Announcements 10/19/12
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Prayer
Learn Smart – info in email
Term project proposal due tomorrow
Labs 4-5 also due tomorrow
Exam 2: starting next Thurs
a. Review session: Tues 5:30-7 pm, place TBA
From warmup
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Extra time on?
a. The math
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Other comments?
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. So im reading the proposal instructions. Can you explain a little
more what calculations and predictions you would need?
c. Do I need a log of hours spent on the term project?
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. Just the fundamental frequency, f0 = 1/period
b. f0 and potentially all integer multiples of f0
c. A finite number of discrete frequencies centered on f0
d. An infinite number of frequencies near f0, spaced
infinitely close together
Summary of last time

 2 nx 
f ( x)  a0 
an cos 


 L 
n1


 2 nx 
bn sin 

L


n1

The series
Written another way f ( x)  a0  a1 cos  k0 x   b1 sin  k0 x 
with k0 = 2/L
 a2 cos  2k0 x   b2 sin  2k0 x 
 a3 cos  3k0 x   b3 sin  3k0 x   ...
a0 
1
L
2
an 
L
2
bn 
L
L
 f ( x)dx
0
L

0
L

0
 2 nx 
f ( x) cos 
 dx
L


 2 nx 
f ( x)sin 
dx

 L 
How to find the coefficients
The graphical “proof” that the functions
integrate to zero
(called “orthogonal” functions)
Mathematica: Graphical “proof”
Sawtooth Wave, like HW 22-2
1

2

1
 2 nx 
sin 

n
 L 
(The next few slides from Dr. Durfee)
N 0
N 1
N 2
N 3
N  10
N  500
The Spectrum of a Saw-tooth Wave
0.6
Amplitude [m]
0.4
0.2
0
-0.2
-0.4
0
10
20
30
k [rad/m]
40
50
60
The Spectrum of a Saw-tooth Wave
0.6
0
0.4
-pi/4
0.3
0.2
-pi/2
0.1
0
0
10
20
30
k [rad/m]
40
50
60
Phase [rad]
Amplitude [m]
0.5
Electronic “Low-pass filter”
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“Low pass filter” = circuit which preferentially lets
lower frequencies through.
What comes out?
Circuit
?
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
0.6
0
0.5
0.4
0.3
-pi/2
0.2
-3 pi/4
0.1
0
0
10
20
30
k [rad/m]
40
50
-pi
60
Phase [rad]
Amplitude [m]
-pi/4
Low-Pass Filter – after filter
0.6
0
0.5
0.4
0.3
-pi/2
0.2
-3 pi/4
0.1
0
0
10
20
30
k [rad/m]
40
50
-pi
60
Phase [rad]
Amplitude [m]
-pi/4
Low Pass Filter (Theory)
1
y and y
filtered
[m]
0.8
0.6
0.4
0.2
0
0
0.5
1
1.5
x [m]
2
2.5
3
Actual Data from Oscilloscope
Periodic?
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“Any function periodic on a distance L can be
written as a sum of sines and cosines like this:”

 2 nx 
f ( x)  a0 
an cos 


 L 
n1



 2 nx 
bn sin 

L


n1

What about nonperiodic functions?
a. “Fourier series” vs. “Fourier transform”
b. Special case: functions with finite domain
HW 23-1
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“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(w1t) --standing wave
 Asin(x/L) is the initial shape
 It oscillates sinusoidally in time at frequency w1
 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.
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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
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What is the main goal of PpP section 6.6? Explain in your
own words what the point is.
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. [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. Rectified sine wave
b. Sawtooth wave
c. Sine wave
d. Square wave
e. Triangle wave
What was section 6.6 all about, anyway?
initial shape: h
L
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What will guitar string look like at some later time?
(assume h, L, and velocity v are known)
Plan:
a. Figure out the frequency components in terms of
“harmonic modes of string”
b. Figure out how each component changes in time
c. Add up all components to get how the overall string
changes in time
Step 1: figure out the frequency components
h
2
h
L
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a0 = ?
an = ?
bn = ?
3
L
1
integrate from –L to L:
three regions
2
bn 
" L"
L

L
 2 nx 
f ( x)sin 
dx

 " L" 
L2
L
L 2

2 
2

nx


 2 nx 
 2 nx  
bn 
dx 
dx   mx  b region 3 sin 
dx
 mx  b region1 sin 
 mx  b region 2 sin 
2L 
2 L 
2 L 
2 L  



L 2
L2
 L




L2
L
L 2

1
 2h
   nx 
 2h
   nx 
 2h
   nx  
bn 
 L x  2h  sin  L  dx 
 L x  0  sin  L  dx   L x  2h  sin  L  dx 
L

 


 


 

L 2
L2
 L




Step 1: figure out the frequency components
h
h
L
L
 n  3  n 
32h cos 
sin 

4 
4 


bn 
n 2 2
1 ( n 1)
8h
bn  2 2  1 2
; odd n
n
Step 2: figure out how each component changes
h
L
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Fundamental: y = b1sin(x/L)cos(w1t)
3rd harmonic: y = b3sin(3x/L)cos(w3t)
5th harmonic: y = b5sin(5x/L)cos(w5t)
w1 = ? (assume velocity and L are known)
= 2f1 = 2(v/l1) = 2v/(2L) = v/L
wn = ?
Step 3: put together

L

1 ( n 1) 
 8h
 n x 
2
f ( x, t  0) 
 2 2  1
 sin  L 
 

n 1  n 

h
odd
Each harmonic has
y(x,t) = Asin(nx/L)cos(nw1t)
= Asin(nx/L)cos(nvt/L)

1 ( n 1) 
 8h
 n x 
 n vt 
2
f ( x, t ) 
 2 2  1
 sin  L  cos  L 
 



n 1  n 

odd
What does this look like?  Mathematica!
Step 3: put together

L

1 ( n 1) 
 8h
 n x 
2
f ( x, t  0) 
 2 2  1
 sin  L 
 

n 1  n 

h
odd
Each harmonic has
y(x,t) = Asin(nx/L)cos(nw1t)
= Asin(nx/L)cos(nvt/L)

Experiment!!
1 ( n 1) 
 8h
 n x 
 n vt 
2
f ( x, t ) 
 2 2  1
 sin  L  cos  L 
 



n 1  n 

odd
What does this look like?  Mathematica!
How about the pulse from HW 23-1?
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Any guesses as to what will happen?
How about the pulse from HW 23-1?
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Any guesses as to what will happen?
Experiment!!