Announcements 10/17/11 Prayer Saturday: Term project proposals, one proposal per group… but please CC your partner on the . 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.A periodic change in amplitude of a wave b. b.Interference between overtones c. c.The first Fourier component of a wave d. d.The reflection of a wave from a rigid barrier e. e.What the musical “Hairspray” says you can’t stop

Beats Demo: Tuning forks; Spectrum lab software “beat frequency”: f beat = |f 1 – f 2 | “beat period” (or  beat = |  1 –  2 | )

Beats, cont. Stokes Video (1:33)

Beats: Quick Math 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 Wave What is its wavelength? What is its location? What is its frequency? When does it occur? Animations courtesy of Dr. Durfee

Beats in Time What is its wavelength? What is its location? What is its frequency? When does it occur?

Localization in Position/Wavenumber What is its wavelength? What is its location? What is its frequency? 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) Power Spectrum Uncertainty in x = ______ Uncertainty in k = ______ In general: (and technically,  = std dev)

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. a.True b. b.False Reading Quiz

Uncertainty Relationships Position & k-vector Time &  Quantum Mechanics: momentum p =  k energy E =   “  ” = “h bar” = Plank’s constant /(2  )

Transforms A one-to-one correspondence between one function and another function (or between a function and a set of numbers). a. a.If you know one, you can find the other. b. b.The two can provide complementary info. Example: e x = 1 + x + x 2 /2! + x 3 /3! + x 4 /4! + … a. a.If you know the function (e x ), you can find the Taylor’s series coefficients. b. 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 x 0 term there is, the second tells you how much of the x 1 term there is, etc. c. 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. a.my car stereo b. b.my computer’s music player c. c.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 = 0 k = 0.02: amplitude = 0 … k = 0.90: amplitude = 1 k = 0.91: amplitude = 1 k = 0.92: amplitude = 1 …”