# The Superposition of Waves

## Transcription

The Superposition of Waves
```Optics Course
(Phys 311)
Wave Optics
The Superposition of Waves (2 of 2)
Lecturer:
Dr Zeina Hashim
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 1
Objectives covered in this lesson :
1. Standing waves (Continue : two waves of the same frequency).
2. Weiner’s Experiment.
3. Complex Waves (The addition of waves of different frequencies along the
same direction).
4. Beats.
5. Group velocity.
6. Addition of waves at right angles.
Phys
311
Wave Optics: The Superposition of Waves
Standing Waves :
also called:
“Stationary Waves”
Two waves with the same frequency, but travelling
in opposite directions will superimpose as follows:
Let
,
,
Standing wave
equation
Lesson 2 of 2
Slide 2
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 3
Standing Waves :
Conditions for a standing wave to occur:
1. The waves have to travel in opposite directions.
2. they must have the same frequency.
3. they must be coherent.
4. they must be 180º out-of-phase.
Reflected waves from totally reflecting surfaces always have a phase-shift of 180º.
Its profile does not travel through space, but it oscillates between a straight line position.
It is not of the form
Standing waves have nodes and antinodes.
Phys
311
Wave Optics: The Superposition of Waves
Standing Waves :
Food for thought:
Q: Why does a microwave oven require a turning table ?
Lesson 2 of 2
Slide 4
Phys
311
Lesson 2 of 2
Slide 5
Wave Optics: The Superposition of Waves
Standing wave
equation
Standing Waves :
Nodes occur when:
𝐸 = 0  amplitude of E = 0  sin 𝑘𝑥 = 0  𝑘𝑥 = 0 , 𝜋 , 2𝜋 , 3𝜋 , …
𝜆
 𝑥 = 0 ,2 ,𝜆 ,
or  cos 𝜔𝑡 = 0  𝜔𝑡 =
𝜋
2
𝜋
,32
𝜋
,52
,…  𝑡 =
𝜋
2𝜔
𝜋
, 3 2𝜔
𝜋
, 5 2𝜔
3𝜆
2
, 2𝜆 , …
,… 𝑡 =
𝜏
4
𝜏
,34
𝜏
,54
,…
Antinodes occur when:
𝜋
2
,
3𝜋
2
,
3𝜆
4
,
5𝜆
4
,…
𝐸 = 𝑚𝑎𝑥𝑖𝑚𝑢𝑚  amplitude of E = 2𝐸𝑜𝐼  sin 𝑘𝑥 = 1  𝑘𝑥 =
𝜆
𝑥=4 ,
𝜋
and  cos 𝜔𝑡 = 1  𝜔𝑡 = 0 , 𝜋 , 2𝜋 , 3𝜋 , …  𝑡 = 0, 𝜔 ,
2𝜋
𝜔
,
3𝜋
𝜔
5𝜋
2
,…
𝜏
𝜏
, …  𝑡 = 0, 2 , 𝜏, 3 2 , 2𝜏, …
Phys
311
Wave Optics: The Superposition of Waves
Standing Waves :
Nodes occur when:
𝑥=𝑚
𝜆
2
or
𝑡 = (2𝑚 + 1)
𝜏
4
, 𝑚 = 0, 1, 2 , …
𝑥 = (2𝑚 + 1)
𝜆
4
, 𝑚 = 0, 1, 2 , …
Antinodes occur when:
𝑡=𝑚
𝜏
2
and
Lesson 2 of 2
Slide 6
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 7
Standing Waves :
Q: Does a standing wave violate the law of conservation of energy (having zero
energy at the nodes) ?
Phys
311
Wave Optics: The Superposition of Waves
Standing Waves :
Lesson 2 of 2
Slide 8
Wiener’s Experiment
Weiner detected standing waves as follows:
1. Quasimonochromatic waves were sent
perpendicular to a mirror. Reflected back.
2.
𝜆
A 20
thin film was put at an angle so
that nodes and antinodes cross the film.
3. Energy in non-nodal positions on the
film caused darkening, so the film had
dark bands separated by non-dark lines.
4. Decreasing the angle of the film caused
the lines to be separated further apart.
5. This experiment also proved that the waves have nodes at the reflecting surface.
Phys
311
Wave Optics: The Superposition of Waves
Complex Waves :
is a wave resulting from the
superposition of waves having different frequencies.
Constituents: different 𝜔 , different 𝐸𝑜 , different 𝛼.
Resultant:
a complex wave. It is not harmonic (not a sine wave).
Constituents: same 𝜔 , different 𝐸𝑜 , same 𝛼.
Resultant:
NOT a complex wave. It is harmonic (sine wave).
Its amplitude depends on both amplitudes and on 𝛿.
When 𝛿 = 0  maximum amplitude.
Smallest amplitude = 𝐸𝑜1 − 𝐸𝑜2
Lesson 2 of 2
Slide 9
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 10
1
Constituents: 𝜔1 = 𝜔2 , same 𝐸𝑜 , same 𝛼.
2
Resultant:
A complex wave. It is not a sine wave.
Changing the phase of one wave with respect to the
other results in a totally different profile.
If these waves were of visible light, the eye will
sense a mixture of the two colors regardless of the
phase difference.
1
𝜔
2 2
Constituents: 𝜔1 =
, same 𝐸𝑜 , different 𝛼.
Resultant:
A complex wave. It is not a sine wave.
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 11
Constituents: 𝜔1 ≪ 𝜔2 , any 𝐸𝑜 , any 𝛼.
Resultant:
A complex wave.
Constituents: 𝜔1 ≅ 𝜔2 , any 𝐸𝑜 , any 𝛼.
Resultant:
A complex wave.  the phenomena of beats.
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 12
Complex waves and visible light:
The frequency of visible light determines its color.
Complex waves of light are produced when beams of light of different colors are used.
The “impure” colors, which are not found in the spectrum (e.g. brown)  have a
complex form.
“white light” is also a complex wave (in the form of beats or pulses)  it consists of
waves differing by only infinitesimal (i.e. very small) amounts.
Fourier analysis is used to decompose a complex wave into a number of simple ones.
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 13
Beats :
The beat’s wave equation of two
waves having the same amplitudes.
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 14
The beat’s wave equation of two
waves having the same amplitudes.
Time-varying
(i.e. modulated)
amplitude
average
The resultant wave has a wavelength = average
wavelength of the two constituents. But the
amplitude is modulated to form groups.
Beat frequency
modulation
angular frequency
propagation number
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 15
A beat resulting from the superposition of two waves with different frequencies and
different amplitudes:
Phys
311
Wave Optics: The Superposition of Waves
Group Velocity :
Lesson 2 of 2
Slide 16
Envelope moves with group velocity
Carrier moves with phase velocity
Any pulse of light can be viewed as a superposition of harmonic waves with different
frequencies.
The duration of the pulse is inversely proportional to the range of frequencies.
EM waves with different frequencies travel with different speeds through a given
medium (dispersion).
Phase velocity: is the velocity of the resultant harmonic wave that constitute the signal.
Group velocity: is the velocity at which the positions of maximal constructive
interference propagates.
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 17
Envelope moves with group velocity
Group Velocity :
Carrier moves with phase velocity
Because the velocities of the constituent waves are very similar; the resultant wave
has the same phase velocity as either of them:
𝜔
𝑣𝑝 =
𝑘
However, the group velocity is:
𝑑𝜔
𝑣𝑔 =
𝑑𝑘
The group velocity may be greater than, equal to, or less than the phase velocity.
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 18
Group Velocity :
The relationship between the group and phase velocities is:
𝑑𝑣𝑝
𝑣𝑔 = 𝑣𝑝 + 𝑘
𝑑𝑘
In terms of the wavelength:
𝑑𝑣𝑝
𝑣𝑔 = 𝑣𝑝 − 𝜆
𝑑𝜆
In a non-dispersive medium (vacuum): 𝑣𝑔 = 𝑣𝑝
The group velocity is the one important for light because it is the one that can be
observed experimentally.
Phys
311
Wave Optics: The Superposition of Waves
Addition of two waves perpendicular to each other :
We will take this when we study “Polarization”:
Pages to cover later on:
1. Jenkins: p.253 – 256.
2. Hecht: first pages of chapter 8.
3. Jenkins: parts of chapter 24.
Lesson 2 of 2
Slide 19
Phys
311
Lesson 2 of 2
Slide 20
Wave Optics: The Superposition of Waves
Homework :
Q1: Standing waves are produced by the superposition of two waves travelling in opposite
directions:
𝑡
𝐸1 = 7 sin 2𝜋(𝜏 −
2𝑥
)
𝜋
,
𝑡
𝐸2 = 7 sin 2𝜋(𝜏 +
2𝑥
)
𝜋
Find (a) the amplitude, (b) the wavelength, (c) the length of one antinode, (d) the velocity of
the waves, and (e) the period.
Q2: The phase velocity of a resultant wave in a light pulse in a certain medium is represented
by:
𝑣𝑝 = 𝐶1 + 𝐶2 𝜆
where 𝐶1 and 𝐶2 are constants. What is the velocity of the pulse?
Phys
311
Wave Optics: The Superposition of Waves
Homework :
Q3:
Lesson 2 of 2
Slide 21
Phys
311
Wave Optics: The Superposition of Waves
Lesson 2 of 2
Slide 22 (last)
Summary:
1. Standing waves (Continue : two waves of the
same frequency).
2. Weiner’s Experiment.
Next lesson will
cover:
3. Complex Waves (The addition of waves of
Interference (1)
different frequencies along the same direction).
4. Beats.
5. Group velocity.
6. Addition of waves at right angles.
Any Questions?
```