Announcements 10/18/10
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Prayer
Found: Physics phor Phynatics book, still unclaimed
Term project proposals due on Saturday night! Email to
me: proposal in body of email, 650 word max. See website
for guidelines, grading, ideas, and examples of past
projects.
Resonator boxes and the Beatles
Flame standing wave video from website
http://www.physics.byu.edu/faculty/colton/courses/phy123-fall10/
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Colton “Fourier series summary” handout
Demo: PVC pipe vs. “Spectrum Lab”
Beats
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Demo: Tuning forks; Spectrum lab software
“beat period”
“beat frequency”: fbeat = |f1 – f2|
(or wbeat = |w1 – w2| )
Beats, cont.
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Video:
http://stokes.byu.edu/beats_script_flash.html
Beats: Quick Math
 a b   a b 
cos a  cos b  2cos 
cos 


2
2

 
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Can be proved
with trig identities
cos(30t )  cos(31t )  2cos 30.5t  cos  0.5t 
carrier
“envelope” (beat)
Wait… is beat
frequency 0.5 rad/s
or is it 1 rad/s?
(class poll)
Review: 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)
Review: How did I create this?
Still mesmerizing… if someone wants a
few extra credit points you could post it
to Wikipedia’s group and/or phase
velocity articles as an example of group
& phase velocities being in opposite
directions.
Cos 1.23457 t
0.9 x
Cos 1.20758 t
0.91 x
Cos 1.18147 t
0.92 x
Cos 1.13173 t
0.94 x
Cos 1.10803 t
0.95 x
Cos 1.08507 t
Cos 1.06281 t
0.97 x
Cos 1.04123 t
0.98 x
Cos 1.0203 t
Cos 1.1562 t
0.96 x
0.99 x
Cos 1. t
Cos 0.980296 t
1.01 x
Cos 0.961169 t
1.02 x
Cos 0.942596 t
1.03 x
Cos 0.924556 t
1.04 x
Cos 0.907029 t
1.05 x
Cos 0.889996 t
1.06 x
Cos 0.873439 t
1.07 x
Cos 0.857339 t
1.08 x
Cos 0.84168 t
1.09 x
10
1500
1000
500
500
10
20
1000
1. x
Cos 0.826446 t
20
What I didn’t show you:
(zoomed out)
0.93 x
1500
1.1 x
Sine Wave
What is its wavelength?
What is its frequency?
What is its location?
When does it occur?
Animations courtesy of Dr. Durfee
Beats in Time
What is its wavelength?
What is its frequency?
What is its location?
When does it occur?
Localization in Position/Wavenumber
What is its wavelength?
What is its frequency?
What is its location?
When does it occur?
Beats in Both...
Pure Sine Wave
y=sin(5 x)
Power Spectrum
“Shuttered” Sine Wave
y=sin(5 x)*shutter(x)
Uncertainty in x = ______
In general:
Power Spectrum
Uncertainty in k = ______
1
xk 
2
(and technically,
 = std dev)
Uncertainty Relationships
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Position & k-vector
1
xk 
2
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Time & w
1
t w 
2
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Quantum Mechanics:
momentum p = k
xp 
“” = “h bar”
= Plank’s constant /(2p)
energy E = w
Et 
2
2
What’s a “transform”?
A one-to-one correspondence between one function and
another (or between a function and a set of numbers).
a. If you know one, you can find the other.
b. Why? One representation might give you more insight
into the function than the other.
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Example: ex = 1 + x + x2/2! + x3/3! + x4/4! + …
a. If you know the function (ex), you can find the Taylor’s
series coefficients.
b. If you have the Taylor’s series coefficients (1, 1, 1/2!,
1/3!, 1/4!, …), you can re-create the function. The first
number tells you how much of the x0 term there is, the
second tells you how much of the x1 term there is, and
so forth.
c. Why use a Taylor’s series? Sometimes it’s useful.
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“Fourier” transform
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The coefficients of the transform give information
about what frequencies are present
Example:
a. my car stereo
b. my computer’s music player
Fourier Transform
20
10
600
400
200
200
400
Do the transform (or
have a computer do it)
600
10
Cos 0.9 x
20
Cos 0.91 x
Cos 0.92 x
Cos 0.93 x
Cos 0.94 x
Cos 0.95 x
Cos 0.96 x
Cos 0.97 x
Cos 0.98 x
Cos 0.99 x
Cos 1. x
Cos 1.03 x
Cos 1.04 x
Cos 1.05 x
Cos 1.06 x
Cos 1.07 x
Cos 1.08 x
Cos 1.09 x
Cos 1.1 x
Cos 1.01 x
Cos 1.02 x
Answer from computer: “There
are several components at
different values of k; all are
multiples of k=0.01.
k = 0.01: amplitude = 0
k = 0.02: amplitude = 0
…
…
k = 0.90: amplitude = 1
k = 0.91: amplitude = 1
k = 0.92: amplitude = 1
…”