Announcements 2/28/11

1. Announcements 2/28/11 • Prayer • My office hours this week: I’ll likely be in my lab, room U130, just down the hall from normal office hour location. Find me there. • Exam 2 starts on Saturday • Exam review session, results of voting: • Friday 3:30 – 5 pm. Room: C261 • Next week: I’ll be out of town on Mon. You’ll have Dr. Gus Hart as a substitute.

2. Summary of last time The series How to find the coefficients

3. Fourier Transform (review) 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 …” How does computer know all components will be multiples of k=0.01?

4. Periodic? • “Any function periodic on a distance L can be written as a sum of sines and cosines like this:” • What about nonperiodic functions? • “Fourier series” vs. “Fourier transform” • Special case: functions with finite domain

5. HW 23-1 • “Find y(x) as a sum of the harmonic modes of the string” • Why?  Because you know how the string behaves for each harmonic—for fundamental mode, for example: y = Asin(px/L)cos(w1t) --standing wave  Asin(px/L) is the initial shape  It oscillates sinusoidally in time at frequency w1  If you can predict how each frequency component will behave, you can predict the overall behavior! (You don’t actually have to do that for the HW problem, though.)

6. (a) (b) HW 23-1, cont. • So, how do we do it? • Turn it into part of an infinite repeating function! • Thought question: Which of these two infinite repeating functions would be the correct choice? …and what’s the repetition period?

7. Reading Quiz • Section 6.6 was all about the motion of a guitar string. What was the string’s initial shape? • Rectified sine wave • Sawtooth wave • Sine wave • Square wave • Triangle wave

8. h L What was section 6.6 all about, anyway? • What will guitar string look like at some later time? • Plan: • Figure out the frequency components in terms of “harmonic modes of string” • Figure out how each component changes in time • Add up all components to get how the overall string changes in time initial shape:

9. h L Step 1: figure out the frequency components h • a0 = ? • an = ? • bn = ? 2 3 L 1 integrate from –L to L: three regions

10. h L Step 1: figure out the frequency components h L

11. h L Step 2: figure out how each component changes • Fundamental: y = Asin(px/L)cos(w1t) • 3rd harmonic: y = Asin(3px/L)cos(w3t) • 5th harmonic: y = Asin(5px/L)cos(w5t) • w1 = ? (assume velocity and L are known) = 2pf1 = 2p(v/l1) = 2pv/(2L) = pv/L • wn = ?

12. h L Step 3: put together • Each harmonic has y(x,t) = Asin(npx/L)cos(nw1t) = Asin(npx/L)cos(npvt/L) What does this look like?  Mathematica!

13. h L Step 3: put together • Each harmonic has y(x,t) = Asin(npx/L)cos(nw1t) = Asin(npx/L)cos(npvt/L) What does this look like?  Mathematica!

14. How about the pulse from HW 23-1? • Any guesses as to what will happen?

15. How about the pulse from HW 23-1? • Any guesses as to what will happen?