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 ( ed 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! a. a.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

Reading Quiz In the Fourier transform of a periodic function, which frequency components will be present? a. a.Just the fundamental frequency, f 0 = 1/period b. b.f 0 and potentially all integer multiples of f 0 c. c.A finite number of discrete frequencies centered on f 0 d. d.An infinite number of frequencies near f 0, spaced infinitely close together e. e.1320 KFAN (1320 kHz), home of the Utah Jazz… if there’s a season 

Fourier Theorem Any function periodic on a distance L can be written as a sum of sines and cosines like this: Notation issues: a. a.a 0, a n, b n = how “much” at that frequency b. b.Time vs distance c. c.a 0 vs a 0 /2 d. d.2  /L = k (or k 0 )2  /T =  (or  0 ) e. e.2  n /L = n  fundamental The trick: finding the “Fourier coefficients”, a n and b n

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

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

Example: square wave f(x) = 1, from 0 to L/2 f(x) = -1, from L/2 to L (then repeats) a 0 = ? a n = ? b 1 = ? b 2 = ? b n = ? 0 0 4/  Could work out each b n individually, but why? 4/(n  ), only odd terms

Square wave, cont. Plots with Mathematica:

Deriving the coefficient equations To derive equation for a 0, just integrate LHS and RHS from 0 to L. To derive equation for a n, multiply LHS and RHS by cos(2  mx/L), then integrate from 0 to L. (To derive equation for b n, multiply LHS and RHS by sin(2  mx/L), then integrate from 0 to L.) Recognize that when n and m are different, cos(2  mx/L)  cos(2  nx/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(2  mx/L)  cos(2  nx/L) always integrates to 0.

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