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Announcements 10/17/11

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.

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Announcements 10/17/11

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  1. 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

  2. Demos • Trumpet, revisited • Gas-lit standing wave

  3. Reading Quiz • As discussed in the reading assignment, a “beat” is: • A periodic change in amplitude of a wave • Interference between overtones • The first Fourier component of a wave • The reflection of a wave from a rigid barrier • What the musical “Hairspray” says you can’t stop

  4. Beats • Demo: Tuning forks; Spectrum lab software “beat frequency”: fbeat = |f1 – f2| “beat period” (or wbeat = |w1 – w2| )

  5. Beats, cont. • Stokes Video (1:33) http://stokes.byu.edu/beats_script_flash.html

  6. Beats: Quick Math Can be proved with trig identities carrier “envelope” (beat) Wait… is beat frequency 0.5 rad/s or is it 1 rad/s? (class poll)

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

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

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

  10. Beats in Both...

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

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

  13. Reading Quiz • The equation that says DxDk  ½ means that if you know the precise location of an electron you cannot know its momentum, and vice versa. • True • False

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

  15. Transforms • A one-to-one correspondence between one function and another function (or between a function and a set of numbers). • If you know one, you can find the other. • The two can provide complementary info. • Example: ex = 1 + x + x2/2! + x3/3! + x4/4! + … • If you know the function (ex), you can find the Taylor’s series coefficients. • 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. • Why Taylor’s series? Sometimes they are useful.

  16. “Fourier” transform • The coefficients of the transform give information about what frequencies are present • Example: • my car stereo • my computer’s music player • your ear (so I’ve been told)

  17. Fourier Transform 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 …” Do the transform (or have a computer do it)

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