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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

Suppose the master cylinder in a hydraulic system is at a greater height than the cylinder it is controlling. Explain how this will affect the force produced at the cylinder that is being controlled.
Louise Reply
Why is popo less than atmospheric? Why is popo greater than pipi?
Louise
The old rubber boot shown below has two leaks. To what maximum height can the water squirt from Leak 1? How does the velocity of water emerging from Leak 2 differ from that of Leak 1? Explain your responses in terms of energy.
Louise
David rolled down the window on his car while driving on the freeway. An empty plastic bag on the floor promptly flew out the window. Explain why.
Louise Reply
the pressure differential exerted a force on the bag greater than the gravitational force holding it on the floor.
gtitboi
what is angular velocity
Sthandazile Reply
The rate of change in angular displacement is defined as angular velocity.
Manorama
a length of copper wire was measured to be 50m with an uncertainty of 1cm, the thickness of the wire was measured to be 1mm with an uncertainty of 0.01mm, using a micrometer screw gauge, calculate the of copper wire used
Nicole Reply
What is the answer please
Mustapha
If centripetal force is directed towards the center,why do you feel that you're thrown away from the center as a car goes around a curve? Explain
Maira Reply
if there is a centripetal force it means that there's also a centripetal acceleration, getting back to your question, just imagine what happens if you pull out of a car when it's quickly moving or when you try to stop when you are running fast, anyway, we notice that there's always a certain force..
Lindomar
... that tends to fight for its previous direction when you try to attribute to it an opposite one ou try to stop it.The same thing also happens whe a car goes around a curve, the car it self is designed to a"straight line"(look at the position of its tyres, mainly the back side ones), so...
Lindomar
... whenever it goes around a curve, it tends to throw away its the occupiers, it's given to the fact that it must interrupt its initial direction and take a new one.
Lindomar
Which kind of wave does wind form
Matthias Reply
calculate the distance you will travel if you mantain an average speed of 10N m/s for 40 second
Abdulai Reply
400m/s
Feng
hw to calculate the momentum of the 2000.0 elephant change hunter at a speed of 7.50 m/s
Kingsley Reply
how many cm makes 1 inches
Hassan Reply
2.5
omwoyo
2.54cm=1inche
omwoyo
how do we convert from m/s to km/hr
Toni Reply
When paddling a canoe upstream, it is wisest to travel as near to the shore as possible. When canoeing downstream, it may be best to stay near the middle. Explain why?
SANA Reply
Explain why polarization does not occur in sound
Nuradeen
one ship sailing east with a speed of 7.5m/s passes a certain point at 8am and a second ship sailing north at the same speed passed the same point at 9.30am at what distance are they closet together and what is the distance between them then
Kuber Reply
density of a subtance is given as 360g/cm,put it in it s.i unit form
Linda Reply
if m2 is twice of m1. find the ration of kinetic energy in COM system to lab system of elastic collision
Raman Reply
What is a volt equal to?
Clifton 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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