Announcements 10/20/10

1. Announcements 10/20/10 • Prayer • Term project proposals due on Saturday night! Email to me: proposal in body of email, 650 word max. See website for guidelines, grading, ideas, and examples of past projects. • If in a partnership, just one email from the two of you • Exam 2 starts a week from tomorrow. • Exam 2 optional review session: vote on times by tomorrow evening. Survey link sent out this morning. • Anyone not get the Fourier transform handout last lecture?

2. Quick Writing • Ralph doesn’t understand what a transform is. As discussed last lecture and in today’s reading, how would you describe the “transform” of a function to him?

3. 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

4. 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)… compare 2p/T = w (or w0 ) • Durfee: • an and bn reversed • Uses l0 instead of L • The trick: finding the “Fourier coefficients”, an and bn

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: http://www.physics.byu.edu/faculty/colton/courses/phy123-fall10/lectures/lecture 22 - square wave Fourier.nb

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 integrates to (1/2)L (and m=n). (Same for sines.) • Recognize that sin(2pmx/L)cos(2pnx/L) always integrates to 0.

9. Sawtooth Wave, like HW 22-1 (The next few slides from Dr. Durfee)

10. The Spectrum of a Saw-tooth Wave

11. The Spectrum of a Saw-tooth Wave

12. Electronic “Low-pass filter” • “Low pass filter” = circuit which preferentially lets lower frequencies through. What comes out? ? Circuit • How to solve: • Decompose wave into Fourier series • Apply filter to each freq. individually • Add up results in infinite series again

13. Low-Pass Filter – before filter

14. Low-Pass Filter – after filter

15. Low Pass Filter

16. Actual Data from Oscilloscope