Announcements 10/7/11
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Prayer
We’re likely not going to finish dispersion today, so
you might want to wait until after Monday before
doing Lab 3 (Dispersion).
I just got the exams from the Testing Center, TA & I
will work on grading them today & this weekend.
Non
Sequitur
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 in reflected or
transmitted waves?
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
Quick Writing
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We saw that A1cos(kx
+ 1) + A2cos(kx + 2)
gives you a cosine
wave with the same k,
and hence wavelength.
If you add a third,
fourth, fifth, etc., such
cosine wave, you still
get a simple cosine
wave. See 
How can you then add
together cosine waves
to get more
complicated shapes?
Or can you?
Wave packets
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HW 17-5
Wave packets, cont.
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Results:
a. To localize a wave in space, you need lots of spatial
frequencies (k values)
b. To remove neighboring localized waves (i.e. to make it
non-periodic), you need those frequencies to spaced
close to each other. (infinitely close, really)
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 transform” lectures