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Announcements 10/21/11

Announcements 10/21/11. Prayer Chris: no office hours today Due tomorrow: Labs 4-5; term project proposal Due Monday: HW 22 and HW 23 See email for hint on HW22-3

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Announcements 10/21/11

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  1. Announcements 10/21/11 • Prayer • Chris: no office hours today • Due tomorrow: Labs 4-5; term project proposal • Due Monday: HW 22 and HW 23 • See email for hint on HW22-3 • If you can’t get Mathematica to plot things, please come find me (after giving it a serious attempt). I can usually spot Mathematica errors in 2 minutes or less. • Exam 2 review session: Tuesday 5-6 pm. Room: C460 (probably) • Exam 2 starts on Thursday

  2. Summary of last time The series Written another way with k0 = 2p/L How to find the coefficients

  3. Building a function by specifying an, bn Mathematica: build a function…

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

  5. The Spectrum of a Saw-tooth Wave

  6. The Spectrum of a Saw-tooth Wave

  7. 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

  8. Low-Pass Filter – before filter

  9. Low-Pass Filter – after filter

  10. Low Pass Filter

  11. Actual Data from Oscilloscope

  12. 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

  13. 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.)

  14. (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?

  15. 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

  16. h L What was section 6.6 all about, anyway? • What will guitar string look like at some later time? (assume h, L, and velocity v are known) • 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:

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

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

  19. h L Step 2: figure out how each component changes • Fundamental: y = b1sin(px/L)cos(w1t) • 3rd harmonic: y = b3sin(3px/L)cos(w3t) • 5th harmonic: y = b5sin(5px/L)cos(w5t) • w1 = ? (assume velocity and L are known) = 2pf1 = 2p(v/l1) = 2pv/(2L) = pv/L • wn = ?

  20. 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!

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

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

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

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