Announcements 10/19/12 Prayer Learn Smart – info in Term project proposal due tomorrow Labs 4-5 also due tomorrow Exam 2: starting next Thurs a. a.Review session: Tues 5:30-7 pm, place TBA

From warmup Extra time on? a. a.The math Other comments? a. a.As i almost always do the homework by myself i haven't had the opportunity to find a partner for the term project. can we spend a minute or two in class and make sure there aren't others in my situation. b. b.So im reading the proposal instructions. Can you explain a little more what calculations and predictions you would need? c. c.Do I need a log of hours spent on the term project? d. d.I'd like it to be noted that there are now two discrepancies in Dr. Durfee's book. The first being that he switched the coefficients for sin and cos, being the only person (as far as we can tell) who does this. Second, he believes the guitar is the ideal musical instrument. This is impossible seeing as the piano is the ideal musical instrument.

Review clicker question: 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

Summary of last time The series How to find the coefficients Written another way with k 0 = 2  /L

The graphical “proof” that the functions integrate to zero Mathematica: Graphical “proof”Graphical “proof” (called “orthogonal” functions)

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

The Spectrum of a Saw-tooth Wave

Electronic “Low-pass filter” “Low pass filter” = circuit which preferentially lets lower frequencies through. ? Circuit What comes out? How to solve: (1)Decompose wave into Fourier series (2)Apply filter to each frequency individually (3)Add up results in infinite series again

Low-Pass Filter – before filter

Low-Pass Filter – after filter

Low Pass Filter (Theory)

Actual Data from Oscilloscope

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

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(x,t) = Asin(  x/L)cos(  1 t) --standing wave  Asin(  x/L) is the initial shape  It oscillates sinusoidally in time at frequency  1  What’s y(x,t) for the second harmonic?  If you can predict how each frequency component will behave, you can predict the behavior for any shape of wave! (You don’t actually have to do that for the HW problem, though.)

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? (a)(b) …and what’s the repetition period?

From Warmup What is the main goal of PpP section 6.6? Explain in your own words what the point is. a. a.The point in this section was to come up with what the string of a guitar looks like at any point in time [for a particular starting position] How does Dr. Durfee achieve that goal? Explain in your own words what he is doing, mathematically. a. a.[my answer] The process is to 1) write the initial shape as a sum of Fourier components, 2) attach the appropriate time dependence to each component, then 3) add the components back together to get the time dependence of the string as a whole.

Clicker question: Section 6.6 was all about the motion of a guitar string. What was the string’s initial shape? a. a.Rectified sine wave b. b.Sawtooth wave c. c.Sine wave d. d.Square wave e. e.Triangle wave

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: a. a.Figure out the frequency components in terms of “harmonic modes of string” b. b.Figure out how each component changes in time c. c.Add up all components to get how the overall string changes in time h L initial shape:

Step 1: figure out the frequency components a 0 = ? a n = ? b n = ? h L h L integrate from –L to L: three regions 1 23

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

Step 2: figure out how each component changes Fundamental: y = b 1 sin(  x/L)cos(  1 t) 3 rd harmonic: y = b 3 sin(3  x/L)cos(  3 t) 5 th harmonic: y = b 5 sin(5  x/L)cos(  5 t)  1 = ? (assume velocity and L are known) = 2  f 1 = 2  (v/ 1 ) = 2  v/(2L) =  v/L  n = ? h L

Step 3: put together Each harmonic has y(x,t) = Asin(n  x/L)cos(n  1 t) = Asin(n  x/L)cos(n  vt/L) h L What does this look like?  Mathematica!

Step 3: put together Each harmonic has y(x,t) = Asin(n  x/L)cos(n  1 t) = Asin(n  x/L)cos(n  vt/L) h L What does this look like?  Mathematica! Experiment!!

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

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