PH 221-1D Fall 2013

Waves - II

Lectures 41-43

Chapter 17(Halliday/Resnick/Walker, Fundamentals of Physics 9th edition)

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Chapter 17 Waves II

In this chapter we will study sound waves and concentrate on the following topics:

Speed of sound waves Relation between displacement and pressure amplitude Interference of sound waves Sound intensity and sound level Beats The Doppler effect

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Sound waves are mechanical longitudinal waves that propagate in solids liquids and gases. Seismic waves used by oil explorers propagate in the earth’s crust. Sound waves generated by a sonar system propagate in the sea. An orchestra creates sound waves that propagate in the air.

The locus of the points of a sound wave that has the same displacement is called a “wavefront”. Lines perpendicular to the wavefronts are called “rays” and they point along the direction which the sound wave propagates. An example of a point source of sound waves is given in the figure. We assume that the surrounding medium is isotropic i.e. sound propagates with the same speed for all directions. In this case the sound wave spreads outwards uniformly and the wavefronts are spheres centered at the point source. The single arrows indicate the rays. The double arrows indicate the motion of the molecules of the medium in which sound propagates. 3

Δp

If we apply an overpressure on an objectof volume , this results in a change of volume as shownin the figure. The bulk modulus of the compressed material

is defined as:

p

p

V V

B

Bulk modulus

The negative sign denotes the in volumewhen is positive

SI unit: the Pasca

.

/l

V

p

V

Note : decrease

Using the above definition of the bulk modulus and combining it withNewton's second law one can show that the speed of sound in a homogeneous isotropic medium with bulk modulus andB

The speed of sound

density

is given by the equation:

Bulk modulus is smaller for more compressible

media. Such media exhibit lower speed of sound. Denser materials (higher )

p

v

VB

B

V

Note 1 :

Note 2 : have lower speed of sound

Bv

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The Speed of Sound

5

6

Example: Thunder, lightning and a rule of thumb

There is a rule of thumb for estimating how far away a thunderstorm is. You can estimate your distance from a bolt of lightning by counting the seconds between seeing the flash and hearing the thunder and then dividing by 3 to obtain the distance in km. Why does this rule work?

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The Speed of SoundProblem : Both krypton (Kr) and neon (Ne) can be approximated at monatomic ideal gases. The atomic mass of krypton is 83.8 u, while that of neon is 20.2 u. A loudspeaker produces a sound whose wavelength in krypton is 1.25m. If the loudspeaker were used to produce sound of the same frequency in neon at the same temperature, what would be the wavelength?

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Consider the tube filled with air shown in the figure.We generate a harmonic sound wave traveling to theright along the axis of the tube. One simple methodis to place a speaker

Traveling sound waves.

at the left end of the tube anddrive it at a particular frequency. Consider an airelement of thickness which is located at position

before the sound wave is generated. This is known as the "e

xx

qulibrium position" of the element. Underthese conditions the pressure inside the tube is constantIn the presence of the sound wave the elementoscillates about the equlibrium position. At the sametime the pressure at the location of the element oscillates about its static value. The sound wave in the tube can be described using one of two parameters:

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One such parameter is the distance , of theelement from its equilibrium position

. The constant is the of the wave. The angular w

, co

av

sm m

s x t

ss x t s kx t

Traveling sound waves.

displacement amplitude enumber and the angular frequency

have the same meaning as in the case of the transversewaves studied in chapter 16. The second possibility is to use the pressure variation

from the static value.

k

p

The constant p is the wave's pressureThe two amplitudes are connected by the equation:

The displacement and the pressure variationhave a phase dif

, si

fe

n

r

m

m m

m

p x t p kx t

p v s

amplitude.

Note :ence of 90 . As a result when

one parameter has a maximum the other has a minimum and vice versa.

m mp v s

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1 2Consider two point sources of sound waves S and Sshown in the figure. The two sources are in phase and emit sound waves of the same frequency.Waves from both sources arrive at point P wh

Interference

1 2 1 2

ose distance from S and S is and respectively. The two waves interfere at point P.

L L

1 1 1

2 2 2

At time the phase of sound wave 1 arriving from S at point P is At time the phase of sound wave 2 arriving from S at point P is In general the two waves at P have a phase diffe

t kL tt kL t

2 1 2 1 2 1 2 1

2 1

rence 2

The quantity is known as the " "

between the two waves. Thus

Here is the wavelength of the two wav s.

2

e

kL t kL t k L L

L

L L

L L L

path length difference

2 L

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1 2

The wave at P resulting from the interference of the two waves that arrive from S and S has a maximum amplitude when the phase difference 2

2 0,1, 2,... . 2

m

m L m

Constructive intereference.

0, , 2 ,. ..L

L m

1 2

The wave at P resulting from the interference of the two waves that arrive from S and S has a minimum amplitude when the phase difference

2 1 0,1, 2,... . m m

Destructive intereference.

1

2 2 1

/ 2, 3 / 2, 5 / 2,... 2

L m

L m

L

Δ equal to an integral multiple of λ L constructive interference→

Δ equal to a half-integral multiple of λL destructive interference12

At an open-air concert on a hot day (Tc = 25°C, Vs = 346.5 m/s), a person sits at a location that 7.0 m and 9.1 m respectively from speakers at each side of the stage. A musician, warming up, plays a single 494 Hz tone. What does the spectator hear?

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Constructive and Destructive Interference of Sound WavesAssume that two loudspeakers in the figure are vibrating out of phase instead of in phase. (see example 2 from the text) The speed of sound is 343 m/s. What is the smallest frequency that will produce destructive interference at point C?

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Constructive and Destructive Interference of Sound WavesSpeakers A and B are vibrating in phase. They are directly facing each other, are 7.80 m apart, and are each playing a 73.0 Hz tone. The speed of sound is 343 m/s. On the line between the speakers there are three points where constructive interference occurs. What are the distances of these three points from speaker A?

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Consider a wave that is incident normally on a surfaceof area . The wave transports energy. As a result power (energy per unit time) passes through . We define the wave i

AP A

Intensity of a sound wave

2

nte

nsity

as the rat

SI units: /W

o

m

i /IPA

P A

I

22 2

The intensity of a harmonic wave with displacement amplitude is given by:

In terms of the pressure amplitude

Co

1

ns

ider a point source S emitting

. 2 2

a power in t

m

m m

s

P

vI s I pv

he form of sound waves

of a particular frequency. The surrounding medium is isotropic so the waves spread uniformly. The corresponding wavefronts are spheres that have S as

their center. The sound i

2

2ntensity at a distance from S is:

1The intensity of a sound wave for a point sources is proportional

t

4

o

r

r

PIr

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The auditory sensation in humans is proportional to the logarithm of the sound intensity . This allows the ear to percieve a wide range of sound intensities. The threshold of hearing o

II

The decibel

12 20

is defined as the lowest

sound intensity that can be detected by the human ear. 10 W/mThe sound level is defined in such a way as to mimic the response

of the human 10loear. go

I

I

I

/10

is expressed in decibels (dB)

We can invert the equation above and express in terms of as:

10For we have: 10 log1 0

increases by 10 decibels every time increa

o

o

I

I II I

I

Note 1 :

Note2 : 4

ses by a factor of 10For example 40 dB corresponds to 10 oI I

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Example 1: Express the threshold of hearing (2.5 x 10-12 W/m2) and the threshold of pain (1 W/m2) in decibels.

Example 2: At a distance of 60m from a jet airline the intensity is 1 W/m2. I180 = ?

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The intensity of sound near a loud rock band is 120dB. What is the intensity of sound near two such rock bands playing together?

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Problem (Decibels): Two identical rifles are shot at the same time and the sound intensity level is 800dB. What would be the sound intensity level if only one rifle were shot? (hint: the answer is not 400db)

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Problem : Two sound sources each emit sound power uniformly in all directions. There are no reflections. Both sources are located on the x axis, one at the origin and the other at x = +123m. The source of the origin emits four times more power than the other source. Where on the x axis is the intensity of each sound equal? Note there are 2 answers.

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Consider a pipe filled with air that is open at both ends.Sound waves that have walengths that satisfy a particularrelation with the length L of the pipe setup standing wa

Sound standing waves in pipes

vesthat have sustained amplitudes.

The simplest pattern can be set up in a pipe that is open at both ends as shown in fig.a.In such a pipe standing waves have a antinode (maximum) in the dispacement amplitudeThe amplitude of the standing wave is plotted as function of distance in fig.b.The pattern has an node at the pipe center since two adjacent antinodes are separated by a node (minimum). The distance between two adjacent antinodes is /2.

Thus / 2 Its frequency

The standing wave of fig.b is known as the " "or " " of the tube.

Antinodes in the displacement amplitude c r

2 2

o

L v vL fL

Note :

fundamental modefirst harmonic

respond to nodes in the pressure amplitude. This is because and are 90 out of phase.m ms p 22

The next three standing wave patterns are

shown in fig.a. The wavelength

where 1, 2, 3, ... The integer n is known as the

The corr po

2

es

n

n

Ln

Standing waves in tubes open at both ends

harmonic number

nding frequencies 2nnvfL

2n

Ln

The first four standing wave patterns areshown in fig.b. They have an antinode at the open end and a node at the closed end.

The wavelengt

Standing waves in tubes open at one endand closed at the other

4h nLn

for n=1,3,5,74 , ...nLn

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Longitudinal Standing WavesProblem A person hums into the top of a well and finds that standing waves are established at frequencies of 42, 70.0 and 98 Hz. The frequency of 42 Hz is not necessarily the fundamental frequency. The speed of sound is 343 m/s. How deep is the well?

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1 2 1 2 1 2

1 2

If we listen to two sound waves of equal amplitude and frequencies

and and we perceive them as a sound of frequency

. in addition we also perceive "beats" which a2av

f f f f f ff ff

Beats.

1 2

1 1 2 2

re variations in the

intensity of the sound with frequency . The displacements of the two sound waves are given by the equations: cos , and cos .These are plotted in fig.a

beat

m m

f f fs s t s s t

1 2 1 21 2 1 2

1

and fig.b.Using the principle of superposition we can determine the resultant displacement as:

cos cos 2 cos cos2 2

2 cos cos where

m m

m

s s s s t t s t t

s s t t

2 1 2

1 2

and 2 2

Since

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Tbeat

T'

1 2 1 22 cos cos where and 2 2

The displacement s is plotted as function of time in the figure. We can regardit as a cosine function whose amplitude is equal to 2 cos . The a

m

m

s s t t

s t

mplitude is time dependent but varies slowly with time. The amplitudeexhibits a maximum whenever cos is equal to either +1 or -1 which happenstwice within one period of the cos function.

Thus the

tt

1 21 2

1 2 1 2

angual frequency of the beats 2 22

The frequency of the beats 2 2 2

beat

beat beatf f f

1 2 beatf f f

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Consider the source and the detector of sound wavesshown in the figure. We assume that the frequencyof the source is equal to .f

The Doppler effect

We take as the reference frame that surrounding air through which the sound wavespropagate. If there is relative motion between the source and the detector then the detector perceives the frequency of the sound as . If the source or the detector move towards to each other . if on the other hand the source or the detector move away from each other . This is known as the " "

f ff f

f f

Doppler

effect. The frequecy is given by the equation: . Here and

are the speeds of the source and detector with respect to airWhen the motion of the detector or so

, respscturce i

ively.

D

SS D

v vf fv

v vv

f

s each other the sign of the speedmust give an shift in frequency. If on the other hand the motion is from each other the sign of the speed must give a shift in frequency

towards upward away

downwardThe four possible combinantions are illustrated in the next p

.age.

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vS

vS

vS

vS

vD

vD

vD

vD

D

S

v vf f f fv v

D

S

v vf f f fv v

D

S

v vf fv v

D

S

v vf fv v

D

S

v vf fv v

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The Doppler EffectA train approaching a siren. The train encounters more wave fronts per unit time than when stationary.

A train receding from the siren. The train encounters fewer wave fronts per unit time than when stationary.

f = f + additional # of condensationsadditional # of condensations in a time t= (Vr · t)/λ

in 1 sec additional # = Vr/λ-> f = f + Vr/λ = f(1 + Vr/fλ) = f(1 + Vr/V)

A receiver on the train will detect a higher frequency when approaching a siren, and a lower frequency when receding.

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The source (train) is in motion, the receiver is stationary

A stationary receiver will detect a higher frequency when it is front of the train and a lower frequency when behind the train.

The wavelength ahead of the train is shorter λ’ = λ – VE/f and behind the train is longer λ’ = λ+ VE/f when the train is stationary.

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Problem: The trucks travel at the same speed. They are far apart on adjacent lanes and approach each other essentially head-on. One driver hears the horn of the other truck at a frequency that is 1.20 times the frequency he hears when the truck is stationary. The speed of sound is 343 m/s. At what speed is each truck moving?

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