Announcements 10/21/11
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Prayer
Chris: no office hours today
Due tomorrow: Labs 4-5; term project proposal
Due Monday: HW 22 and HW 23
a. See email for hint on HW22-3
b. 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
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
Building a function by specifying an, bn
Mathematica: build a function…
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 freq. 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
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 = Asin(x/L)cos(w1t) --standing wave
 Asin(x/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.)
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?
Reading Quiz
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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)
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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!!