Announcements 2/23/11
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Prayer
Deron Williams just got traded to the Nets (what???)
Due Saturday night:
a. Labs 4-5
b. First extra credit papers - Can do each type of paper
once in first half of semester, once in second half
c. Term project proposals
– Email to me: proposal in body of email, 650 word max.
– One proposal per group… but please CC your
partner(s) on email.
– See website for guidelines, grading, ideas, and
examples of past projects.
Thanks to those who filled out the mid-semester survey
Colton “Fourier series summary” handout. Caution about
notation!!
Demos
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Trumpet
Gas-lit standing wave
Reading Quiz
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As discussed in the reading assignment, a “beat”
is:
a. A periodic change in amplitude of a wave
b. Interference between overtones
c. The first Fourier component of a wave
d. The reflection of a wave from a rigid barrier
e. What the musical “Hairspray” says you can’t
stop
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

 

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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HW 17-5
Review of wave packets, cont.
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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… and extra credit still
up for grabs
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)
Reading Quiz
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The “Uncertainty Principle” from quantum
mechanics (if you know exactly where a particle
is, you can’t know exactly what its momentum is,
and vice versa) is a result of the xk equation
just discussed.
a. True
b. False
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
Transforms
A “transform” is: a one-to-one correspondence between
one function and another function (or between a function
and a set of numbers).
a. If you know one, you can find the other.
b. The two can provide complementary info.
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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, etc.
c. Why Taylor’s series? Sometimes they are 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
c. your ear (so I’ve been told)
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
…”