position x , we integrate over the period, I = β k ω s max 2 2 . Using Δ p max = β k s max , v = β ρ , and v = ω k , we obtain

I = β k ω s max 2 2 = β 2 k 2 ω s max 2 2 β k = ω ( Δ p max ) 2 2 ( ρ v 2 ) k = v ( Δ p max ) 2 2 ( ρ v 2 ) = ( Δ p max ) 2 2 ρ v .

That is, the intensity of a sound wave is related to its amplitude squared by

I = ( Δ p max ) 2 2 ρ v .

Here, Δ p max is the pressure variation or pressure amplitude in units of pascals (Pa) or N/m 2 . The energy (as kinetic energy 1 2 m v 2 ) of an oscillating element of air due to a traveling sound wave is proportional to its amplitude squared. In this equation, ρ is the density of the material in which the sound wave travels, in units of kg/m 3 , and v is the speed of sound in the medium, in units of m/s. The pressure variation is proportional to the amplitude of the oscillation, so I varies as ( Δ p ) 2 . This relationship is consistent with the fact that the sound wave is produced by some vibration; the greater its pressure amplitude, the more the air is compressed in the sound it creates.

Human hearing and sound intensity levels

As stated earlier in this chapter, hearing is the perception of sound. The hearing mechanism involves some interesting physics. The sound wave that impinges upon our ear is a pressure wave. The ear is a transducer    that converts sound waves into electrical nerve impulses in a manner much more sophisticated than, but analogous to, a microphone. [link] shows the anatomy of the ear.

Picture is a drawing of an ear. It shows the ear canal finishing with the eardrum. Hammer connected to the anvil is in the in the contact with the eardrum. Behind the eardrum is the hammer and the anvil. The anvil is connected to the stirrup which is attached to the oval window. Cochlea, cochlear nerve and vestibular nerve are in contact with the stirrup.
The anatomy of the human ear.

The outer ear, or ear canal, carries sound to the recessed, protected eardrum. The air column in the ear canal resonates and is partially responsible for the sensitivity of the ear to sounds in the 2000–5000-Hz range. The middle ear converts sound into mechanical vibrations and applies these vibrations to the cochlea.

Watch this video for a more detailed discussion of the workings of the human ear.

The range of intensities that the human ear can hear depends on the frequency of the sound, but, in general, the range is quite large. The minimum threshold intensity that can be heard is I 0 = 10 −12 W/m 2 . Pain is experienced at intensities of I pain = 1 W/m 2 . Measurements of sound intensity (in units of W/m 2 ) are very cumbersome due to this large range in values. For this reason, as well as for other reasons, the concept of sound intensity level was proposed.

Questions & Answers

who discover periodic table?
lovet Reply
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
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.
what is the Greek name for calcium
Oniyide Reply
different types of wave
Yog Reply
longitudinal and transverse waves
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 ?
Gobinda Reply
want is meant by the term solar system
jafar Reply
it refers to the sun and all heavenly bodies revolving around it.
excatly...for sure
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.
what is meant by total internal reflection
Akshay Reply
what iw meant by total internal reflection
Lorentz force?
study fibre optics. .you will get total internal reflection
what is Lorentz force?
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
motion in strat line where is this chapter
Vijaybhai Reply
motion in straight line is kinematic's part
what is Lorentz force?
jyotirmayee Reply
what is maxwell electromagnetic law?
at what angle should the two forces 2p and root 2p acts so that the resultant force is p root 10
Akshay Reply
what answer fir this
what's the working difference between a dynamo and a pump?
Piyali Reply
a dynamo is basically a dc generator while pump is usually equipped with a motor
a dynamo converts mechanical energy to electrical while a pump is opposite to that
why sea water looks bluish?
Piyali Reply
cuz the sky is blue...
you see the reflexion of the "blue" sky in the water
somewhere sea water turns green why?
never seen bro... are u sure ?
ur answer was correct but due to the presence of phytoplankton color can be changed near the shore
waaaww... you re awesome
because of the reflection of the sky
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
how can someone identify sea water from rain water
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
i need solutions of unuversity pbysucs volume 1
Vimla Reply
me too
Help us if anyone knows
bring questions
actually if u wanted whole book solution then u should buy the solution book
from where
where do u live ,,,,,if u live in Delhi then at bellsarayeiii or from stationary store ,,,,chatarpurrr ,,,,,there is a popular books store,,,,,u have to buy from there
Kota Rajasthan
so strange,,,,,r u preparing for pmt,,,?
for IIT
then concern near book store
in Rajasthan
or ask questions here
thanks for your help
its fine,,most welcome
what is three dimensional coordinate system?
Sachindra Reply
considering the change of vectors in all three dimensions of space
direction co-sign of vector questions
Vipin Reply
opposite mean
what is the different between action and reaction?
Asali Reply
action is external force. Reaction exists because of the external action. Reaction is mostly an internal force
The difference is said in the word itself. There is no existance of reaction without an application of action. So action occur first then reaction. Reaction may produce in body itself or to other body i.e., it may be internal or external.
or we can say , reaction is the result of action
Practice Key Terms 8

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Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
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