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Oscillations and Waves
Wave Characteristics
Waves
A wave is a means of transferring energy and
momentum from one point to another without
there being any transfer of matter between the
two points.
This illustrates that there is no net transfer of the
medium through which the wave travels, only
energy moves from place to place.
In many examples, the wave carrying medium will
oscillate with simple harmonic motion (i.e. a  -x).
Progressive Waves
E.g. A duck on water:
Wave direction
Duck oscillation
Any wave that moves through or across a medium
(e.g. water or even a vacuum) carrying energy away
from its source is a progressive (travelling) wave.
As the wave passes the duck, the water (and duck)
only oscillate vertically.
Describing Waves
1. Mechanical or Electromagnetic
Mechanical waves are made up of particles vibrating.
e.g. sound – air molecules; water – water molecules
All these waves require a substance for transmission and so
none of them can travel through a vacuum.
Electromagnetic waves are made up of oscillating electric and
magnetic fields.
e.g. light and radio
These waves do not require a substance for transmission and
so all of them can travel through a vacuum.
Describing Waves
2. Progressive or Stationary
Progressive waves are waves where there is a net transfer of
energy and momentum from one point to another.
e.g. sound produced by a person speaking; light from a lamp
Stationary waves are waves where there is a NO net transfer of
energy and momentum from one point to another.
e.g. the wave on a guitar string
Describing Waves
3. Longitudinal or Transverse
Longitudinal waves are waves where
the direction of vibration of the
particles is parallel to the direction in
which the wave travels.
e.g. sound
vibrations
wave direction
LONGITUDINAL WAVE
Describing Waves
Transverse waves are waves where the
direction of vibration of the particles or
fields is perpendicular to the direction
in which the wave travels.
e.g. water and all electromagnetic
waves
Test for a transverse wave:
Only TRANVERSE waves undergo
polarisation.
vibrations
wave direction
TRANSVERSE WAVE
Polarisation
The oscillations within a transverse wave and the direction
of travel of the wave define a plane. If the wave only
occupies one plane the wave is said to be plane polarised.
Polarisation
Light from a lamp is unpolarised. However, with a
polarising filter it can be plane polarised.
Polarisation
If two ‘crossed’ filters are used then no light will
be transmitted.
Aerial alignment
Radio waves (and microwaves) are
transmitted as plane polarised waves. In the
case of satellite television, two separate
channels can be transmitted on the same
frequency but with horizontal and vertical
planes of polarisation.
In order to receive these transmissions the
aerial has to be aligned with the plane
occupied by the electric field component of
the electromagnetic wave.
The picture shows an aerial aligned to
receive horizontally polarised waves.
Measuring waves
Displacement, x
This is the distance of an oscillating particle from its undisturbed or
equilibrium position.
Amplitude, a
This is the maximum displacement of an oscillating particle from its
equilibrium position. It is equal to the height of a peak or the depth of a
trough.
amplitude a
undisturbed
or equilibrium
position
Measuring waves
Phase, φ
This is the point that a particle is at within an
oscillation.
Examples: ‘top of peak’, ‘bottom of trough’
Phase is sometimes expressed in terms of an angle up
to 360°. If the top of a peak is 0° then the bottom of a
trough will be 180°.
Measuring waves
Phase difference, Δφ
This is the fraction of a cycle between two particles within one
or two waves.
Example: the top of a peak has a phase difference of half of one
cycle compared with the bottom of a trough.
Phase difference is often expressed as an angle difference. So
in the above case the phase difference is 180°. Also with phase
difference, angles are usually measured in radians.
Where: 360° = 2π radian; 180° = π rad; 90° = π/2 rad
Measuring waves
Wavelength, λ
This is the distance between two consecutive particles at
the same phase.
Example: top-of-a-peak to the next top-of-a-peak
unit – metre, m
wavelength λ
Measuring waves
Period, T
This is equal to the time taken for one complete oscillation in of
a particle in a wave.
unit – second, s
Frequency, f
This is equal to the number of complete oscillations in one
second performed by a particle in a wave.
unit – hertz, Hz
NOTE: f = 1 / T
The wave equation
For all waves:
speed = frequency x wavelength
c=fλ
where speed is in ms-1 provided frequency is in hertz and
wavelength in metres
Complete
Wave speed
Frequency
Wavelength
Period
600 m s-1
100 Hz
6m
0.01 s
10 m s-1
2 kHz
0.5 cm
0.5 ms
340 ms-1
170 Hz
2m
5.88 ms
3 x 108 ms-1
200 kHz
1500 m
5 x 10-6 s
Each bright line in this diagram represents a crest and can be regarded as a WAVEFRONT.
Wavefronts and rays
.
A RAY can be thought of as a locus of one point on a wavefront showing the direction in
which energy is travelling.
Download from http://phet.colorado.edu/simulations/index.php?cat=Sound_and_Waves
In the late 19th century physicists had been working
extensively with electricity and magnetic fields. A
great many discoveries in these fields were being
made. At the same time it became universally
accepted that the best model for light was the wave
model.
James Clerk Maxwell summarised, synthesised and
unified these ideas. He came up with the idea that all
of these phenomena, including light, were simply
different forms of
ELECTROMAGNETIC RADIATION
Electromagnetic waves are created by accelerating
charges which result in a rapidly changing magnetic
field and electric field travelling at right angles to each
other and to their direction of travel.
EM Wave applet: http://micro.magnet.fsu.edu/primer/java/scienceopticsu/electromagnetic/index.html
Although the previous image showed only two
transverse waves, EM waves don’t look like these in
reality. There are actually many planes of electricmagnetic field oscillations.
Source: http://sol.sci.uop.edu/~jfalward/physics17/chapter11/chapter11.html
It is the FREQUENCY of the waves that determines the
type of electromagnetic wave and the different
frequencies make up the ELECTROMAGNETIC SPECTRUM
Source: http://outreach.atnf.csiro.au/education/senior/astrophysics/images/em_spectrumextended.jpg
Note that VISIBLE LIGHT only makes up a small part of
the spectrum
Source: http://imgs.xkcd.com/comics/electromagnetic_spectrum_small.png
All electromagnetic waves travel with the same speed in free space. It is worthwhile to
recall the orders of magnitude of the wavelengths of the principal radiations in the
electromagnetic spectrum, such as the following
Wave Pulses
• A pulse wave is a sudden distortion or
disturbance that travels through a material or
medium
Reflection from fixed or free end
Reflection of Pulses
String with a fixed end
• If a pulse travels along a string that is fixed to a rigid
support, the pulse is reflected with a phase change of
180º
• The shape of the pulse stays the same, except that it
is inverted and travelling in the opposite direction
• The amplitude of the pulse is slightly less as some
energy is absorbed at the fixed end
• When the (upward) pulse reaches the fixed end, it
exerts an upward force on the support, the support
then exerts and equal and opposite downward force
on the string (reaction force), causing the inverted
pulse to travel back along the string
•
http://rt210.sl.psu.edu/phys_anim/waves/indexer_waves.html
Reflection of Pulses
String with a free end
• If a pulse travels along a string that is tethered
to a pole but free to move, the pulse is
reflected with no phase change
• The shape of the pulse stays the same, except
that it is travelling in the opposite direction
Superposition
Superposition is seen when two waves of the same type cross.
It is defined as “the vector sum of the two displacements of
each wave”:
Superposition of waves
This is the process that occurs
when two waves of the same
type meet.
The principle of superposition
When two waves meet, the
total displacement at a point
is equal to the sum of the
individual displacements at
that point
reinforcement
cancellation
Superposition
Law of Superposition (interference)
Whenever two waves of the same type meet at the
same point, the total amplitude (displacement) at
that point equals the sum of the amplitudes
(displacements) of the individual waves.
(You tube link1 and link2)
For constructive interference at any point, wavefronts must be
‘in phase’ and their path difference must be a whole number of
wavelengths:
path difference = nλ
For destructive interference at any point, wavefronts are ‘π out
of phase’ and their path difference is given by:
path difference = (n + ½) λ
Superposition of Sound Waves
Constructive interference i.e. Loud
or bright. Waves are in phase
Destructive interference i.e.
dark or quiet. Waves are π
rads out of phase.
Path Difference
Q
2nd subsidiary
maximum
P
1st subsidiary
maximum
S1
O
Central maximum
P’
1st subsidiary
maximum
Q’
2nd subsidiary
maximum
S2
At O : Zero path
difference
At P and P’ path
difference = 1λ
At Q and Q’ Path
difference = 2λ
So in the above example…
S2Q – S1Q = 2λ
For constructive interference at any point, wavefronts must be ‘in phase’ and their path
difference must be a whole number of wavelengths:
path difference = nλ
For destructive interference at any point, wavefronts are ‘π out of phase’ and their path
difference is given by:
path difference = (n + ½) λ
Task:
i.
On your interference diagram…
Draw in lines of constructive and destructive
interference
ii. Indicate the lines that join points…
a. in phase
b. 2π out of phase (path difference = λ)
c. 4π out of phase (path difference = 2λ)
d. 3π out of phase (path difference = 1.5λ)
Coherent waves
A stable pattern of interference is only obtained if
the two wave sources are coherent.
Two coherent wave sources…
i. have a constant phase difference,
ii. thus produce waves with equal frequency.
Interference
Interference occurs when two waves of the same type (e.g. both water, sound,
light, microwaves etc.) occupy the same space.
Wave superposition results in the formation of an interference pattern made
up of regions of reinforcement and cancellation.
Double slit interference with light
This was first demonstrated by Thomas Young in 1801.
The fact that light showed interference effects supported
the theory that light was a wave-like radiation.
Thomas Young 1773 1829
Experimental details
Light source:
This needs to be monochromatic (one colour or frequency).
This can be achieved by using a colour filter with a white light. Alternatives
include using monochromatic light sources such as a sodium lamp or a laser.
Single slit:
Used to obtain a coherent light source. This is not needed if a laser is used.
Double slits:
Typical width 0.1mm; typical separation 0.5mm.
Double slit to fringe distance:
With a screen typically 1.0m.
The distance can be shorter if a microscope is used to observe the fringes.
Interference fringes
Interference fringes are formed where the two
diffracted light beams from the double slit overlap.
A bright fringe is formed where the light from one slit
reinforces the light from the other slit.
At a bright fringe the light from both slits will be in
phase.
They will have path differences equal to a whole
number of wavelengths: 0, 1λ, 2λ, 3λ etc…
A dark fringe is formed due to cancelation where the
light from the slits is 180° out of phase.
They will have path differences of: 1/2λ, 3/2 λ , 5/2 λ
etc..
Young’s slits equation
fringe spacing, w = λ D / s
where:
s is the slit separation
D is the distance from the slits to
the screen
λ is the wavelength of the light
w
Question 1
Calculate the fringe spacing obtained from a double slit
experiment if the double slits are separated by 0.50mm
and the distance from the slits to a screen is 1.5m with
(a) red light (wavelength 650nm and (b) blue light
(wavelength 450nm).
fringe spacing w = λ D / s
Question 2
Calculate the wavelength
of the green light that
produces 10 fringes over
a distance of 1.0cm if the
double slits are separated
by 0.40mm and the
distance from the slits to
the screen is 80cm
1.0 cm
Demonstrating interference with a laser
A laser
(Light Amplification by Stimulated Emission of Radiation)
is a source of coherent monochromatic light.
0.5m to 2m
Oscillations and Waves
Wave Properties
Wave diagrams
1) Reflection
2) Refraction
3) Refraction
4) Diffraction
Refraction
• Refraction is when waves bend as they travel
from one medium to another
• When a wave travels into a different medium:
– the wave speed changes
– the wavelength changes
– the frequency stays the same
– If the wave hits the new medium at an angle, the
wave direction will change
Refraction
Refraction occurs when a
wave passes across a
boundary at which the
wave speed changes.
The change of speed
usually, but not always,
results in the direction of
travel of the wave
changing.
A wave slowing down on
crossing a media boundary
Refraction of light
(a) Less to more optical dense transition (e.g. air to glass)
AIR
GLASS
normal
angle of
incidence
Light bends TOWARDS the normal.
The angle of refraction is LESS than the angle of incidence
angle of
refraction
(b) More to less optical dense transition (e.g. water to air)
angle of
refraction
normal
angle of
incidence
WATER
AIR
Light bends AWAY FROM the normal.
The angle of refraction is GREATER than the angle of incidence
Refractive index (n)
This is equal to the ratio of the wave speeds.
refractive index, ns = c / cs
ns = refractive index of the second medium relative to
the first
c = speed in the first region of medium
cs = speed in the second region of medium
Question 1
When light passes from air to glass its speed falls from
3.0 x 108 ms-1 to 2.0 x 108 ms-1.
Calculate the refractive index of glass.
ns = c / c s
= 3.0 x 108 ms-1 / 2.0 x 108 ms-1
refractive index of glass = 1.5
Question 2
The refractive index of water is 1.33.
Calculate the speed of light in water.
ns = c / c s
→ cs = c / ns
= 3.0 x 108 ms-1 / 1.33
speed of light in water = 2.25 x 108 ms-1
Examples of refractive index
Examples of ns for light (measured with respect to a
vacuum as the first medium)
vacuum = 1.0 (by definition)
air = 1.000293 (air is usually taken to be = 1.0)
ice = 1.31
water = 1.33
alcohol = 1.36
glass = 1.5 (varies for different types of glass)
diamond = 2.4
The law of refraction
Medium of
refractive
index, n1
θ1
n2
θ2
When a light ray passes from a
medium of refractive index n1 to
another of refractive index n2
then:
n1 sin θ1 = n2 sin θ2
where:
θ1 is the angle of incidence in the
first medium
θ2 is the angle of refraction in the
second medium
In the data booklet the angles of incidence and
refraction are called θ1 and θ2.
It can further be shown that…
1n 2
= sinθ1 = v1
sinθ2
v2
= n2
n1
Note that this is written in the data booklet as…
sinθ2 = v2
sinθ1
v1
= n1
n2
E.g. 1
A wave travelling at 12cms-1 is incident upon a
surface at an angle of 55° from the normal.
a. If the angle of refraction is 40°, determine the
speed of the wave in the second medium.
sinθ2 = v2
sinθ1
v1
sin 40 = v2
sin 55
12
v2 = 9.4 cms-1
b. If the initial wavelength is 6cm determine the
frequency of the wave in the second medium.
In first medium:
v = fλ
f = v/λ = 0.12 / 0.06 = 2.0 Hz
Frequency does not change during refraction

f = 2.0 Hz
E.g. 2
For light travelling from water into glass, r=20°.
If nw = 1.33 and ng = 1.50, determine i (θ1).
sinθ2 = n1
sinθ1
n2
sin20 = 1.33
sinθ1
1.50
sinθ1 = 0.34 / 0.89
= 0.38
θ1 = sin-1 0.38
= 22.5°
Question
Calculate the angle of refraction when light passes from
air to glass if the angle of incidence is 30°.
n1 sin θ1 = n2 sin θ2
→ 1.0 x sin 30° = 1.5 x sin θ2
1.0 x 0.5 = 1.5 x sin θ2
→ sin θ2 = 0.5 / 1.5 = 0.333
→ angle of refraction, θ2 = 19.5°
Complete:
Answers
medium
1
n1
θ1
medium
2
n2
θ2
air
1.00
50o
water
1.33
35.2o
glass
1.50
30o
air
1.00
48.6o
water
1.33
59.8o
glass
1.50
50o
air
1.00
50o
diamond
2.4
18.6o
air
1.00
50o
unknown
1.53
30o
Total internal reflection
θ1 > c
n1
n2 ( < n1 )
θ1
Total internal reflection (TIR)
occurs when light is incident
on a boundary where the
refractive index DECREASES.
And the angle of incidence is
greater than the critical
angle, c for the interface.
Critical angle (c)
This is the angle of incidence,
θ1 that will result in an angle of
refraction, θ2 of 90 degrees.
n1 sin θ1 = n2 sin θ2
becomes in this case:
n1 sin c = n2 sin 90°
n1 sin c = n2 (sin 90° = 1)
Therefore: sin c = n2 / n1
θ1 = c
θ1
n1
n2 (<n1)
θ 2 = 90o
Finding the Critical Angle…
1) Ray gets refracted
3) Ray still gets refracted (just!)
THE CRITICAL
ANGLE
2) Ray still gets refracted
4) Ray gets
internally reflected
Question 1
Calculate the critical angle of glass to air.
(nglass = 1.5; nair =1)
sin c = n2 / n1
→ sin c = 1.0 / 1.5
= 0.667
→ critical angle, c = 41.8°
Question 2
Calculate the maximum refractive index of a medium if light is to
escape from it into water (nwater = 1.33) at all angles below 30°.
sin c = n2 / n1
→ sin 30° = 1.33 / n1
→ 0.5 = 1.33 / n1
→ n1 = 1.33 / 0.5
→ refractive index, n1 = 2.66
Optical fibres
Optical fibres are an application of
total internal reflection.
Step-index optical fibre consists of
two concentric layers of transparent
material, core and cladding.
The core has a higher refractive
index than the surrounding cladding
layer.
core
cladding
Total internal reflection takes
place at the core - cladding
boundary.
The cladding layer is used to
prevent light crossing from
one part of the fibre to
another in situations where
two fibres come into contact.
Such crossover would mean
that signals would not be
secure, as they would reach
the wrong destination.
Question
A step-index fibre consists of a core of refractive index
1.55 surrounded by cladding of index 1.40. Calculate
the critical angle for light in the core.
sin c = n2 / n1
→ sin c = 1.40 / 1.55
= 0.9032
→ critical angle, c = 64.6°
Optical fibres in communication
A communication optical fibre allows
pulses of light to enter at one end,
from a transmitter, to reach a
receiver at the other end. The fastest
broadband systems use optical fibre
links.
The core must be very narrow to
prevent multipath dispersion.
This occurs in a wide core because
light travelling along the axis of the
core travels a shorter distance per
metre of fibre than light that
repeatedly undergoes total internal
reflection.
Such dispersion would cause an
initial short pulse to lengthen as it
travelled along the fibre.
Multipath dispersion causing
pulse broadening
input
pulse
output
pulse
The Endoscope
The medical endoscope contains two bundles of fibres.
One set of fibres transmits light into a body cavity and
the other is used to return an image for observation.
Optical fibres
Uses of Total Internal Reflection
Optical fibres:
An optical fibre is a long, thin, _______ rod made of
glass or plastic. Light is _______ reflected from one
end to the other, making it possible to send ____
chunks of information
Optical fibres can be used for _________ by sending
electrical signals through the cable. The main advantage
of this is a reduced ______ loss.
Words – communications, internally, large, transparent, signal
Diffraction
Diffraction occurs when waves spread out after passing
through a gap or round an obstacle.
Sea wave diffraction
Diffraction becomes more
significant when the size of
the gap or obstacle is
reduced compared with
the wavelength of the
wave.
i) Diffraction by a
"large" object
ii) Diffraction at a
"large" aperture
iii) Diffraction by a
"small" object
iv) Diffraction by a
"narrow" aperture
Interference
Interference occurs when two waves of the same type (e.g. both water, sound,
light, microwaves etc.) occupy the same space.
Wave superposition results in the formation of an interference pattern made
up of regions of reinforcement and cancellation.
Coherence
For an interference pattern to be
observable the two overlapping
waves must be coherent.
This means they will have:
1. the same frequency
2. a constant phase difference
If the two waves are incoherent the
pattern will continually change
usually too quickly for observations
to be made.
Two coherent waves can
be produced from a single
wave by the use of a
double slit.
Path difference
Path difference is the difference in distance travelled by two
waves.
Path difference is often measured in ‘wavelengths’ rather than
metres.
Example:
Two waves travel from A to B along different routes. If they
both have a wavelength of 2m and the two routes differ in
length by 8m then their path difference can be stated as ‘4
wavelengths’ or ‘4 λ’
Double slit interference with light
This was first demonstrated by Thomas Young in 1801.
The fact that light showed interference effects supported
the theory that light was a wave-like radiation.
Thomas Young 1773 1829
Experimental details
Light source:
This needs to be monochromatic (one colour or frequency).
This can be achieved by using a colour filter with a white light. Alternatives
include using monochromatic light sources such as a sodium lamp or a laser.
Single slit:
Used to obtain a coherent light source. This is not needed if a laser is used.
Double slits:
Typical width 0.1mm; typical separation 0.5mm.
Double slit to fringe distance:
With a screen typically 1.0m.
The distance can be shorter if a microscope is used to observe the fringes.
Interference fringes
Interference fringes are formed where the two
diffracted light beams from the double slit overlap.
A bright fringe is formed where the light from one slit
reinforces the light from the other slit.
At a bright fringe the light from both slits will be in
phase.
They will have path differences equal to a whole
number of wavelengths: 0, 1λ, 2λ, 3λ etc…
A dark fringe is formed due to cancelation where the
light from the slits is 180° out of phase.
They will have path differences of: 1/2λ, 3/2 λ , 5/2 λ
etc..
Standing Waves & Resonance
A standing wave is created from two traveling waves, having the same frequency and
the same amplitude and traveling in opposite directions in the same medium.
Using superposition, the net displacement of the medium is the sum of the two waves.
When 180° out-of-phase with each other, they cancel (destructive interference).
When in-phase with each other, they add together (constructive interference).
jw
Fundamentals of Physics
99