Announcements 10/17/11
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Prayer
Saturday: Term project proposals, one proposal per group…
but please CC your partner on the email. See website for
guidelines, grading, ideas, examples.
Chris: not here on Friday for office hours
Colton “Fourier series summary” handout. Notation warning!
xkcd
Demos
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Trumpet, revisited
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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Stokes Video (1:33)
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)
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 equation that says xk  ½ means that if
you know the precise location of an electron you
cannot know its momentum, and vice versa.
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 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
600
10
20
Cos 0.9 x
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
Do the transform (or
have a computer do it)
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
…”