Physics%20Beyond%202000

1
Physics Beyond 2000

• Chapter 9
• Wave Phenomena

2
Glossary

• Wave It is a periodic disturbance in a medium or
  in space.
• There must be source(s) in periodic oscillation.
• There is a transfer of energy from the source.

3
Glossary

• Progressive waves (or travelling waves)
• The wave pattern is moving forwards.
• Stationary waves (or standing waves)
• The wave pattern remains at the same
  position.

Progressive and stationary transverse
waves http//www.geocities.com/yklo00/2Waves.html

Stationary longitudinal wave http//www.fed.cuhk.
edu.hk/sci_lab/Simulations/phe/stlwaves.htm
4
Glossary

• Mechanical waves with medium for the vibration.
• Electromagnetic waves no medium is required. It
  is the electric field and magnetic field in
  vibration.

5
Glossary

• Transverse wave the direction of vibration is
  parallel to the direction of travel of the wave.
• Longitudinal wave the direction of vibration is
  perpendicular to the direction of travel of the
  wave.

http//www.fed.cuhk.edu.hk/sci_lab/ntnujava/waveTy
pe/waveType.html
http//www.kettering.edu/drussell/Demos/waves/wav
emotion.html
http//webphysics.davidson.edu/Applets/TaiwanUniv/
waveType/waveType.html
6
Glossary

• Sinusodial wave
• Square wave
• Saw-toothed wave

Observe these wave patterns on a CRO.
7
Progressive Waves

• The disturbance is propagating from a source to
  places.
• Energy is transferred from the source to places.
• Matter of medium is not transferred. It is
  vibrating.

http//members.nbci.com/surendranath/Applets.html
8
water ripples
Ripple is an example of progressive wave that
carries energy across the water surfaces.
http//www.colorado.edu/physics/2000/waves_particl
es/waves.html
http//www.freehandsource.com/_test/ripple.html
9
water ripples
Ripple is an example of progressive wave that
carries energy across the water surfaces.
wavefront
ray
source
10
Mechanical waves

• In need of a material medium for the propagation
  of wave.
• Examples are wave in a string, sound wave, water
  wave etc.

11
Electromagnetic waves

• They can travel through a vacuum.
• Speed of electromagnetic wave in vacuum is 3
  108 ms-1.
• Electromagnetic spectrum

http//www.colorado.edu/physics/2000/waves_particl
es/
12
Electromagnetic waves

• Electromagnetic waves are transverse waves.
• Electric field and magnetic field are changing
  periodically at right angle to each other and to
  the direction of propagation.

http//www.fed.cuhk.edu.hk/sci_lab/Simulations/phe
/emwave.htm
Oscillation of charges http//www.colorado.edu/ph
ysics/2000/waves_particles/wavpart4.html
13
Analytical Description of Progressive Waves

• Speed of propagation (c)
• Frequency (f)
• Wavelength (?)
• Amplitude (a)
• Period (T)

14
Analytical Description of Progressive Waves

• T
• c f. ?

where s the distance travelled in time t.
15
Example 1

• c f. ?

16
Graphical Representation andPhase Relation

• Each particle in the wave is performing a simple
  harmonic motion, oscillating about its
  equilibrium position.

http//www.fed.cuhk.edu.hk/sci_lab/ntnujava/wave/w
ave.html
http//members.nbci.com/surendranath/Applets.html
17
Graphical Representation andPhase Relation

• There is a phase difference between any two
  particles.
• In phase (or phase difference 0)

18
Graphical Representation andPhase Relation

• There is a phase difference between any two
  particles.
• In phase (or phase difference 0)

19
Graphical Representation andPhase Relation

• There is a phase difference between any two
  particles.
• In phase (or phase difference 0)

20
Graphical Representation andPhase Relation

• There is a phase difference between any two
  particles.
• In phase (or phase difference 0)

21
Graphical Representation andPhase Relation

• There is a phase difference between any two
  particles.
• In antiphase (or phase difference p)

22
Graphical Representation andPhase Relation

• There is a phase difference between any two
  particles.
• In antiphase (or phase difference p)

23
Graphical Representation andPhase Relation

• There is a phase difference between any two
  particles.
• In antiphase (or phase difference p)

24
Graphical Representation andPhase Relation

• There is a phase difference between any two
  particles.
• In antiphase (or phase difference p)

25
Notation in the textbook

• y displacement of a vibrating particle.
• x distance of the particle from the source.

y-x graph for a transverse wave (points along
the path at a particular instant)
c speed of propagation
?
a
26
Notation in the textbook
y-x graph for a transverse wave
C crest T trough
direction of propagation
C
T
27
Notation in the textbook

• y displacement of a vibrating particle.
• In transverse wave,
• y is positive if the particle is above the
  equilibrium position
• y is negative if the particle is below the
  equilibrium position.

Equilibrium position
28
Notation in the textbook
y-x graph for a longitudinal wave
Ccompression Rrarefaction
C
R
29
Notation in the textbook

• y displacement of a vibrating particle.
• In longitudinal wave,
• y is positive if the particle displaces along
  the direction of the travel of the wave
• y is negative if the particle displaces in the
  opposite direction of the travel of the wave.

30
Wave Speed and Speed of Particle

• Wave speed c
• Speed of a particle vy

The wave is moving forward with a constant speed
c.
The particle is vibrating in a SHM with a
changing speed.
31
Example 2

• Find the maximum speed of a vibrating particle.
• The particle is in SHM with
• y a.sin(?t ?o)

32
Phase Relationship

• For the motion of a vibrating particle in SHM,
• y a. sin(?t ?o)
• with ?o the initial phase.
• In a wave, different particles have the different
  initial phase.

33
Phase difference between two points in a y-x
graph
P
Q
For two points with separation ?x in a wave with
wavelength ?, their phase difference is
in radian
34
Phase difference between two points in a y-x
graph
P
Q
Which point leads the other?
35
Phase difference between two points in a y-x
graph
y
P
direction of propagation
0
x
R
Compare points P and R. Which point leads the
other?
36
Phase difference between two points in a y-x
graph
y
P
direction of propagation
0
x
R
Draw the new wave pattern after a time t. The
whole wave pattern moves to the right.
37
Phase difference between two points in a y-x
graph
y
P
direction of propagation
0
x
R
Point P will move back to the equilibrium
position from the crest after . Point R will
move to the position of the crest from the
equilibrium position.
38
Phase difference between two points in a y-x
graph
y
P
direction of propagation
0
x
R
After , P moves to the equilibrium position
and R moves up to the position of the crest. So P
leads R by
39
Phase difference between two waves in a y-x graph
y
y1
direction of propagation
y2
0
x
Two waves y1 and y2 of the same frequency are
moving to the right simultaneously. What is the
phase difference between these two waves?
40
Phase difference between two waves in a y-x graph
y
y1
direction of propagation
y2
0
x
x
Measure the separation x of their crests ( choose
x lt ?/2 ).
41
Phase difference between two waves in a y-x graph
y
y1
direction of propagation
y2
P
0
x
Choose a point P in front of the waves. Which
wave has its crest reach the point first?
The first wave leads the second wave by ?.
42
Density variation along a longitudinal wave
C

• The centres of compression have the highest
  density and highest pressure (for gas).

43
Density variation along a longitudinal wave
R

• The centres of rarefaction have the lowest
  density and lowest pressure (for gas).

44
Density variation along a longitudinal wave
x
x

• The crest of the density/pressure leads that of
  displacement by ?/2.

45
Example 3

• Ripple in water is transverse wave.
• Hint

46
Example 4

• Which one leads?
• Keep the phase difference lt ?.

47
Displacement-time graph

• Describe a particle of the wave at different time.

48
Displacement-time graph

• For a particle performing simple harmonic motion,
• y a.sin(?t ?o)
• It is a sinusoidal wave.

49
Displacement-time graph

• Other particles along the path are performing SHM
  at the same frequency but with a different phase
  ?o

50
Displacement-time graph

• Two points P and Q are in the path of a wave.

51
Displacement-time graph

• Wave P becomes crest earlier than wave Q . ( tP lt
  tQ).
• ? Wave P leads wave Q.

52
Displacement-time graph

• Time difference ?t tQ tP
• Phase difference

53
Example 5

• y a.sin(?t ?o)
• yA 0.1 sin (2?t ?/3) ? ?A ?/3
• yB 0.1 sin (2?t ?/4) ? ?B ?/4
• ? phase of wave A ?A gt phase of wave B ?B
• ? A leads B.

Note keep the phase difference lt p
54
Traces on CRO

• Use a double beam oscilloscope.
• The two input waves must be of the same
  frequency.
• It is a y-t graph on the screen of the CRO.

to signal generator 1
to signal generator 2
Y1
Y2
55
y-x and y-t graphs

• A typical example

At t 0,
http//www.geocities.com/yklo00/Wave_yx-yt-graphs.
htm
At point P,
56
Wave speed of a mechanical wave

• c

57
Wave speed of a mechanical wave

• c
• Longitudinal wave in a solid

where E is the Young modulus of the solid and ?
is the density of the solid.

• Example 6

58
Wave speed of a mechanical wave

• c
• Transverse wave in a string

where T is the tension in the string and ? is
the mass per unit length of the string.

• Example 7

59
Wave speed of a mechanical wave

• c
• Sound wave in air

where P is the air pressure, ? is the density of
air and ? is a constant.
60
Sound wave in air
where P is the air pressure, ? is the density of
air and ? is a constant.
By gas law PV nRT ? ?
Speed of sound in air is independent of the
pressure. It increases with temperature.
61
Measure speed of sound

• Use a double beam oscilloscope.
• Compare the phase difference between the input
  from the signal generator and that from the
  microphone.

62
Measure speed of sound

• Place the microphone at a position such that both
  inputs are in phase.

63
Measure speed of sound

• Move the microphone towards the loudspeaker to
  locate the next position with both inputs in
  phase again.

64
Measure speed of sound

• The distance x wavelength ?
• For accurate measurement, we usually move the
  microphone more than one wavelength.

65
Measure speed of sound

• Find the frequency f from the signal generator or
  the period T from the time base of the CRO.

f
T
66
Measure speed of sound

• c f.? or c

f
T
67
Water waves

• Ripple tank for studying water wave
• Stroboscope for freezing wave pattern.

68
Water waves

• Ripple tank
• Production of circular waves by a dipper.

69
Water waves

• Ripple tank
• Production of plane waves by a straight line.

70
Water waves

• Hand Stroboscope for freezing the motion.

slit
71
Water waves

• Hand Stroboscope
• Strobe frequency fs is the number of slits
  passing the viewer per second.
• No. of slits on the hand strobe n
• No. of revolutions per second m
• ? fs m.n

72
Freezing a periodic motion

• A periodic motion with frequency fo and period
  To.
• To freeze the motion,
• - the strobe should rotate in the same
  direction as the periodic motion and
• - the strobe frequency should be
• fs fo, 2.fo, 3.fo, or
• fs fo/2, fo/3, fo/4,

Viewing a rotating arrow.
73
Freezing a periodic motion

• A periodic motion with frequency fo and period
  To.
• fs 2.fo, 3.fo, or Ts To/2, To/3,
• Though the motion is frozen, the observed
  pattern becomes too dense. The motion is viewed
  before the arrow completes one cycle.
• Take fs 2.fo as an example,
• (Ts To/2)

74
Freezing a periodic motion

• A periodic motion with frequency fo and period
  To.
• fs fo/2, fo/3,or Ts 2.To, 3.To,
• The motion is viewed when the motion has
  completed more than one cycle.
• Take fs fo/2 as an example
• (Ts 2.To)

75
Freezing a periodic motion

• A periodic motion with frequency fo and period
  To.
• fs fo or Ts To
• The motion is viewed when the motion has just
  completed one cycle.

76
Example 8

• 4 arrows are seen.
• Think of the time and periods.

77
Find the frequency of motion

• Freeze the motion without increasing its density.
  
• ? fs fo, fo/2, fo/3,.
• Increase fs to a maximum so that it still can
  freeze the motion without increasing its density.
• ? fs f0

78
Find the frequency of water wave in a ripple tank

• Freeze the wave pattern without decreasing its
  wavelength.
• ? fs fo, fo/2, fo/3,.
• Increase fs to a maximum so that it still can
  freeze the wave pattern without decreasing its
  wavelength.
• ? fs f0

79
Example 9

• Freeze the wave pattern.

80
Typical Example stroboscope

• 4 identical arrows on a disc which is rotating at
  12 revolutions per second.
• Use a strobe with 12 slits.
• What are the possible strobe speeds at which 4
  frozen arrows are seen?

81
Example 9

• Find the speed of water wave.

82
Forward and backward motions

• If fs is slightly larger than fo (Ts lt To), the
  motion seems moving forward.
• The arrow rotates more than one cycle in Ts.

83
Forward and backward motions

• If fs is slightly less than fo (Ts gt To), the
  motion seems moving backward.
• The arrow rotates less than one cycle in Ts.

84
General Wave Properties

• Wave properties
• Reflection
• Refraction
• Diffraction
• Interference
• Water wave is an example in this section.

85
Huygens principle
Straight wavefront

• Suppose that there is a
• straight wavefront.
• Take every point on the wavefront as a source of
  secondary waves.
• Draw the secondary waves.
• The new wavefront is the envelope of these
  secondary waves.

c.t
86
Huygens principle
Curved wavefront

• Suppose that there is a curved wavefront.
• Take every point on the wavefront as a source of
  secondary waves.
• Draw the secondary waves.
• The new wavefront is the envelope of these
  secondary waves.

ct
87
Reflection

• It is the return of wavefront when it encounters
  the boundary between two media.

http//www.fed.cuhk.edu.hk/sci_lab/ntnujava/propag
ation/propagation.html
88
Reflection

• It is the return of wavefront when it encounters
  the boundary between two media.
• Straight reflector and plane wave

89
Reflection

• It is the return of wavefront when it encounters
  the boundary between two media.
• Straight reflector and circular wavefront

90
Radar and sonar

• Radar RAdia Detection And Ranging

91
Radar and sonar

• Radar RAdia Detection And Ranging

The station sends a pulse.
92
Radar and sonar

• Radar RAdia Detection And Ranging

The station receives the pulse.
93
Radar and sonar

• Radar RAdia Detection And Ranging

d
If the time difference is t, find the distance d.
94
Radar and sonar

• Sonar SOund Navigation And Ranging

seabed
95
Radar and sonar

• Sonar SOund Navigation And Ranging

Send a pulse of ultrasound
seabed
96
Radar and sonar

• Sonar SOund Navigation And Ranging

Receive the pulse of ultrasound
seabed
97
Refraction

• It is the change of direction of wavefront as it
  passes obliquely from one medium to another.
• The wave speeds must be different in the two
  media.
• The wavelength also changes.
• The frequency does not change.

http//www.fed.cuhk.edu.hk/sci_lab/ntnujava/propag
ation/propagation.html
http//www.fed.cuhk.edu.hk/sci_lab/download/projec
t/Lightrefraction/LightRefract.html
98
Snells law
99
Snells law
100
Snells law
In time t, Y moves to Y with YY c1.t. At the
same time t, X moves to X with XX c2.t
101
Snells law
Prove Snells law. Hint Study triangles XYY and
XXY.
102
Total internal reflection

• Total internal reflection occurs when
• Wave moves from medium of low speed to another
  medium of high speed.
• The angle of incidence is equal to or larger than
  the critical angle.

medium 2 (wave with high speed)
medium 1 (wave with slow speed)
normal
103
Total internal reflection

• Note that there is not any wave into the second
  medium.

104
Diffraction

• It is the spreading or bending of waves as they
  pass through a gap or round the edge of a
  barrier..

105
Diffraction

• The extent of diffraction depends on the relative
  sizes of the gap and the wavelength.

narrow gap
wide gap
106
Interference

• Principle of superposition
• The total displacement of a point is equal to
  the algebraic sum of the individual displacements
  at that point.

Superposition of pulses http//www.fed.cuhk.edu.h
k/sci_lab/ntnujava/wave/impulse.html
Superposition of continuous waves http//www.fed.
cuhk.edu.hk/sci_lab/ntnujava/waveSuperposition/wav
eSuperposition.html
107
Interference of water waves

• Two dippers produce water waves of same
  frequency.
• The two dippers are in phase.

108
Interference of water waves

• Along lines of reinforcement (anti-nodal lines),
  the amplitude of each point is large.
• Along lines of cancellation (nodal lines), the
  amplitude of each point is almost zero.
• The interference pattern depends on the relative
  sizes of the source separation and wavelength.

http//www.fed.cuhk.edu.hk/sci_lab/ntnujava/waveIn
terference/waveInterference.html
109
Conditions to produce an interference pattern

• The two sources are coherent.
• They produce waves of same frequency
• The sources of waves maintain a constant phase
  difference.
• The separation between the sources is only
  several wavelengths.

110
Path difference

• We always express the path of a wave train in
  terms of the number of wavelengths.
• A continuous wave travels from S1 to A. If the
  distance S1A 10 m and the wavelength is 0.2 m.
  What is the path S1A in terms of the number of
  wavelengths?

50?
10m
S1
A
111
Path difference

• What is the path difference ? of S1 and S2 at
  point A in terms of the number of wavelengths?
  The wavelength is 0.2 m.

? S1A S2A
?10?
112
Path difference and interference
The two coherent sources S1 and S2 are Path difference at A m. ? where m 0, 1, ... Path difference at A (m ). ? where m 0, 1, ...
in phase constructive interference at A destructive interference at A
in anti-phase destructive interference at A constructive interference at A
113
Determine the kind of interference

• Are the two sources coherent?
• Check the phase difference between the two
  sources.
• Find the path difference at the point.
• Determine the kind of interference at the point.

? 0.2m. S1 and S2 are coherent sources which
are in phase.
114
Examples

• Coherent sources S1 and S2 are in phase.
• MN is the perpendicular bisector of S1S2.
• If there are constructive interference at points
  A, B and C, find their respective path difference
  in terms of ?.

115
Examples

• Coherent sources S1 and S2 are in phase.
• MN is the perpendicular bisector of S1S2.
• If there are destructive interference at points X
  and Y, find their respective path difference in
  terms of ?.

116
Example 10

• Points with maximum sound are points with
  constructive interference.
• Points with minimum sound are points with
  destructive interference.

117
Energy and power of a mechanical wave

• Example a transverse wave along a string with
  frequency ? and amplitude a.

118
Energy and power of a mechanical wave

• Example a transverse wave along a string with
  frequency ? and amplitude a.
• Each particle on the string is performing simple
  harmonic motion.
• The energy of each particle is

119
Energy and power of a mechanical wave

• The energy of each particle is
• If the mass per unit length of the string is ?,
• each wavelength ? of the string will carry
  energy

120
Energy and power of a mechanical wave

• If the mass per unit length of the string is ?,
• each wavelength ? of the string will carry
  energy
• This is the amount of energy transferred in one
  period of time T.

121
Energy and power of a mechanical wave

• If the mass per unit length of the string is ?,
• each wavelength ? of the string will carry
  energy
• Power of the wave is P
• where c is the wave speed.

122
Energy and power of a mechanical wave

• If the mass per unit length of the string is ?,
• each wavelength ? of the string will carry
  energy
• Power of the wave is P
• where c is the wave speed.

123
Intensity

• Definition
• Intensity I is defined as the power received
  per unit area.
• I ? a2.

124
Intensity of spherical waves

• The wave spreads out from a point source.
• The wave energy is distributed over a spherical
  area.

Point source
125
Intensity of spherical waves

• From a point source, the wave spreads out.
• The wave energy is distributed over a spherical
  area.

r
Area 4?r2
126
Intensity of spherical waves

• Assume that there is not any loss of energy
• (attenuation).
• If the power of the source is Po, I

127
Cylindrical waves

• The wave spreads out from a line source.
• The wave energy is distributed on a cylindrical
  surface.

line source
128
Cylindrical waves

• The wave spreads out from a line source.
• The wave energy is distributed on a cylindrical
  surface.

129
Cylindrical waves

• Assume that there is not any loss of energy
• (attenuation).
• If the power of the source is Po, I

130
Reflection andPhase Change

• Transverse wave
• Fixed at one end phase change ?.
• Incident crest changes into trough on reflection.
• Incident trough changes into crest on reflection.

http//www.physics.nwu.edu/ugrad/vpl/waves/waveref
lection.html
131
Reflection andPhase Change

• Transverse wave
• Free at one end no phase change
• Incident crest is still a crest on reflection.
• Incident trough is still a trough on reflection.

http//www.physics.nwu.edu/ugrad/vpl/waves/waveref
lection.html
132
Reflection andPhase Change

• Longitudinal wave
• Fixed at one end ? phase change.
• Incident compression is still a compression on
  reflection.
• Incident rarefaction is still a rarefaction on
  reflection.

133
Reflection andPhase Change

• Longitudinal wave
• Fixed at one end ? phase change.
• Note that the displacement in the compression of
  the incident pulse is to the right side (ve) and
  the displacement in the compression of the
  reflected pulse is to the left side (-ve). There
  is a ? phase change

incident pulse
reflected pulse
134
Reflection andPhase Change

• Longitudinal wave
• Free at one end no phase change
• Incident compression becomes a rarefaction on
  reflection.
• Incident rarefaction becomes a compression on
  reflection.

135
Stationary wave

• Principle of superposition is applied wave
  meets wave.
• Energy is localized.
• The waveform is confined within some boundaries.
• The progressive wave still moves at speed c.

136
Stationary wave in a string

• Transverse wave.
• Incident wave and reflected wave meet.
• At some frequencies , nodes and antinodes are
  formed at some points of the string.

http//www2.biglobe.ne.jp/norimari/science/JavaEd
/e-wave4.html
137
Stationary wave in a string

• Fundamental the lowest frequency with a node at
  either side and an antinode in the middle.
• Overtones some higher frequencies with node at
  either side and some antinodes between the ends.

http//members.nbci.com/_XMCM/surendranath/Harmoni
cs/Harmonics.html
138
Frequency of stationary waves in a string

• When a stationary wave is formed, the length
  of the string must be a whole number of
  half-wavelength .

where n 1, 2,
139
Frequency of stationary waves in a string
From
and
140
Note of a vibrating string

• The air is also set into vibration by the
  vibrating string.
• The frequency of air is the same as that of the
  vibrating string.
• If the frequency is within audible frequency
  range, a note may be heard.

141
Properties of stationary wave

• Waves are travelling in both directions but
  energy is confined in the boundaries.
• All parts of the string (except at the nodes) are
  vibrating with the same frequency but with
  different amplitudes.

http//members.nbci.com/_XMCM/surendranath/Harmoni
cs/Harmonics.html
142
Properties of stationary wave

• Within one loop, all points are moving in phase.
• Points between adjacent loops are in anti-phase.

http//members.nbci.com/_XMCM/surendranath/Harmoni
cs/Harmonics.html
143
Properties of stationary wave

• If the frequency of the stationary wave is equal
  to the natural frequency of the string, resonance
  occurs.
• Distance between adjacent anti-nodes is

http//members.nbci.com/_XMCM/surendranath/Harmoni
cs/Harmonics.html
144
Measuring the wavelength of microwave

• A stationary wave is formed between the
  transmitter and the reflector.
• The probe shows a large current at positions of
  nodes and small current at positions of antinodes.

to power supply
to microammeter
reflector
transmitter
probe
145
Measuring the wavelength of microwave
Move the probe to detect the positions of the
nodes Count the number of nodes n. Measure the
distance moved d
A
A
A
A
N
N
N
N
to power supply
to microammeter
reflector
transmitter
probe
146
Formation of stationary waves

• It is a result of superposition of two waves with
  the same frequency and amplitude and are
  travelling in the same medium in opposite
  directions.

http//www.fed.cuhk.edu.hk/sci_lab/ntnujava/waveSu
perposition/waveSuperposition.html
147
Standing Longitudinal Wave

• http//www.fed.cuhk.edu.hk/sci_lab/Simulations/phe
  /stlwaves.htm

148
Beats

• Two waves with slightly different frequencies
  propagating in the same direction overlap with
  each other.
• The resultant amplitude varies periodically.

http//webphysics.ph.msstate.edu/jc/library/15-11/
index.html
149
Beats

• Two loudspeakers emit sound waves of slightly
  different frequencies.
• The microphone receives both sound.
• The beat is seen on the CRO.

150
Formation of beats

• Two waves with frequency f1 and f2 respectively
  with f1 slightly higher than f2.
• Take them as rotating vectors with angular
  velocity ?1 and ?2 with ?1 gt ?2.

? 2?f
t 0
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Formation of beats

• At t 0, the two vectors coincide.

? 2?f
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Formation of beats

• At t 0, the two vectors coincide.
• Vector 1 rotates faster and leads vector 2.

? 2?f
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Formation of beats

• Find the time Tb when vector 1 catches up with
  vector 2. (Hint vector 1 runs one cycle more
  than vector 2.)

? 2?f
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Formation of beats
?1Tb - ?2Tb 2?
Tb is the period of the beat.
? 2?f
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Formation of beats
Tb is the period of the beat.
The beat frequency fb f1 f2
? 2?f
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Frequency of the resultant wave

• The resultant wave frequency is

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Example 11

• Checking frequency.
• e.g. fine tuning a piano.

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Analytical treatment of beats

• The equations of the two waves reaching the same
  position
• y1 a.sin(2?f1t) and
• y2 a.sin(2?f2t)
• The equation of the resultant wave is
• y y1 y2

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Analytical treatment of beats
A varying amplitude
frequency of the resultant wave
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Analytical treatment of beats
A varying amplitude
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Analytical treatment of beats
A varying amplitude
As our ear responds to intensity ? a2, the number
of maximum heard per second 2.fA
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Polarization

• A wave is unpolarized if the vibration is not
  confined to any direction.
• An unpolarized wave can be made to be
  plane-polarized by a process called polarization.
• Only transverse wave can be polarized.

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Polarizer and analyzer

• An unpolarized transverse wave passes through a
  polarizer and becomes polarized.
• The plane of vibration of the wave is parallel to
  the slit of the polarizer.
• The intensity of the polarized wave is half that
  of the unpolarized wave.

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Polarizer and analyzer

• Use an analyzer to change the polarization of the
  incident wave.

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Polarizer and analyzer

• Use an analyzer to change the polarization of the
  incident wave.
• If the intensity of the emerging wave is
  unchanged, the plane of polarization of the wave
  is parallel to the analyzer.

analyzer
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Polarizer and analyzer

• Use an analyzer to change the polarization of the
  incident wave.
• If the intensity of the emerging wave is zero,
  the plane of polarization of the wave is
  perpendicular to the analyzer.

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Polarizer and analyzer

• Use an analyzer to change the polarization of the
  incident wave.
• If the intensity of the emerging wave is reduced,
  the plane of polarization of the wave neither
  parallel nor perpendicular to the analyzer.

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Polarizer and analyzer

• If a is the amplitude of the polarized wave
  emerging from the 1st slit and the 2nd slit
  makes an angle ? to the 1st slit,
• the amplitude a of the wave emerging from
  the 2nd slit is a a.cos ?

a
a.cos ?
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Polarizer and analyzer

• The amplitude a of the wave emerging from the
  2nd slit is a a.cos ?

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Polarizer and analyzer

• The amplitude a of the wave emerging from the
  2nd slit is a a.cos ?
• The intensity of the emerging wave Io.cos2 ?
  where Io is the intensity of wave from the 1st
  slit.

Io.cos2 ?
Io
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Polaroid filters

• Light can be polarized by using a polaroid
  filter.
• Use two polaroid filters to analyze the
  polarization of light.