Announcements 10/17/11 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 Trumpet, revisited Gas-lit standing wave
Reading Quiz As discussed in the reading assignment, a
“beat” is:a. A periodic change in amplitude of a waveb. Interference between overtonesc. The first Fourier component of a waved. The reflection of a wave from a rigid
barriere. What the musical “Hairspray” says you
can’t stop
Beats Demo: Tuning forks; Spectrum lab software
“beat frequency”: fbeat = |f1 – f2|“beat period”
(or beat = |1 – 2| )
Beats, cont. Stokes Video (1:33)
http://stokes.byu.edu/beats_script_flash.html
Beats: Quick Math
cos cos 2cos cos2 2
a b a ba b
cos(30 ) cos(31 ) 2cos 30.5 cos 0.5t t t t
carrier “envelope” (beat)
Wait… is beat frequency 0.5 rad/s or is it 1 rad/s? (class poll)
Can be proved with trig identities
Sine WaveSine Wave
What is its wavelength?
What is its location?
What is its frequency?
When does it occur?
Animations courtesy of Dr. Durfee
Beats in TimeBeats in Time
What is its wavelength?
What is its location?
What is its frequency?
When does it occur?
Localization in Position/WavenumberLocalization in Position/Wavenumber
What is its wavelength?
What is its location?
What is its frequency?
When does it occur?
Beats in Both...Beats in Both...
Pure Sine WavePure Sine Wave
y=sin(5 x) Power Spectrum
““Shuttered” Sine WaveShuttered” Sine Wave
y=sin(5 x)*shutter(x) Power Spectrum
Uncertainty in x = ______ Uncertainty in k = ______
1
2x k In general: (and technically,
= std dev)
The equation that says xk ½ means that if you know the precise location of an electron you cannot know its momentum, and vice versa.
a. Trueb. False
Reading Quiz
Uncertainty Relationships Position & k-vector
Time &
Quantum Mechanics: momentum p = k
energy E =
1
2x k
1
2t
“” = “h bar” = Plank’s constant /(2)
2x p
2E t
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.
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.
“Fourier” transform The coefficients of the transform give
information about what frequencies are present
Example: a. my car stereob. my computer’s music playerc. your ear (so I’ve been told)
Fourier Transform
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 = 0k = 0.02: amplitude = 0……k = 0.90: amplitude = 1k = 0.91: amplitude = 1k = 0.92: amplitude = 1…”
Cos0.9 x Cos0.91 x Cos0.92 x
Cos0.93 x Cos0.94 x Cos0.95 x
Cos0.96 x Cos0.97 x Cos0.98 x
Cos0.99 x Cos1. x Cos1.01 x Cos1.02 x
Cos1.03 x Cos1.04 x Cos1.05 x Cos1.06 x
Cos1.07 x Cos1.08 x Cos1.09 x Cos1.1 x
600 400 200 200 400 600
20
10
10
20
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