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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

What is the equation illustrating Williamsons ether synthesis
Queen Reply
what is simple harmonic motion
Faith Reply
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?
DAVID Reply
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.
Dodou Reply
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?
Adhar Reply
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?
auson Reply
two electric lines of force never interested each other. why?
Sujit Reply
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
Eric Reply
what is physics?
Abdulaziz Reply
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?
Enyia Reply
tell me conceptual quetions of mechanics
Syeda Reply
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
Kou Reply
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
Enocy Reply
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
what a answer bro
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
ALI Reply
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
about force
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
official Reply
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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