note - chapter 16

8/11/2019 Note - Chapter 16

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Chapter 16

Sound and Hearing

16.1 Sound Waves

Sounds are longitudinal waves produced by the vibrations

of material objects. Your voice results from the vibrations

of your vocal cords.

The frequency of the sound waves equals the frequency of

the vibrating source.

The audible range of frequencies (for a loud tone of

intensity level 80 dB) by a human of good hearing is from

about 20 Hz to about 20,000 Hz .

(ii) infrasonic waves: Hz f 20! (earthquakes, thunder)

(iii) ultrasonic waves: Hz f 000,20!

Acoustics is the branch of physics that deals with the

study of sound.

We can describe sound waves either as

(i) changes in the local pressure in the medium or as

(ii) displacements of the air molecules from their

equilibrium positions.



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As the source of sound vibrates, it produces a periodic

series of compressions and rarefactions in the medium

surrounding it. Compressions are regions of high density

and pressure (higher than average), while rarefactions areregions of low density and pressure (lower than average).

Ears and microphones detect sound by sensing pressure

differences, not displacements, so it is useful to describe

sound in terms of pressure fluctuations.

Let p(x,t) be an instantaneous pressure fluctuation in a

sound wave at any point x at time t . That is, p(x,t) is the

amount by which pressure differs from normal

atmospheric pressure P o , so p(x,t) is a gauge pressure.

One may write the pressure fluctuation p(x,t) in a medium(solid, liquid, or gas) in a sound wave propagating in the

positive x-direction as

p( x,t ) = B k A sin(kx "# t )



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16.2 Speed of Sound Waves

As a sound wave travels along a medium, the

compressions and rarefactions travel along the medium.

A. Speed of sound in a fluid:

The speed of sound in a fluid (gas or liquid) is given by

v = B

"

where

B = Bulk modulus of the fluid

! = density of fluid

specifically, the speed of sound in an ideal gas may be

written as

v =" RT

M

where

! = ratio of specific heat capacities. This is a quantity that

characterizes the thermal properties of the gas

R = universal gas constant = 8.314 J mol-1

K M = molar mass of the gas

The speed of propagation v of a sound wave in air

depends on wind conditions and air temperature. The



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speed of sound waves in air at temperature T c in Celsius is

given by

v " 331+ 0.6 T c( ) in meters/second

Note that sound travels faster in warm air than in cold air.

This can lead to the refraction of sound. Sound refraction

refers to the bending in the direction of sound travel when

sound travels through a medium of uneven temperature.

The speed of propagation of a sound wave in air does NOT depend on loudness or frequency.



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B. Speed of sound in a solid:

The speed of sound in a solid rod is given by

v =Y

"

where

Y = Young’s modulus of the solid

! = density of fluid

An echo is a reflected sound wave.

Sound energy dissipates into thermal energy as the sound

travels in air. The energy of a high frequency sound wave

is transformed more rapidly into thermal energy than theenergy of a low frequency sound wave. Thus, sound

waves of low frequency travel farther (not faster) through

air than sound of high frequencies. This is the reason why

the foghorns of ships emit low frequency sounds!

Remember that a wave is any disturbance from an

equilibrium condition, which travels or propagates with

time from one region of space to another.



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16.3 Sound Intensity: Decibels

An essential aspect of wave propagation is energy

transfer. The intensity I of a traveling wave is defined asthe average rate at which energy is transported by the

wave, per unit area, across a surface perpendicular to the

direction of wave propagation. That is, the intensity is the

average power transported per unit area.

At the threshold of hearing , the human ear can detect

sounds with an intensity of as low as 10-12 W/m2. At the

threshold of pain, the intensity of sound is 1 W/m2.

Because of the wide range of sound intensities over which

the human ear is sensitive, a logarithmic intensity scale

rather than a linear intensity scale is convenient. The

intensity level " of a sound wave of intensity I is defined

as

" =10 log I

I o

#

$ %

&

' (

where I o is an arbitrary reference intensity, takes as

I o = 10-12

W/m2, corresponding to the intensity of the

faintest sound which can be heard. Intensity level ismeasured in units of decibels (abbreviated dB).



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If the intensity level of a sound wave is I o , or 10-12

W/m2,

its intensity level is 0 dB. The maximum intensity which

the ear can tolerate is 1 W/m2 , which corresponds to an

intensity level of 120 dB.

We can express the intensity I of the sound wave in terms

of the pressure amplitude P max of the sound wave by

I =

Pmax( )

2

2" v



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16.4 Standing Sound Waves and Normal Modes

A. String Instruments

As discussed in the previous chapter, the frequencies ofthe normal modes of vibrations may be calculated using

!"

#$%

&=

L

vn f n

2 ( ),...4,3,2,1=n

Becauseµ

= Tensionv , one can express the natural

frequencies of vibration of a stretched string as

f n =n

2 L

Tension

µ ( ,...4,3,2,1=n

The lowest allowed natural frequency of vibration is

called the fundamental frequency. Any integer multiple of

the fundamental frequency is called a harmonic. Thus,

!1

f fundamental frequency

µ

=!

tension

L

f

2

1

1

!=12

2 f f second harmonic

!=13

3 f f third harmonic, and so on.



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All the even and odd harmonics are present.

B. Open Pipe (pipe open at both ends)



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! same results as stretched string clamped at both ends:

!"

#

$%

&=

L

v

n f n 2 ( ),...4,3,2,1=n

where now v is the speed of sound in the air column.

C. Stopped Pipe (column of air in a pipe open at one

end, closed at the other end)

In general, L = " n#

4where ( " n =1,3,5,7,...) so that



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" =4 L

# n

and the normal mode frequencies f = v"

are thus

f n = " nv

4 L

#

$ %

&

' ( " n =1, 3, 5,...( )

!1

f fundamental frequency L

v f

41 =!

!=13

3 f f third harmonic

!=

15 5

f f fifth harmonic, and so on.

Only the odd harmonics are present.

16.7 Beats

Beats are variations in loudness. They occur whenever

two waves of slightly different frequencies interfere.



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The number of beats heard per second =12

f f ! , that is,

the difference between the frequencies of the two

interfering waves.

16.8 The Doppler Effect for Sound

Christian Doppler (1803 – 1853)

The positive reference direction is always taken from thelistener to the source!

Let,

v = speed of sound in air = 343 m/s at room temperature

(always positive).

v s ! speed of the source

v L ! speed of the listener (observer) f s " frequency of the sound emitted by the source

f L " frequency of the sound heard by the listener.

The master equation for the Doppler effect for sound is

f L

v ± v L

=

f s

v ± vs



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When you solve a problem having to do with the Doppler

effect, there would be only four possibilities. These are:

(a) source and listener traveling toward each other inopposite directions

f L

v + v L

=

f s

v " vs

(b) source and listener traveling away from each other inopposite directions

f L

v " v L

=

f s

v + vs

(c) source and listener traveling in the same direction

with the listener following the source

f L

v + v L

=

f s

v + vs

(d) source and listener traveling in the same direction

with the source following the listener

f L

v " v L

=

f s

v " vs



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Doppler Effect for Electromagnetic Waves

In the frame of reference in which the receiver (listener or

observer) is at rest, the source of EM waves (light) ismoving relative to the receiver with speed v.

f S = frequency of the EM waves emitted by the source

f R = frequency of the EM waves measured or received by

the receiver

c = speed of the EM waves (light)

v = speed of the source relative to the receiver

There are two possibilities:

(a) Source approaching the receiver: here f R > f S(Blue Shift)

f R = f S c+ v

c" v

(b) Source receding away from the receiver: here f R < f S

(Red Shift)

f R = f S c" v

c+ v



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We show in problem 16.78 that when v << c, one can

write the Doppler effect formula as

f R " f S 1±vc

# $ % &

' (

where we use the positive sign when we have a blue shift

(approaching), and the negative sign when we have a red

shift (moving away).

note - chapter 16

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