Announcements 10/18/10 Prayer Found: Physics phor Phynatics book, still unclaimed Term project proposals due on Saturday night! to me: proposal in body of , 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 Colton “Fourier series summary” handout Demo: PVC pipe vs. “Spectrum Lab”

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

Beats, cont. Video:

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

Review: Wave packets Adding cosines together with Mathematica, “sum of cosines.nb” fall10/lectures/lecture%2017%20-%20sum%20of%20cosines.nb What did we learn? a. a.To localize a wave in space, you need lots of frequencies b. 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? What I didn’t show you: (zoomed out) 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.

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)

Uncertainty Relationships Position & k-vector Time &  Quantum Mechanics: momentum p =  k energy E =   “  ” = “h bar” = Plank’s constant /(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. a.If you know one, you can find the other. b. b.Why? One representation might give you more insight into the function than the other. 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, and so forth. c. c.Why use a Taylor’s series? Sometimes it’s 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

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 …”