1. Announcements 2/23/11 Prayer Deron Williams just got traded to the Nets (what???) Due Saturday night: a. a.Labs 4-5 b. b.First extra credit papers - Can do each type of paper once in first half of semester, once in second half c. 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!!

2. Demos Trumpet Gas-lit standing wave

3. 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

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

5. Beats, cont. Video: http://stokes.byu.edu/beats_script_flash.html http://stokes.byu.edu/beats_script_flash.html

6. 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

7. Review: Wave packets HW 17-5

8. Review of wave packets, cont. 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)

9. Review: How did I create this? What I didn’t show you: (zoomed out) Still mesmerizing… and extra credit still up for grabs

10. Sine Wave What is its wavelength? What is its location? What is its frequency? When does it occur? Animations courtesy of Dr. Durfee

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

12. Localization in Position/Wavenumber What is its wavelength? What is its location? What is its frequency? When does it occur?

13. Beats in Both...

14. Pure Sine Wave y=sin(5 x) Power Spectrum

15. “Shuttered” Sine Wave y=sin(5 x)*shutter(x) Power Spectrum Uncertainty in x = ______ Uncertainty in k = ______ In general: (and technically,  = std dev)

16. 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. a.True b. b.False Reading Quiz

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

18. 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. 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.

19. “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)

20. 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 …”