<< Chapter < Page Chapter >> Page >
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

definition of inertia
philip Reply
the reluctance of a body to start moving when it is at rest and to stop moving when it is in motion
charles
An inherent property by virtue of which the body remains in its pure state or initial state
Kushal
why current is not a vector quantity , whereas it have magnitude as well as direction.
Aniket Reply
why
daniel
the flow of current is not current
fitzgerald
bcoz it doesn't satisfy the algabric laws of vectors
Shiekh
The Electric current can be defined as the dot product of the current density and the differential cross-sectional area vector : ... So the electric current is a scalar quantity . Scalars are related to tensors by the fact that a scalar is a tensor of order or rank zero .
Kushal
what is binomial theorem
Tollum Reply
hello are you ready to ask aquestion?
Saadaq Reply
what is binary operations
Tollum
What is the formula to calculat parallel forces that acts in opposite direction?
Martan Reply
position, velocity and acceleration of vector
Manuel Reply
hi
peter
hi
daniel
hi
Vedisha
*a plane flies with a velocity of 1000km/hr in a direction North60degree east.find it effective velocity in the easterly and northerly direction.*
imam
hello
Lydia
hello Lydia.
Sackson
What is momentum
isijola
hello
Saadaq
A rail way truck of mass 2400kg is hung onto a stationary trunk on a level track and collides with it at 4.7m|s. After collision the two trunk move together with a common speed of 1.2m|s. Calculate the mass of the stationary trunk
Ekuri Reply
I need the solving for this question
philip
is the eye the same like the camera
EDWIN Reply
I can't understand
Suraia
same here please
Josh
I think the question is that ,,, the working principal of eye and camera same or not?
Sardar
yes i think is same as the camera
muhammad
what are the dimensions of surface tension
samsfavor
why is the "_" sign used for a wave to the right instead of to the left?
MUNGWA Reply
why classical mechanics is necessary for graduate students?
khyam Reply
classical mechanics?
Victor
principle of superposition?
Naveen Reply
principle of superposition allows us to find the electric field on a charge by finding the x and y components
Kidus
Two Masses,m and 2m,approach each along a path at right angles to each other .After collision,they stick together and move off at 2m/s at angle 37° to the original direction of the mass m. What where the initial speeds of the two particles
MB
2m & m initial velocity 1.8m/s & 4.8m/s respectively,apply conservation of linear momentum in two perpendicular directions.
Shubhrant
A body on circular orbit makes an angular displacement given by teta(t)=2(t)+5(t)+5.if time t is in seconds calculate the angular velocity at t=2s
MB
2+5+0=7sec differentiate above equation w.r.t time, as angular velocity is rate of change of angular displacement.
Shubhrant
Ok i got a question I'm not asking how gravity works. I would like to know why gravity works. like why is gravity the way it is. What is the true nature of gravity?
Daniel Reply
gravity pulls towards a mass...like every object is pulled towards earth
Ashok
An automobile traveling with an initial velocity of 25m/s is accelerated to 35m/s in 6s,the wheel of the automobile is 80cm in diameter. find * The angular acceleration
Goodness Reply
(10/6) ÷0.4=4.167 per sec
Shubhrant
what is the formula for pressure?
Goodness Reply
force/area
Kidus
force is newtom
Kidus
and area is meter squared
Kidus
so in SI units pressure is N/m^2
Kidus
In customary United States units pressure is lb/in^2. pound per square inch
Kidus
who is Newton?
John Reply
scientist
Jeevan
a scientist
Peter
that discovered law of motion
Peter
ok
John
but who is Isaac newton?
John
a postmodernist would say that he did not discover them, he made them up and they're not actually a reality in itself, but a mere construct by which we decided to observe the word around us
elo
how?
Qhoshe
Besides his work on universal gravitation (gravity), Newton developed the 3 laws of motion which form the basic principles of modern physics. His discovery of calculus led the way to more powerful methods of solving mathematical problems. His work in optics included the study of white light and
Daniel
and the color spectrum
Daniel
Practice Key Terms 8

Get the best University physics vol... course in your pocket!





Source:  OpenStax, University physics volume 1. OpenStax CNX. Sep 19, 2016 Download for free at http://cnx.org/content/col12031/1.5
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 1' conversation and receive update notifications?

Ask