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Announcements 10/19/11

Announcements 10/19/11. Prayer Chris: today: 3-5 pm, Fri: no office hours Labs 4-5 due Saturday night Term project proposals due Sat night (emailed to me) One proposal per group; CC your partner(s) See website for guidelines, grading, ideas, and examples of past projects.

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Announcements 10/19/11

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  1. Announcements 10/19/11 • Prayer • Chris: today: 3-5 pm, Fri: no office hours • Labs 4-5 due Saturday night • Term project proposals due Sat night (emailed to me) • One proposal per group; CC your partner(s) • See website for guidelines, grading, ideas, and examples of past projects. • HW 22 due MONDAY instead of Friday. (HW 23 also due Monday) • We’re half-way done with semester! • Exam 2 starts a week from tomorrow! • Review session: either Monday, Tues, or Wed. Please vote by tomorrow night so I can schedule the room on Friday. • Anyone need my “Fourier series summary” handout? Pearls Before Swine

  2. Reading Quiz • In the Fourier transform of a periodic function, which frequency components will be present? • Just the fundamental frequency, f0 = 1/period • f0 and potentially all integer multiples of f0 • A finite number of discrete frequencies centered on f0 • An infinite number of frequencies near f0, spaced infinitely close together • 1320 KFAN (1320 kHz), home of the Utah Jazz… if there’s a season 

  3. Fourier Theorem • Any function periodic on a distance L can be written as a sum of sines and cosines like this: • Notation issues: • a0, an, bn = how “much” at that frequency • Time vs distance • a0 vs a0/2 • 2p/L = k (or k0) 2p/T = w (or w0 ) • 2pn/L = nfundamental • The trick: finding the “Fourier coefficients”, an and bn

  4. Applications (a short list) • “What are some applications of Fourier transforms?” • Electronics: circuit response to non-sinusoidal signals • Data compression (as mentioned in PpP) • Acoustics: guitar string vibrations (PpP, next lecture) • Acoustics: sound wave propagation through dispersive medium • Optics: spreading out of pulsed laser in dispersive medium • Optics: frequency components of pulsed laser can excite electrons into otherwise forbidden energy levels • Quantum: wavefunction of an electron in “particle in a box” situations, aka “infinite square well”

  5. How to find the coefficients • What does mean? • What does mean? Let’s wait a minute for derivation.

  6. Example: square wave • f(x) = 1, from 0 to L/2 • f(x) = -1, from L/2 to L (then repeats) • a0 = ? • an = ? • b1 = ? • b2 = ? • bn = ? 0 0 4/p Could work out each bn individually, but why? 4/(np), only odd terms

  7. Square wave, cont. • Plots with Mathematica:

  8. Deriving the coefficient equations • To derive equation for a0, just integrate LHS and RHS from 0 to L. • To derive equation for an, multiply LHS and RHS by cos(2pmx/L), then integrate from 0 to L. (To derive equation for bn, multiply LHS and RHS by sin(2pmx/L), then integrate from 0 to L.) • Recognize that when n and m are different, cos(2pmx/L)cos(2pnx/L) integrates to 0. (Same for sines.) Graphical “proof” with Mathematica Otherwise, if m=n, then integrates to (1/2)L (Same for sines.) • Recognize that sin(2pmx/L)cos(2pnx/L) always integrates to 0.

  9. Sawtooth Wave, like HW 22-2 (The next few slides from Dr. Durfee)

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