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HKAL Test bank (Essay) : Part 3 Waves
Chapter 9 Wave Phenomena
9.1 Progressive Waves
9.1.1
Longitudinal and Transverse Waves
9.1.1.1
Wave Form
R
C
R
C
R
displacement
t=0
distance
to R
distance
to L
t=0
R
C
R
C
t = T/4
distance
to R
distance
to L
t = T/4
C
velocity v
R
C
R
C
t = T/2
distance
to R
to L
distance
t = T/2
C
t = 3T/4
distance
C
pressure graph
R
to R
to L
Progressive transverse wave
R
distance
t = 3T/4
Progressive longitudinal wave
9.1.2
Sonar
Sonar is similar to radar but employs ultrasonic waves (i.e. sound waves of frequency above 20 kHz)
which are sent towards the bottom of the sea. It makes use of the principle of reflection to measure
the depth of the sea (i.e. in echo sounding) and to detect shoals of fish.
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9.2 Phase Relationship
double beam
oscilloscope
loudspeaker
L
optical bench
y1
microphone
y2
M
Y2
Y1
a.f.
oscillator
scale
(a) Microphone M is moved along bench away from loudspeaker L.
(b) Distance is measured on scale.
Y2
T
in phase
Y1
Y2
M moved by /4
 = /2
Y1
Y2
M moved by /2
=
Y1
(c) If voltage waveform in phase at a particular location of M then on moving through a distance 
(wavelength) they would again be in phase.

T
(d) Period T is measured using time-base of C.R.O. Then propagation velocity = .
9.3 Wave Speed
The speed of mechanical waves depends on the elasticity and the density of the medium for
propagation.
9.4 Water Waves
9.4.1
Tidal Energy
(a) Renewable energy
(i) These energy sources cannot be exhausted.
(ii) Tidal energy can be harnessed by building a barrage (barrier) containing water turbines
and sluice gates, across the mouth of a river, large gates are opened during the incoming
(flood) tide, allowing the water to pass until high tide, when they are closed. On the
outgoing (ebb) tide, when a sufficient head of water was built up, small gates are opened
letting the potential energy of the trapped water drive the turbines and generate electricity.
(b) Difficulties
(i) The tidal range has to be considered together with the demand, both domestic and
industrial.
(ii) The initial and maintenance costs are great.
(iii) There are important environmental implications for nearby mudflats, water supply,
sewage disposal, shipping, fishing and wild life.
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9.5 General Wave Properties
9.5.1
Huygen’s principle
(a) Huygen’s principle states that each point on the existing wavefront of a wave acts as a source
of secondary wavelets. The plane tangential to these wavelets is the new wavefront.
A
C
Constructed
wavefront
Secondary
source
ct
rays showing
the direction
of propagation
ct
ct
ct
First position
of wavefront
D
B
Secondary
wavefront
(b) At several points on the initial wavefront, draw spheres of radius r = ct, representing the
distance traveled by the secondary wavelets in time t (c = velocity of wave).
(e) The new wavefront displaced a distance ct from the initial wavefront.
(f) Thus, from the diagram the wave always propagates at right angles to the wavefronts.
Alternative Description
(a) All points on a wavefront can be considered as point sources for the production of
spherical/circular secondary wavelets.
(b) After a time t the new position of the wavefront will be the surface of tangency to these
secondary wavelets.
9.5.2
Refraction of wave
9.5.2.1
Explanation by Huygen’s Principle
(a) According to Huygen’s wave theory waves are sent out from centres of disturbance along the
interface line OD and these produce a plane wavefront BD in the water.
(b) Since the time t taken A  D is the same as that O  B,
AD OD sin i ct


OB OD sin r vt
and, n = sin i/sin r = c/v, Since n > 1 for water v < c.
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9.5.2.2
Wavefront Approach
v1t
incident
wavefront
Normal
incident
ray
B
v1
medium
A
B’
v2t
refracted
wavefront
A’
medium
refracted
ray
v2
(a) The incident wavefront has just reached the boundary between the two media at point A. Points
A and B act as sources of secondary waves.
(b) The wave from B, travelling at speed v1 in medium 1, moves a distance v1t to point B’.
(c) Within the same time, the wave from A, travelling at a slower speed v2 in medium 2, moves a
smaller distance v2t to point A’.
(d) The new wavefront in medium 2 is given by the line through A’B’ which has clearly been
deflected, refraction of waves results.
Alternative Description
Primary
N wavefront
C
v1
v1 > v2
v2
B
1
1
A
2
A'
2
1
B' 2
Refracted
wavefront
N'
Secondary wavelet from A
(a) Let v1 and v2 (v1 > v2) be the speeds of light in air (medium 1) and in the denser medium
(medium 2) respectively.
(b) Let λ1 andλ2 be the wavelengths of light in air and in the denser medium respectively.
(c) Consider the time t during which a Huygens wavelet from point B moves to include point B’.
BB’ = v1t
(d) Light from point A, traveling through the denser medium at a reduced speed will move a
shorter distance to A’ within this time.
AA’ = v2t
v
sin  1
BB '
BB ' / AB'
(e) From the diagram,
=
= 1 =
.
AA'
AA' / AB'
sin  2
v2
(f)
By Shell’s law of refraction n =
Essay_notes_Waves_09_12
sin  1

v
(= 1 ) = 1 .
sin  2
2
v2
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Alternative Description
A
i
v1
P
P'
air
D
O
water
v2
B
r
(a) On the wavefront OA, the points P becomes centres of disturbance and send out spherical
waves, while in turn the points P’ become centres of disturbance along the interface. (If the
time taken to travel A  D is t, distance AD is v1t.
(b) Since the speeds of waves differ in the water (v2 < v1) the distance traveled OB < AD. The
tangent to the wavefronts of the secondary disturbances BD is the wavefront of the wave
travelling in the water and since r < i, refraction occurs.
9.5.2.3
Dispersion of Light
C
A
air
glass
P
D
B
R
(a) White light consists of a continuous spectrum of light of various wavelengths, and for these the
speeds of the waves differ - blue light, v1 < red light, v2.
(b) Consider the centre of disturbance A on the interface, in the time t taken for the wave in air to
travel CD the waves sent out into the glass will have traveled AB for the blue light and AR for
the red light. BD and RD are the new wavefronts - and clearly dispersion occurs.
9.5.2.4
Daily Example
NIGHT
v  T1/2
cool air 
warmer air 
cool air 
v2 > v 1
ground
DAY
warm air 
v1 > v 2
ground
(a) Refraction is a bending of the propagation path of the waves due to change of speed
experienced when either moving into another medium - or in the example given, due to a
change of temperature.
(b) Snell’s Law of refraction holds good, sin i / sin r = n2/n1 = v1/v2.
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9.5.3
9.5.3.1
Diffraction of Wave
Wavefront Approach
P
obstacle
(a) Diffraction is a bending of the waves around the edge of obstacles.
(b) There are centres of disturbance on the wave front in plane of obstacle, such as P, and these
send out waves (spherical) as shown propagating behind obstacle.
9.5.3.2
Comparison between the Diffraction of Sound and Light
(a) As sound waves (wavelength ~ 0.5 m) can be effectively diffracted through the classroom door
(~1.0 m), the voice of a person can be heard.
(b) As the quality of sound is not the same for different persons, the voice of a particular person
can be easily distinguished without seeing his face.
9.5.4
Interference of Waves
9.5.4.1 Principle of Superposition
When two waves travel through a medium, their resultant displacement at any point is the sum of
the separate displacements due to the two waves.
Alternative Description
When waves pass through a single point the total displacement is the vector sum of the individual
displacements which would be produced by each wave acting along.
9.5.4.2 Demonstrating Experiment
(a) A long slinky spring is laid flat on a horizontal table and slightly stretched. Two transverse
pulses move along the spring in opposite directions from each end
(i)
(ii)
(iii)
(iv)
(v)
(i)
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(ii)
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(i) Both ends of the spring are momentarily moved sideways at the same time.
(ii) One end of the spring is attached to a rigid support and the other end given a momentarily
‘flick’ as in (i). After reflection amplitude of return pulse is reversed and a further pulse
is produced from the free end as in (i).
(b) When the pulses pass through each other :
(i) they remain unaffected (in amplitude).
(ii) Where they cross the total displacement is the vector sum of the individual displacements.
9.5.4.3 Conditions for Observing Interference of Sounds
At observing location the interfering waves must have a constant (or slowly observable changing)
phase difference.
9.5.4.4 Difference Performance of a Radio Receiver
(a) The radio signals from radio stations are electromagnetic waves and they undergo reflection
inside a room.
(b) As a result the radio receiver receives all the reflected waves through different paths and they
may interfere destructively at certain locations of the room.
Alternative Description
(a) Absorption by adjacent conductors
(b) Influence of conductors carrying an a.c. current.
9.6 Stationary Waves
9.6.1
Stationary Waves in a String
9.6.1.1 Formation
A stationary wave is formed when there is superposition of two waves of nearly equal amplitude
and equal frequency travelling in opposite directions.
Alternative Approach
N
N
N
N
N
stationary wave resulting
from superposition
The superposition of two identical waves travelling in opposite directions gives a stationary wave.
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Alternative Approach
Stationary wave
N
N
N
N
N
N
N
N
N
1
t = __
4f
Progressive
wave
1
t = __
2f
3
t = __
4f
1
t = __
f
N
N
N
N
N
N
N
N
N
(a) Stationary waves are produced when the progressive wave reflected from a boundary interferes
(amplitudes adding) (superposition) with the wave travelling towards the boundary and the
resulting wave has to fit into the system, i.e. for particular system nodes are formed at each end
of the wire.
(b) The conditions for this are given by L =
n
, where L is the length of wire and n = 1, 2, 3 ....
2
Alternative Approach
(a) Stationary waves result from the superposition of two trains of progressive waves of equal
amplitude and frequency travelling with the same speed in opposite directions.
(b) At the nodes (N), the interference of the two waves is always destructive and the resultant
amplitude is zero. At the antinodes, the interference of the two waves is constructive and the
resultant amplitude is largest.
N
N
N
N
N
N
Stationary wave
N
N N
Progressive
waves
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9.6.1.2 Demonstrating Experiment
(a) Nearby loudspeaker connected to oscillator (f).
(b) A.c. current passed through wire (f) with nearby magnet (antinode position best).
(c) Frequency f varied to give different ‘oscillation’ modes.
9.6.1.3 Comparison between Progressive and Stationary Waves
progressive wave
stationary waves
(a) Waveform advances as time goes on.
(a) Waveform does not advance.
(b) Energy is transmitted along the direction
(b) Energy is confined within the
of travel of the wave.
region of the stationary wave.
(c) Particles within one wavelength have
(c) All particles between two adjacent
different phases.
nodes are in phase.
(d) All particles are vibrating.
(e) All vibrating particles have the same
amplitude.
(d) Some particles (at nodes) have no vibration.
(e) Different particles have different amplitudes,
in particular, amplitude is maximum at
anti-nodes.
9.7 Beats
9.7.1
Formation of Beats
y1
Displacement
T2
f1 f 2
T1
T3
time
0
Time
y2
T1
A
B
Variation of
amplitude
Resultant
Displacement
T2
C
0
T3
time
Variation of
amplitude
y3
resultant
T2
T1
T3
Time
Time
T
Beat period = T
Beats (not to scale)
Sound beats are heard when two waves of slightly different frequency pass into one’s ear. In
diagram amplitudes reinforce at times A and C while at B they cancel.
Alternative Description
Following from the principle of superposition, in a region where 2 wave-trains meet the amplitudes
either reinforce or cancel, depending upon whether they are in-phase or anti-phase - giving rise to
an interference pattern (or fringes).
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Alternative Description
(a) At an instant such as A, the waves from the sources arrive in phase and reinforce to produce a
loud sound.
(b) The phase difference then increases until a compression (or rarefaction) from one source arrive
at the same time as a rarefaction (or compression) from the other.
(c) The observer hears little or nothing, point B. Later the waves are in phase again (point C) and a
loud note is heard.
(d) Beats are thus due to interference but because the sources are of slightly different frequencies
there is sometimes reinforcement at a given point and at other times cancellation.
Alternative Description
(a) At T1, the waves arrive in phase to produce a loud sound.
(b) Phase difference then increases between the waves due to different frequencies.
(c) At T2, the waves are completely out of phase, little or no sound is heard.
(d) Later at T3, the waves are in phase again and a loud note is heard.
9.7.2
Beat Frequency
If time between successive maximas is T and one wave train of frequency f1 makes one cycle more
than the other of frequency f2, f1T - f2T = 1 and beat frequency is 1/T = f1 - f2.
Alternative Description
(a) Suppose the beat period = T, then in time T :
number of cycles of f1 = f1T,
number of cycles of f2 = f2T.
(b) Assume f1 greater than f2, then :
f1T - f2T = 1
f1 - f2 =
 beat frequency =
1
T
1
= f1 - f2 .
T
9.7.3
Daily Example
The notes produced by two violin players might differ slightly in frequency due to, say, a small
difference in tension. As a result of superposition, beats are formed and the intensity of the sound
varies.
Alternative Description
Beats produced between two sources of slightly different frequencies - as in music if one instrument
is slightly out of tune.
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Chapter 10 Sound Waves
10.1 Nature of Sound
10.1.1
Properties of Sound
(A) Propagation
(i) Energy passed on from one region of air (e.g. compression) to adjacent region in actual
direction of propagation (longitudinal progressive wave), with actual air particles
vibrating with S.H.M. about their mean positions, also longitudinally.
(ii) For this type of wave a medium is necessary. Distance between instantaneous centres of
compression or rare-factions = wavelength , velocity, v = f,
(iii) In diagram : C – compression, R – rarefaction.
R
C
R
C
R
to R
distance
to L
t=0
R
C
R
C
to R
distance
to L
t = T/4
C
velocity v
R
C
R
to R
to L
distance
t = T/2
C
R
C
to R
to L
C
pressure graph
R
distance
t = 3T/4
T = 1/ f
(B) Displacement and Pressure
displacement
(right)
0
time
(left)
pressure
wave
travelling
to the left
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normal air
pressure when
wave absent
time
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(C) Phase relationship
The displacement leads/lags the air pressure by a phase angle of
1

(or period).
4
2
(D) Frequency Range
The audio frequency range is 20 - 20,000 Hz (20 kHz).
Alternative Description
Displacement
(right)
C
R
C
R
Distance
C
R
C
R
0
(left)
Pressure
Normal air
pressure when
wave absent
Distance
(a) These are of much lower frequency (50 - 2  104 Hz) and are due to momentum transfer of
energy from pockets of air vibrating with S.H.M. about their mean locations.
(b) There are associated pressure changes of rarefactions R and compressions C.
(c) Waves propagate outwards from vibrating source and need a medium for propagation. Speed is
much lower ~ 330 m/s in air.
10.1.2
Detection/Intensity Measurement
sound
wave
microphone
audio
amplifier
rectifier
meter
(a) Microphone converts sound into a.c. electrical signal (of same f), this signal is rectified and
produces steady meter reading indicating intensity of sound wave.
(b) Allowance may be needed to compensate for frequency responses of microphone and audio
amplifier.
10.2 Intensity and Loudness
10.2.1
Difference between Sound Intensity and Sound Intensity Level
(a) Sound intensity is the energy per second crossing a unit area normally to the direction of the
sound (unit : W m-2).
(b) Sound intensity level of a source is its intensity relative to some agreed ‘zero’ intensity level
(unit : dB).
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10.2.2
Daily Example
X
loudspeakers
O
Y
(a) Two identical loudspeakers connected to the same signal generator are placed inside a room as
shown. Point O is equidistant from the loudspeakers.
(b) When one loudspeaker is turned off, the amplitude of waves at O is halved, intensity drops to
1
4
1
I
4
of the original value, resulting in a change of intensity level 10 log10
= - 6 dB.
I
10.3 Speed of Sound
10.3.1
Kundt’s Tube
(or cork dust)
(a) The loudspeaker produces progressive longitudinal waves travelling towards the end of the
cylinder where they are reflected to interfere/superpose the incident waves.
(b) The frequency of the sound/signal generator is varied until resonance occurs.
(c) The stationary wave formed is revealed by the lycopodium powder which swirls away from the
antinodes (where the air is vibrating strongly) and heaps are formed at the nodes.
(d) By measuring the average separation between the heaps (or nodes), d.
(e) The speed of sound in air equals f (2d) where f = frequency of the sound waves.
(f) Precautions : (i) the tube should be dry,
(ii) the layer of lycopodium powder should be thin.
10.4 Doppler Effect
10.4.1 Police Radar Speed Check System
(a) Radar installed near the road sends microwaves of frequency f1 to a travelling car, then the
microwaves are reflected back to the radar.
(b) Due to Doppler effect, the observed frequency f2 of the reflected microwaves is slightly
different from f1.
(c) Hence by comparing the transmitted and reflected microwaves, beats are formed.
(d) As the beat frequency (= f1 - f2) depends on the car speed, the car speed can be checked.
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10.5 Interference of Sound
sound intensity
X
loudspeakers
O
position
Y
X
O
Y
(a) Two identical loudspeakers connected to the same signal generator are placed inside a room as
shown. Point O is equidistant from the loudspeakers and line XOY is parallel to the line joining
the loudspeakers. The variation of sound intensity along XOY is shown above.
(b) (i) Sound waves from the two loudspeakers reaching different points on XY have different
path differences, interference thus occurs.
(ii) For the path difference equals nλ, constructive interference occurs and a maximum
intensity results.
1

(iii) For the path difference equals  n   λ, destructive interference occurs and a minimum
2

intensity results.
(c) Waves from the loudspeakers take paths of different lengths to a minimum position, by
inversely square law, amplitude of the waves are different, or
(When waves diffracted from a loudspeaker, the amplitude is maximum at the central position
but decreases when going sideways, so waves from the two loudspeakers reaching a minimum
point would have different amplitudes.)
destructive interference of waves is not complete, hence a non-zero intensity results.
(d) If the frequency of the signal generator increases, the wavelength of the sound waves produced
would decrease, making the separation between positions of maximum/minimum intensity
decrease.
10.6 Standing Waves and Musical Instruments
10.6.1
Wind Instruments (Resonance)
(a) Glass tube is lowered slowly into the beaker of water until the air inside the tube is heard to
vibrate loudly (with the frequency of the tuning fork). [Then a stationary wave motion of the
air in the tube is produced form the superposition of the incident and reflected waves from the
air/ water surface.]
(b) Resonant frequency, f0 = v/4l, where l is the air column length and v velocity of sound.
(c) Damping of the oscillations would reduce sharpness of the frequency response and this could
be produced by e.g. an increase in the humidity of the air.
Alternative Description
(a) In the production of musical sounds from air columns in wind instruments resonance occurs, in
many cases, between the vibrations of air columns and of small vibrating reeds.
(b) The resonant frequency depends on the length of the vibrating air column (f 
1
) in general
L
the shorter the air column is, the higher the resonant frequency results.
(c) The vibrating air gives rise to the damping.
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Chapter 11 Electromagnetic Waves
11.0.1
Nature of Electromagnetic Waves
E
E
v
B
v
B
(a) These are a particular electromagnetic wave of high (1014 Hz) frequency.
(b) They consist of an electric field E and magnetic field B component which undergo time
variations in planes at right angles to each other and the direction of propagation. The changing
magnetic field produces the changing electric field (and vice-versa). Waves do not require any
medium in which to propagate with the relatively high speed of 3  108 m/s.
11.0.2
Comparison with Mechanical Waves
Electromagnetic waves are oscillating electric and magnetic fields, which do not necessarily require
a medium for propagation.
11.0.3
Comparison with Sound Waves
The propagation of light waves differs from that of sound waves in air since :
(a) no medium is necessary.
(b) wave is transverse.
(c) wave energy is carried by the electric and magnetic field of the electromagnetic wave.
E
direction of
propagation
B
11.0.4
The Electromagntic Nature of Light Wave
Light waves can propagate through vacuum to the earth, which is a property of electromagnetic
waves.
11.0.5
Radar
The radar sends out a narrow microwave beam, which is swept continuously through 360 ∘ by a
rotating aerial. The pulses reflected from land, other ships and buoys are shown on a CRO (Plan
Position Indicator), which has the time base origin in the center of the screen and represents the ship.
This assists safe navigation in fog and at night.
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11.1 Electromagnetic Spectrum
11.1.1
V.H.F. Radio Transmission
(A) Properties
E
E
v
B
v
B
(i)

Waves are : polarized and transverse (variations of in-phase electric and magnetic fields 
to direction of propagation).
(ii) In diagram wave moves to the right with time, similar to the sound wave given
previously.
(iii) For this type of wave no medium is necessary - the changing electric field gives rise to a
changing magnetic field and vice-versa, can take place in free space.
(B) Frequency Range is 80 - 100 MHz. (1 MHz = 106 Hz)
(C) Detection/Intensity Measurement
E.M.
wave
correct
polarised
aerial
tuned
circuit
amplifier
rectifier
meter
(i)
Voltage (a.c.) induced in an aerial (1) tuned to V.H.F. band (particular length) and (2)
orientated // to E vector of wave (i.e. correct polarisation).
(ii) Tuned circuit (L/C // circuit) used to select particular frequency from other radio waves.
(D) Long distance transmission
(i) The sky waves emitted from the transmitting aerial towards the sky are returned to earth
by the ionsphere due to gradual refraction.
(ii) On striking the earth the sky waves bounce back to the ionosphere where they are once
again directed to earth and thus in this way long distance propagation is possible.
ionsphere
Lost sky
wave
Sky wave
Space wave
Transmitting
aerial
Ground
waves
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Earth
Skip
distance
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11.2 Polarization of Light
11.2.1
Transverse Nature of Light Wave
Light can be polarized, which implies that it is transverse.
11.2.2
Polarization by Reflection
The glare of the sun reflected from the sea is partially (horizontally) plane-polarised, therefore it can
be effectively reduced by wearing polaroid sunglasses such that its polarising direction (vertical)
is crossed with that of the glare.
11.2.3
Polarization by Scattering
(a) Unpolarised sunlight is incident horizontally on air molecules around O in the earth’s
atmosphere.
y
unpolarised
sunlight
air
molecules
transmitted
light
x
0
z
scattered
light
(b) When unpolarised sunlight impinges on the molecules, the electric field of the EM wave sets
the electric charges within the molecules into motion (or the molecules absorb some of the
(c)
(d)
(e)
(f)
incident radiation).
The molecules then quickly re-emit light since oscillating electric charges produce EM waves.
Because of the transverse nature of light, the electric field of the re-emitted waves should be in
the plane that includes the line of oscillation (i.e., y-z plane), the scattered light is completely
plane polarized with its electric vector in the z-direction.
Observe the scattered light ray through a polaroid.
When the polaroid is rotated, the intensity varies alternately for every rotation of 90∘.
11.2.4
Application of Polarization (Polaroid Sunglasses)
(a) Reflected light is partially or completely polarized.
(b) If Polaroid discs in sunglasses are suitably oriented, much polarized light will be absorbed.
(c) Thus reflected glare is effectively reduced and thereby enabling detail to be seen that would
otherwise be hidden by glare.
11.3 Interference of Light
11.3.1
Explanation by Principle of Superposition
In interference complete cancellation/reinforcement of the 2 waves takes place at particular
locations and these stay constant and do not vary with time.
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Alternative Description
Following from the principle of superposition, in a region where 2 wave-trains meet the amplitudes
either reinforce or cancel, depending upon whether they are in-phase or anti-phase - giving rise to
an interference pattern (or fringes).
11.3.2
Conditions and Difficulties in the Observation of Interference of Light
(a) The necessary conditions are that the phase between the waves remains constant (or zero) and
that the frequencies are the same, with similar amplitudes.
(b) Light is emitted from a normal light source as photons - or discrete packages of wave trains
and since these are emitted in a random manner there is no constant phase relationship between
the individual wave packets. To satisfy the conditions for interference :
(i) A single source used with light directed on to a location via 2 paths of different lengths.
(ii) Since the wavelength of light is small ( 600 nm) (and also the wave train length is short)
the path difference must be small and comparable to the wavelength.
Alternative Description
(a) The emission of light from light source is random.
(b) The problem is that no interference pattern is observable with light from two separate sources
which are incoherent. (or, it is not possible to have two separate sources in constant phase
relationship, i.e. coherent)
11.3.3
Young’s Double-slit Experiment
(A) Set-up
Diffracted beam
from S 1
S1
S
Monochromatic
light source
interference
effects in
region where
beams overlap
S2
(narrow)
Single slit Double
slit
Diffracted beam
from S 2
(i) Instead of using separate sources, Young used two sources derived from a single source.
(ii) The same wavefront from the single slit reaches the two close and narrow slits, S1 and S2.
(iii) Thus the waves reaching the double slits are in constant phase difference, i.e. S1 and S2
become coherent sources.
(B) Observed Interference Pattern
(i) Light waves from the two slits have path difference when reaching a point on the screen.
(ii) When the path difference is such that the two waves superimpose constructively (in
phase), bright fringe is observed.
(iii) When the path difference is such that the two waves superimpose destructively (out of
phase), dark fringe is observed.
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Alternative Description
(a) Set up the apparatus as shown.
(b) The filament lamp acts as a single slit, and the diffracted light beams from the two narrow
slits overlap in the region beyond the slits (accept idea presented using diagram).
(c) Use a filter to obtain monochromatic light so that alternate bright and dark, equally spaced
interference fringes are observed (at the cross-wires of the travelling eyepiece). This shows the
wave nature of light.
(d) The average fringe spacing y is found by measuring across as many fringes as possible with the
travelling eyepiece.
(e) A metre rule is used to measure the separation d between the double slit and the traveling
eyepiece. The slit separation a is measured directly with a traveling microscope.
(f)
The wavelength λ of the monochromatic light is approximated by λ =
ay
.
d
Alternative Description
(a) Young’s two slit interference experiment. S is a monochromatic light source. A and B are two
slits, T being a screen. Alternate dark and bright equi-spaced parallel (to slits) fringes are
observed on the screen due the superposition of waves from A and B.
(b) When these are in phase a bright fringe is produced and when they differ in phase by  (or path
difference /2) cancellation occurs producing a dark fringe.
Alternative Description
R
C
S
*
Q
A
O
P
B
T
(a) Using 2 narrow (~ 1 mm) slits A and B waves from A and B may cancel or reinforce if the
phase difference are  or 0 and the path difference BQ - AQ = (2n + 1)/2 or n, n integer.
(b) This produces alternate bright/dark fringe along RT.
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11.3.3.1 Precautions and Necessary Condition
(a) Light source should be strong (or black out the laboratory as much as possible) and properly
(b)
(c)
(d)
(e)
(f)
shielded so that no stray light falls on the screen.
A monochromatic light source can be used in order to obtain sharper fringes.
Both slits S1 and S2 should be as narrow as possible so that the light emerging from the slits
undergoes significant diffraction.
The slits should be separated by a very small distance (~ 0.5 mm) so that the light from the two
slits overlap somewhere in front of the screen.
The screen should be placed at an appreciable distance (1 ~ 2 m) from the slits so that the
separation of fringes is observable while the intensity is not too low.
Make sure that the filament is parallel to the slits S1 and S2. (or if a source slit is used it should
be parallel to the two slits S1 and S2.)
Alternative Description
(a) Interference for light waves only possible from single source via 2 different path distances.
(b) Light is not coherent (normally) since waves are emitted in random quanta or wave packets. To
maintain phase relation path distance has to be < wave packet length.
11.3.3.2 Disappear of Interference Pattern
(a) Two separate light sources are used.
(b) The light from the two slits is coherent while that from the two light bulbs is not and a
stable/observable interference pattern cannot be formed.
Alternative Description
(a) Photons which consist of short length wave packets are emitted randomly during energy
transitions in atoms.
(b) A constant phase relation between light waves from two independent sources is therefore not
possible and so steady interference conditions where they meet is not possible.
(c) The path difference between light rays from the same light source is too great.
(d) If the path difference is longer than the wave packet length then interference is not possible
since wave packets do not overlap.
11.3.4
Interference Pattern of Microwave
6 cm
Transmitter
3 cm
3 cm
Probe
receiver
Metal
plates
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(a) Arrange the metal plates such that two slots of approximately 3 cm wide are formed with a
separation of about 6 cm.
(b) Interference occurs between the two wave-trains diffracted from the two slots that act as two
coherent sources. The receiver detects the maxima and minima of the interference pattern as it
is moved around.
(c) Constructive and destructive interference occur whenever the path difference of the
1
microwaves from the slots is nλ and ( n  ) respectively.
2
11.3.5
blooming of lenses
1
t
2
Air
(nF) Film
Glass
(a) The amount of light reflected can be reduced by coating the lens surface with a thin film of
transparent material (e.g. a fluoride salt) with suitable thickness t and refractive index nF so
that destructive interference from the reflected light rays and from the interfaces can occur.
(b) Since complete interference is not possible simultaneously for every wavelength of white light,
an average wavelength for λ, such as green - yellow is usually chosen since it is most
sensitive to the eye.
(c) Thus for red and blue (violet) light, the reflection is weakened but not completely eliminated,
so a coated lens appears purple in white light.
11.3.6
Soap Film
Due to gravitational force and evaporation, the cross-section of the film will vary roughly with time
as follows:
t
A
50 mm
B
P
light
soap
film
(1)
(2)
(3)
(a)
1 minute
2 minute
3 minute
(a) There will be interference in the eye between light rays A and B and the path difference will be
~ 2t. Moving down the film horizontal dark fringes will be observed at locations where 2t =
n, n being integer - note a phase inversion takes place at P, on reflection from a denser
medium.
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(b) At a later time the angle of the ‘wedge’ film is decreased, as well as the thickness and the
number of fringes observed will be decreased, with their separation increasing. For small
angles,  = 2w. When w increases,  decreased.
2 fringes

t2
t1
w
(c) When film has reduced to t < /4 rays A and B (in previous diagram) will have a phase
difference of , thus they will interfere destructively – and film will appear black.
11.3.7
Thin Film
eye
N
(i)
A
i
M
(ii)
D
O
C
E
t
r
B
r t
(a) Different colours observed as eye is moved at different angles, viewing a thin oil film on water.
(b) Different path distances for white light (i) and (ii), 2t cos r = n for re-inforcement.  varies
for different coloured light.
11.3.8
Other Application (Testing of optical flatness)
Place the testing surface on top of a standard ‘flat’, any unevenness of the testing surface shows up
as fringes due to air-gaps produced.
11.4 Diffraction of Light
11.4.1
Wavefront Approach
diffracted
light

edge of slit
P1 P2
light
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(a) Due to wave nature of light, light fringes are observed outside the normal geometric image of
slit.
(b) Waves from secondary sources P1, P2 etc. may cancel (in intensity) if waves are in antiphase (
phase difference) or add if in phase giving rise to the observation of dark or bright fringes,
viewed at a distance from slit. e.g. in fig. 2 all such waves from secondary sources X and Y
across slit will cancel in direction 2, since path difference are /2.
11.4.2
Comparison with the Diffraction of Sound
(a) For observable diffraction the wavelength of the waves has to be comparable with the
dimension of the opening. Briefly for a ‘slit’ maxima occur due to re-inforcement of waves
from secondary sources at an angle  to normal of slit when sin  = /a.
(b) ‘a’ being width of slit. For sound  ~ 1 m while for light  ~ 6  10-7 m.
(c) Thus very narrow slits needed.
11.4.3
Diffraction Grating
11.4.3.1 Explanation by Principle of Superposition
Grating
R
V R V W V RV
Second
order
R
First Zero First Second
order order order order
A
R = Red Monochromatic
V = Violet light
W = White
 C
d
B

(a) Waves from secondary sources such as A and B on adjacent slits will reinforce if path
difference AC = dsin  = n, n integer (Brought to focus in eye)
(b) n = 0 zero, n = 1 first, n = 2 second orders of spectra. V < R and so V < R.
(c) Differentiating,
d
n
d

, as  increases
increases - wider spectra.
d d cos 
d
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Alternative Description
(a) Consider the diffracted rays from the slits in a certain direction.

d

(i)
When all these rays are in phase, they will superpose to give a local maximum, i.e. a
principal maximum.
(ii) The path difference between two consecutive rays must be nλ, where n is an integer.
(iii) In the diagram, the path difference between two consecutive rays = d sinθ.
(iv) Therefore, the angular positionθof the principal maxima is given by d sinθ= nλ.
(b) (i) Gratings gives sharper and brighter patterns.
(ii) Also the angular separation of the fringes is larger by using gratings as the slit separation
can be made many times smaller.
11.4.3.2 Measuring the Wavelength of Light
(a) Place two metre rules perpendicular to each other as shown. Hold a diffraction grating against
one end of a metre rule.
(b) View through the grating the vertical filament of the ray-box lamp placed about 1 m from the
metre rules. Ask your partner to move a pencil along the second metre rule until it is in line
with the middle of the red colour of the first order spectrum.
(c) Measure the distance x and hence find sinθ.
(d) Apply the grating formula to calculate the wavelength of the red light.
d sinθ = nλ
where n = 1 and d is the grating separation.
(e) Precautions : Make sure the filament of the lamp is vertical.
(f) The major source of error is the uncertainty in locating the exact position of the red light which
is normally a narrow band instead of a sharp line.
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Chapter 12 Optical Instrument
12.1 Human Eye
R
long distance
moon
L
dollar
O2

O1
a
b

image
(a) The size of the image formed on the retina of the eye only depends upon the angle subtended
by the object at the eye.
(b) This angle is similar for the larger object (moon) at a long distance and the smaller object
(dollar) at a nearer distance.
12.2 Magnifying Glass

L
I'
h'
O'
I

h
F
'
O
C
D
(a) Larger objects at a fixed distance will appear larger since the angle subtended at eye is greater.
(b) Clearly, from ray diagram, the viewed image I formed at the distance of nearest clear vision D
appears greater than the image I’ formed at infinity since  > ’, and hence magnification is
greater for the former arrangement.
(c) Magnification is limited by the focal length of the lens.
12.3 Compound Microscope
D
eyepiece
objective
I2
”
Fe I 1
h
O
h1
Fo
L1
f1
h2
L2
f2
(a) Clearly, from ray diagram, a magnified image I1 is produced for object O by lens L1.
(b) The final image I2 of this ‘object’ I1 produced by lens L2 subtends an angle ”.
(c) Since h1 > h the angle ” > ’ and the magnification is greater than using a single lens.
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12.4 Reflecting Telescope
12.4.1
Normal Adjustment
The telescope is in normal adjustment if the final image is at infinity, so that the eye is in a fully
relaxed state and the observer can view for long periods of time without undue strain. (or so that the
final image is at the position where the observer expects to see it.)
12.4.2
Ray Diagram
Objective
L1
fo
Parallel rays
from point at
top of distant
object


Eyepiece
L
fe 2
Fe
Fo
h1 
I1
To top of final virtual
image at infinity
(a) Objective lens is used to collect large amount of light from distant object and to form a real
intermediate image at its focal plane acting as an object for the eyepiece.
(b) Eyepiece acts as a magnifying glass for magnifying the intermediate real image formed by the
objective and produce a virtual image.
12.4.3
Magnification Power
(a) Magnifying power M is defined as M = β / α where β = angle subtended at the eye by
the final image at infinity (i.e. visual angle of object with aided eye), α = angle subtended at
the eye by the object without the telescope (i.e. visual angle of object with unaided eye).
(b) Magnifying power can be increased by
(i) using objective lens of longer focal length fo,
(ii) using eyepiece of shorter focal length fe .
(c) Limitations on magnifying power :
(i) fo cannot be too long as long tube length (fo + fe) will cause inconvenience.
(ii) fe cannot be too short as too curved lens causes spherical/chromatic aberration.
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12.4.4
Viewing Ground Object
(a) One major disadvantage is that the final image observed is inverted.
(b) Use an erecting lens (convex, focal length f) for an additional inversion by placing it between
the intermediate image and the eyepiece as shown.
(c) The final image observed is erect and not laterally inverted.
To top of final virtual
image at infinity
Eyepiece
Objective
Erecting
lens
Parallel rays
from point at
top of distant
object
I1
I2
fo
or
2f
2f
fe
(Arrange two totally reflecting prisms (45∘– 90∘– 45∘) between the objective and
eyepiece.)
12.5 Spectrometer
12.5.1
Set-up (Optical Instrument Adjustments)
C - collimator, T – Telescope
(a) Telescope
(i) Move the eyepiece until a clear image of the cross-wires is observed to be in focus.
(ii) The length of the telescope is adjusted until a focused image of a distant object can be
observed at cross-wires location.
(b) Collimator
The collimator and telescope are rotated to being in-line, with the prism removed from the
table. The slit of the collimator is illuminated by a monochromatic light source and its position
varied so that a sharp image of this is observed, looking through the telescope.
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(c) Reasons
(i) Telescope is adjusted so that parallel light rays entering it will be brought to a focus on
the cross-wires.
(ii) Collimator is adjusted so that light emerging from the slit will be converted into parallel
light ray-beam, on exit.
Alternative Description
collimator
telescope
*
light
source
crosswire
platform
(a) Usual adjustments are first made on (i) cross-wires, (ii) telescope and (iii) collimator of the
spectrometer so that the collimator produces parallel light and the telescope focuses it at the
cross-wires.
(b) The table with the grating are turned so that the incident light falls on the grating normally.
(c) The telescope is rotated, say to T1, and the reading corresponding to the first-order
image is taken.
(d) The first-order reading on the other side of the normal, say at T2, is also taken.
incident
light
T1
grating


T2
(e) Halve the angle between these two readings givesθ, from which λ can be calculated using n
λ = d sinθ, where d is the grating spacing and n = 1.
12.5.2
Advantages of Using Diffraction Grating
(a) Easier to calibrate spectrometer/measure wavelength - simple relation n = d sin .
(b) By using gratings of different line-spacing (d) a wide spectrum range possible. Absorption
serious for glass lenses/prisms in U.V./I.R. Expensive/ practical difficulties using, e.g. rocksalt
components.
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Extra Essay
Wave nature of matters
(a) The probability of finding a particle is proportional to the square of the amplitude of the ‘wave’
- i.e. the ‘waves’ are not waves of matter or fields but of probability.
(b) Thus in the dark fringes/spots produced by electron diffraction there is a very low probability
that an electron can be found, while for the bright fringes/spots the probability is very high.
Structure of Camera
(a) The intensity of light collected by the camera  aperture area, i.e. d2 and this determines the
intensity of the image.
(b) The recorded brightness (and darkening of chemicals on film) depends upon the time
integration of the light energy received, i.e. t.
(c)
F1
d
F2
f1
f2
For distant objects it can be seen, from ray diagram, that the size of the image is  f and the
area of image  f 2.
 Brightness of image  1/f2 .
(d) The f-stop is defined as f-n, where the aperture diameter d = f/n, f being the focal length of
camera lens.
(e) In a camera the effective aperture diameter can be varied by a mechanical ‘iris’. e.g. by
decreasing the f-stop the time exposure can be reduced which could produce a sharper image
of a moving object. (or any other reasoning)
(f) So the brightness of an image recorded on photographic film in a simple camera is
proportional to
d 2t
.
f2
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Questions haven’t asked during the past 20 years
10.1.1
Pressure and Displacement in Sound Waves
Describe how the sound wave propagate through the air.
(a) When the diaphragm of the loudspeaker vibrates and pushes forward, it compresses the air
near the diaphragm.
(b) The compressed air propagates forward for a short distance. Then it collides with other
molecules.
(c) The momentum of the compressed air is transferred to these air molecules.
(d) Later, these air molecules carry out similar process and transfer their momenta to some other
air molecules. In this way, compressions travel through the air.
(e) When the diaphragm of the loudspeaker pulls back, the air molecules near the diaphragm
(f)
acquire more space.
Then the neighbouring air molecules quickly fill the entire space. An expansion of air is
created and this expansion of air, called rarefaction, will move away from thediaphragm.
10.6 Standing Waves and Musical Instruments
Stringed Instruments
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