Announcements 2/28/11
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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:
a. 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.
Summary of last time

 2 nx 
f ( x)  a0 
an cos 


 L 
n1

a0 
1
L
2
an 
L
bn 
2
L

 2 nx 
bn sin 

L


n1

The series
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
Fourier Transform (review)
20
10
600
400
200
200
400
Do the transform (or
have a computer do it)
600
10
Cos 0.9 x
20
Cos 0.91 x
Cos 0.92 x
Cos 0.93 x
Cos 0.94 x
Cos 0.95 x
Cos 0.96 x
Cos 0.97 x
Cos 0.98 x
Cos 0.99 x
Cos 1. x
Cos 1.03 x
Cos 1.04 x
Cos 1.05 x
Cos 1.06 x
Cos 1.07 x
Cos 1.08 x
Cos 1.09 x
Cos 1.1 x
Cos 1.01 x
Cos 1.02 x
How does computer know all
components will be multiples of
k=0.01?
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
…”
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?
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 = Asin(x/L)cos(w1t)
3rd harmonic: y = Asin(3x/L)cos(w3t)
5th harmonic: y = Asin(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)

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?