Physics E-1ax
April 14, 2015
Waves and Interference
For transverse waves, the disturbance is perpendicular to the direction of propagation.
Other waves are longitudinal: the disturbance is parallel to the direction of propagation.
For each kind of wave, the wave speed is a characteristic of the wave and the medium.
The most important examples are:
c
Electromagnetic waves:
n = index of refraction
v=
n
B = bulk modulus of fluid (compressibility)
B
Sound waves in a fluid:
v=
ρ = mass density
ρ
Waves carry energy as they travel. The rate at which energy travels is the power of the
wave, and the power per unit area is the intensity. All of these properties (energy,
power, and intensity) are proportional to the square of the amplitude of the wave. For a
spherical wave traveling in three dimensions, the intensity decreases in proportion to 1/r2,
where r is the distance from the source.
A sinusoidal wave also requires a phase, φ, so we have y(x,t) = Asin(kx − ωt + φ ) . In
particular, if the phase is equal to π, then the wave is “inverted” from its usual shape. If
two waves are in phase, they will combine favorably, with constructive interference. If
they are exactly out of phase (relative phase π), they will cancel each other out, with
destructive interference.
When a wave reaches a boundary, the reflected wave may have a phase shift. Going
from fast to slow, the reflected wave will be inverted (phase shift of π). Going from slow
to fast, the reflected wave will have no phase shift.
All waves exhibit interference. The most important question is: what is Δ φ ?
If Δφ = 0, ±2π, ±4π, ±6π, … then you have constructive interference.
If Δφ = ±π, ±3π, ±5π, … then you have destructive interference.
2 πΔs
.
λ
For two sources that are in phase (or two slits), with distances s1 and s2 from each source,
a distance d between the slits, and an angle θ from the normal:
Constructive interference occurs for s2 − s1 = mλ = d sin θ
If there is a difference in path length Δs, that contributes a phase difference Δφ =
Destructive interference occurs for s2 − s1 = ( m + 12 ) λ = d sin θ
When light from two slits travels a distance L and strikes a screen at a height y,
mλ L
Constructive interference occurs for y =
d
( m + 12 ) λ L
Destructive interference occurs for y =
d
When light strikes a thin film, such as a soap bubble or oil slick, the reflected light will
show thin-film interference, and different colors will be reflected strongly (or not).
Constructive or destructive interference depends on the usual rules for Δφ. Notably,
reflection off of an interface might result in an additional phase change of Δφ = π, if the
light is going from a region of low index n to an region of high index.
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Physics E-1ax
April 14, 2015
•
Learning objectives: After this lecture, you will be able to…
1.
Describe waves using the various parameters: wavelength, wavenumber, frequency,
angular frequency, period, amplitude, phase.
2.
Understand and use the equation for sinusoidal waves: y(x,t) = Asin(kx − ωt + φ ) .
3.
Determine the wave speed for different kinds of waves (light waves, sound waves, or
waves on a string).
4.
Describe how waves will propagate from a point source, and explain the relationship
between wavefronts and rays.
5.
Calculate the change in wave intensity as a wave propagates out from a source.
6.
Describe the role of phase in characterizing a wave.
7.
Explain how waves can combine (superposition), and describe how they can combine
with constructive or destructive interference.
8.
Describe how waves reflect from a boundary (either “fixed” or “free”)
9.
Describe how waves reflect from an interface where the wave speed changes.
10.
Determine the relative phase Δφ for two waves with a difference in path length.
11.
Identify what kinds of phase difference Δφ leads to constructive vs. destructive
interference.
12.
Use the conditions on Δφ in double-slit interference to find where bright or dark bands
will appear on a screen
13.
Use the pattern of bands on a screen from a double-slit experiment to determine the
wavelength of light
14.
Identify the phase changes that occur when light is reflected from an interface between
two different media
15.
Use the phase changes, along with path length, to find constructive and destructive
interference in thin films (soap, water, oil, etc.)
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Physics E-1ax
April 14, 2015
Activity 1: Waves
•
A wave is any kind of disturbance that travels through a medium. For instance, we can
describe a sinusoidal wave using the equation:
y(x,t) = Asin(kx − ωt)
What are the parameters in this equation? What is this wave doing?
y=
A=
k=
ω=
http://webphysics.davidson.edu/Applets/superposition/default.html
1.
You recall from our earlier discussion that the wave speed v = λf. What is the wave
speed in terms of the parameters ω and k?
2.
I’ll show you waves with the parameters:
A=1
How could you make the wave:
• Taller?
• Shorter?
• Twice the wavelength, but same speed?
• Same wavelength, but twice the speed?
• Move to the left with the same speed and wavelength?
• Same wavelength and speed, but twice the frequency?
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k=1
ω=1
Physics E-1ax
April 14, 2015
Examples of Waves
•
There are many wave phenomena! For each of the following waves, describe the wave,
and identify the characteristic wave speed for that type of wave:
Wave on a string
http://physics.bu.edu/~duffy/semester1/c20_trans_long.html
Electromagnetic Wave
http://www.surendranath.org/Applets/Waves/EMWave/EMWave.html
Sound wave
http://ralphmuehleisen.com/animations.html
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Physics E-1ax
April 14, 2015
Traveling Waves
•
When a wave travels, what is actually traveling with
the wave?
•
Many kinds of waves (light, sound)
can propagate out in three
dimensions. How will they propagate
from a small point source?
http://ralphmuehleisen.com/animations.html
•
For a spherical wave, we can identify wavefronts as well as rays.
•
From far away, a spherical wave will look like a plane wave:
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Physics E-1ax
April 14, 2015
Activity 2: Energy, Power, and Intensity
•
One very important property of waves is that they carry energy. If the wave disturbance
is given by the equation:
y(x,t) = Asin(kx − ωt)
how is that related to the energy carried by the wave?
1
To keep a wave going over time, the source must provide energy to the wave. The rate
of emission of energy is the power, or the energy emitted per unit time. What are the SI
units for power?
•
For a wave traveling in
three dimensions, the
most important
measure of energy is
usually the intensity
of the wave, which is
the power per unit
area. If you have a
spherical wave, the
wave spreads over a
sphere of increasing
area as it propagates.
2.
What is the intensity of a spherical wave at a distance r from a source with power W, if
the spherical wave spreads uniformly in all directions?
Bonus! You drop a stone in a pond. The stone creates circular waves that spread uniformly in all
directions. If the power of the source is W, find an expression for the intensity of the
waves at a distance r from the source.
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Physics E-1ax
April 14, 2015
Activity 3: Waves and Phase
•
We left out an important aspect of sinusoidal waves earlier. The most general expression
for a sinusoidal wave requires a phase, φ: Below is a graph at t = 0 with a phase of zero:
y(x,t) = Asin(kx − ωt + φ )
1.
On the graph above, sketch what the wave would look like with a small, positive phase
(i.e. if φ is a small positive number).
2.
For each of the following graphs, identify the phase. The top graph has a phase of zero:
Bonus! We could also write a wave as y = A cos(kx – ωt + φ). Why don’t we bother to do that?
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Physics E-1ax
April 14, 2015
Superposition and Interference
•
When waves from several sources combine in a medium, the resulting wave disturbances
simply add together. This is the principle of superposition. For instance, what would
happen if two wave pulses approach each other on a string?
•
When waves combine, you can have interference that can either enhance or diminish the
resulting wave amplitude. For interference, the relative phase is important!
If two waves are exactly in phase
(i.e. the relative phase is zero):
If two waves are exactly out of phase
(i.,e. the relative phase is π ):
http://webphysics.davidson.edu/Applets/superposition/default.html
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Physics E-1ax
April 14, 2015
Am I getting it?
1.
A wave travels to the right (towards +x)
along a stretched string with speed
v = 10 cm/s. A graph of the
displacement D(x, t) at time t = 0 is
shown in the figure at right.
Which of the following is the correct equation for D for all x and t?
⎛⎛ 0.5 ⎞ ⎛ 5.0 ⎞ ⎞
A. D(x,t) = (4.0 cm)sin⎜⎜ ⎟ x + ⎜ ⎟t ⎟
⎝⎝ cm ⎠ ⎝ s ⎠ ⎠
⎛⎛ 0.5 ⎞ ⎛ 5.0 ⎞ ⎞
B. D(x,t) = (4.0 cm)sin⎜⎜ ⎟ x − ⎜ ⎟t ⎟
⎝⎝ cm ⎠ ⎝ s ⎠ ⎠
⎛⎛ 0.5 ⎞ ⎛ 5.0 ⎞ ⎞
C. D(x,t) = (4.0 cm)cos⎜⎜ ⎟ x + ⎜ ⎟t ⎟
⎝⎝ cm ⎠ ⎝ s ⎠ ⎠
⎛⎛ 0.5 ⎞ ⎛ 5.0 ⎞ ⎞
D. D(x,t) = (4.0 cm)cos⎜⎜ ⎟ x − ⎜ ⎟t ⎟
⎝⎝ cm ⎠ ⎝ s ⎠ ⎠
⎛⎛ 3.1⎞ ⎛ 31⎞ ⎞
E. D(x,t) = (4.0 cm)cos⎜⎜ ⎟ x + ⎜ ⎟t ⎟
⎝⎝ cm ⎠ ⎝ s ⎠ ⎠
⎛⎛ 3.1⎞ ⎛ 31⎞ ⎞
F. D(x,t) = (4.0 cm)cos⎜⎜ ⎟ x − ⎜ ⎟t ⎟
⎝⎝ cm ⎠ ⎝ s ⎠ ⎠
2.
Here is a Logger Pro microphone reading of the sound pressure from a tuning fork as a
function of time. Estimate the frequency of the tuning fork in Hz.
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Physics E-1ax
April 14, 2015
Phase Changes upon Reflection
•
When a wave reaches a boundary or interface, all or part of the wave can be reflected.
For a wave on a string, the reflection will be different depending on whether the end of
the string is fixed in place, or free to move up and down.
Wave reflection at fixed end:
Wave reflection at free end:
http://physics.bu.edu/~duffy/semester1/c21_int_reflections.html
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Physics E-1ax
April 14, 2015
Activity 4: Phase Changes at a Boundary
1.
When a wave reaches a boundary where the wave speed changes, there is once again
both a reflected part of the wave and a transmitted part of the wave. Where have we seen
before this phenomenon of both reflection and transmission at a boundary?
2.
The phase of the reflected wave depends on the nature of the boundary. If the wave is
traveling to a medium with a slower wave speed, then the slower medium looks a little bit
like a string hitting a fixed end… so what will be the phase of the reflected wave? An
example is a wave traveling from a light string to a heavy string:
http://physics.bu.edu/~duffy/semester1/c21_int_reflections.html
3.
What about the opposite case: can you predict what will happen if you go from a heavy
string to a light string? In the extreme limit, the light string is like having a totally free
end…
•
As we will see later, these phase changes at a boundary have real applications!
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Physics E-1ax
April 14, 2015
Activity 5: Interference: What is the Relative Phase?
•
The most important question to ask about interference is: what is the relative phase
between the waves? We’ll make this a game:
What is Δ φ ?
1.
What values of Δφ will give constructive interference?
2.
What values of Δφ will give destructive interference?
3.
Suppose there is a difference in distance traveled by one wave (compared with another).
If one wave travels an extra distance Δs, what is the difference in phase Δφ? (Hint: Go
back to where we first introduced the concept of phase. What distance is the wave
displaced when its phase is changed by φ?)
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Physical
Physics Sciences
E-1ax 3
April
April12,
14,2012
2015
Interference:
TwoSources,
Sources,InInPhase
Phase
Activity 6: Two
••
What
Whatwill
willwe
wehear
hearfrom
fromtwo
twoloudspeakers
loudspeakersthat
thatare
areproducing
producingthe
thesame
sametone,
tone,ininphase?
phase?
••
What
is thea criterion
vs. destructive
Consider
point thatfor
is constructive
a distance s1 from
one speakerinterference
and s2 fromfor two sources?
What
is Δspeaker.
φ?
the other
1.
What is the difference in path length, Δs?
2.
What should Δs be in order to have constructive interference?
(Hint: Start with the condition Δφ = …)
3.
What should Δs be in order to have destructive interference?
313
Physics E-1ax
April 14, 2015
Constructive vs. Destructive Interference
•
We can do the same thing with light, by shining light through two slits. What do we see?
•
Does it look like light is made up of particles or waves?
•
Let’s see if we can understand the light and dark bands on the screen. What is Δ φ ?
www.falstad.com/ripple
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Physics E-1ax
April 14, 2015
Activity 7: Two Slits: Pattern on the Screen
1.
As shown in the diagram, the difference in path length is Δs = d sin θ. Find an expression
for the angles at which we will find constructive or destructive interference. (You may
assume that θ is small, in which case sin θ ≈ θ.)
2.
Now determine the height y of the bright and dark fringes on the screen. Again, use the
approximation, valid for small θ, that sin θ ≈ tan θ ≈ θ.
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Physics E-1ax
April 14, 2015
Activity 8: Measuring the Wavelength
1.
Light strikes two slits 0.1 mm apart. The pattern on the screen (1.2 m away) shows
constructive interference at 6 mm and 12 mm from the center. What is the wavelength of
the light?
2.
If you make the two slits farther apart, what will happen to the pattern on the screen?
•
Now let’s use what we’ve learned to understand why we see colors in a soap bubble or in
an oil slick on wet pavement…
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Physics E-1ax
April 14, 2015
Thin-Film Interference
•
Let’s consider the soap bubble first. Remember: What is Δ φ ? We need to consider the
phase changes upon reflection and the phase change that accompanies the longer path:
The phase difference between the two reflected rays is:
and we will get constructive interference when:
•
For the oil slick, the results are slightly different:
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Physics E-1ax
April 14, 2015
Am I getting it?
1.
The light from two lasers (A and B) are sent through the same double-slit apparatus.
Two interference patterns are projected on a screen a distance L away. The bright fringes
from laser A are spaced more closely together than the bright fringes from laser B. What
can we say, if anything, about the relative wavelengths of the two lasers?
a) Laser A has a shorter wavelength than laser B.
b) Laser A has the same wavelength as laser B.
c) Laser A has a longer wavelength than laser B.
d) We can’t compare the wavelengths without the slit separation d.
e) We can’t compare the wavelengths without the slit width D.
2.
The graph at right shows the intensity of light as a
function of position on the screen in a two-slit
interference setup. Fill in the blank: At the secondorder maximum on either side, the light from one
slit travels ______ the light from the other slit.
a) the same distance as
b) twice as far as
c) one wavelength further than
d) two wavelengths further than
e) three wavelengths further than
3.
White light reflects at normal incidence off of a very thin soap film with air on either side
of it. If the thickness of the film is much less than the wavelengths of visible light, which
of the following will occur?
a) Constructive interference will be observed for all wavelengths of visible light, so the
film will appear white.
b) Destructive interference will be observed for all wavelengths of visible light, so the
film will appear dark.
c) Constructive interference will be observed for short wavelengths, but destructive
interference for long wavelengths, so the film will appear blue.
d) Constructive interference will be observed for long wavelengths, but destructive
interference for short wavelengths, so the film will appear red.
e) The film will appear dark, but because of total internal reflection, not because of
destructive interference.
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Physics E-1ax
April 14, 2015
One-Minute Paper
Your name: _________________________________
Names of your group members:
_________________________________
_________________________________
•
Please tell us any questions that came up for you today during lecture. Write “nothing”
if no questions(s) came up for you between 6–9pm (or while viewing it online).
•
What single topic left you most confused after today’s class?
•
Any other comments or reflections on today’s class?
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