Announcements 10/8/10
Prayer
Exam: last day = tomorrow!
a. Correction to syllabus: on Saturdays, the Testing
Center gives out last exam at 3 pm, closes at 4 pm.
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Homework problem changes: some extra credit, one
moved. See email.
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Lab 3 starting tomorrow: it’s a computer simulation. See
class website.
a. If we don’t finish the relevant discussion today (which
seems likely), you probably should wait until after
Monday’s class before starting the lab.
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Quick writing assignment while you wait: Ralph is still not
quite grasping this… he asks you, “How are complex
exponentials related to waves on a string?” What should
you tell him to help him understand? (Please actually
write down your answer.)
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Thought Question
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Which of these are the same?
(1) A cos(kx – wt)
(2) A cos(kx + wt)
(3) A cos(–kx – wt)
a.
b.
c.
d.
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(1) and (2)
(1) and (3)
(2) and (3)
(1), (2), and (3)
Which should we use for a left-moving wave: (2) or (3)?
a. Convention: Usually use #3, Aei(-kx-wt)
b. Reasons: (1) All terms will have same e-iwt factor. (2)
The sign of the number multiplying x then indicates
the direction the wave is traveling.
k  k iˆ
Reading Quiz
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Which of the following was not a major topic of
the reading assignment?
a. Dispersion
b. Fourier transforms
c. Reflection
d. Transmission
Reflection/transmission at boundaries: The setup
x=0
Region 1: light string
in-going wave
reflected wave
i ( k1xw1t )
transmitted wave
AI e
i (  k1xw1t )
ARe
i ( k1xw1t )
f1  AI e
Region 2: heavier string
Goal: How much of wave is
transmitted and reflected?
(assume k’s and w’s are known)
i ( k1xw1t )
 ARe
f1  AI cos(k1x  w1t  I )  AR cos(k1x  w1t  R )

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AT ei ( k2 xw2t )
i ( k2 xw2t )
f2  AT e
f 2  AT cos(k2 x  w2t  T )
Why are k and w the same for I and R? (both labeled k1 and w1)
“The Rules” (aka “boundary conditions”)
a. At boundary: f1 = f2
b. At boundary: df1/dx = df2/dx
Boundaries: The math
x=0
i ( k1xw1t )
f1  AI e
B.C.1:
i ( k1xw1t )
i ( k2 xw2t )
 ARe
f1 x0  f 2
f2  AT e
Goal: How much of wave is
transmitted and reflected?
x0
AI ei (k1 0w1t )  ARei (k1 0w1t )  AT ei (k2 0w2t )
AI e
iw1t
 ARe
AI  AR  AT
iw1t
 AT e
and
iw2t
w1  w2
Boundaries: The math
x=0
i ( k1xwt )
f1  AI e
B.C.2:
df1
dx
x 0
i ( k1xwt )
 ARe
df 2

dx
ik1 AI e
f2  AT e
Goal: How much of wave is
transmitted and reflected?
x 0
ik1 AI ei ( k1xwt )  ik1 AR ei (  k1xwt )
iwt
i ( k2 xwt )
 ik1 ARe
x 0
iwt
k1 AI  k1 AR  k2 AT
 ik2 AT ei ( k2 xwt )
 ik2 AT e
iwt
x 0
Boundaries: The math
x=0
Goal: How much of wave is
transmitted and reflected?
AI  AR  AT
k1 AI  k1 AR  k2 AT
2 equations, 3 unknowns??
x y  z
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Like:
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How do you solve?
and
3x  3 y  5z
Can’t get x, y, or z, but can get ratios!
y = -0.25 x
z = 0.75 x
Boundaries: The results
x=0
The results….
Goal: How much of wave is
transmitted and reflected?
AR k1  k2
r

AI k1  k2
“reflection coefficient”
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AT
2k1
t

AI k1  k2
“transmission coefficient”
Recall v = w/k, and w is the same for region 1 and region 2.
So k ~ 1/v
Can write results like this:
AR v2  v1
r

AI v1  v2
AT
2v2
t

AI v1  v2
Special Cases
x=0
The results….
AR v2  v1
r

AI v1  v2
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AT
2v2
t

AI v1  v2
Do we ever have a phase shift?
a. If so, when? And what is it?
What if v2 = 0?
a. When would that occur?
What if v2 = v1?
a. When would that occur?
Power
x=0
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1
Recall: P  mw 2 A2v
2
Region 1: m and v are same
… so P ~ A2
PR
R
 r2
PI
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(A = amplitude)
r,t = ratio of amplitudes
R,T = ratio of power/energy
Region 2: m and v are different… more complicated
…but energy is conserved, so easy way is:
PT
T
 1 r2
PI
Dispersion
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A dispersive medium: velocity is different for different
frequencies
a. Any real-world examples?
Why do we care?
a. Real waves are often not shaped like sine waves.
– Non sine-wave shapes are made up of combinations of
sine waves at different frequencies.
b. Real waves are not infinite in space or in time.
– Finite waves are also made up of combinations of sine
waves at different frequencies.
Focus on (b) for now…
(a) is the main topic of the
“Fourier” lectures of next week.
Wave packets
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Adding cosines together with Mathematica, “sum of cosines.nb”
http://www.physics.byu.edu/faculty/colton/courses/phy123fall10/lectures/lecture%2017%20-%20sum%20of%20cosines.nb
What did we learn?
a. To localize a wave in space, you need lots of frequencies
b. To remove neighboring localized waves, you need those
frequencies to spaced close to each other. (infinitely close,
really)
Back to Dispersion
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What happens if a wave pulse is sent through a
dispersive medium? Nondispersive?
Dispersive wave example:
a. s(x,t) = cos(x-4t) + cos(2 (x-5t))
– What is “v”?
– What is v for w=4? What is v for w=10?
What does that wave look like as time progresses?
(Mathematica “dispersion of two cosines.nb”; on
website. And next slide.)
Mathematica
0.1 seconds
0.7 seconds
1.3 seconds
Femtosecond Laser Pulse
Credit: Dr. Durfee
Et=0=sin(10 x)*exp(-x^2)
Power Spectrum
Note:
frequencies
are infinitely
close together
Initial shape of wave
How much energy is
contained in each
frequency component
(w = vk)
Propagation Of Light Pulse
Credit: Dr. Durfee
E(x,t)
Wave moving in time
Power Spectrum
How much energy is
contained in each
frequency component
Tracking a Moving Pulse
Credit: Dr. Durfee
E(x+vt,t)
Graph “window” is moving
along with speed v
Power Spectrum
How much energy is
contained in each
frequency component
Laser Pulse in Dispersive Medium
Credit: Dr. Durfee
Et=0 = sin(10 x)*exp(-x^2)
Initial shape of wave
Not all frequency
components travel at
same speed
Power Spectrum
How much energy is
contained in each
frequency component
Time Evolution of Dispersive Pulse
Credit: Dr. Durfee
Peak
moves at
about 13
m/s (on
my office
computer)
Wave moving in time
How much energy is
contained in each
frequency component
Tracking a Dispersive Pulse
Credit: Dr. Durfee
E(x+vgt,t)
Graph window moving
along with peak, ~13 m/s