Announcements 2/9/11
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Prayer
Complex numbers next time. The Colton
“Complex numbers summary” handout should be
helpful.
k and w
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What’s the difference between these:
s  cos( x  5t )
s  cos(2( x  5t ))
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General form of cosine wave:
s  A cos(k ( x  vt )   )
…sometimes written as:
w = _______
s  A cos(kx  wt   )
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k = “wavevector”; w = “angular frequency”
Reading Quiz
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A wave pulse traveling on a string hits the end of
the string, which is tied to a post. What happens?
a. The pulse reflects, flipped over
b. The pulse reflects, not flipped over
Demos
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Shive wave machine
Web demo:
http://www.colorado.edu/physics/phet/simulations/stringwave/stringWave.swf
Reminder: the Wave Equation
2 f
2 f
C 2
2
t
x

C  v2
Any function that has
“x-vt” will work!
…or “x+vt”
Why is it called the wave equation?
a. Because traveling waves are solutions of the equation!
f  A cos( x  vt )
f
  A sin( x  vt )
x
f
  A sin( x  vt )  v 
t
  Av sin( x  vt )
2 f
  A cos( x  vt )
2
x
2 f
  Av cos( x  vt )  v 
2
t
  Av 2 cos( x  vt )
Analysis: A section of rope
F
x
T2
1
 0  T1 cos1  T2 cos 2
 = small; cos  1
T1
T1  T2
T2
2
1
x x+Dx
1
T1
 = mass/length
“linear mass density”
F
y
 ma y  T2 sin  2  T1 sin 1  ma y
T  sin 2  sin 1    Dx  a y
T  sin  2  sin 1   2 y
 2



Dx
 t
 = small; sin  tan
T  tan  2  tan 1   2 y
 2



Dx
 t
A section of rope, cont.
T  tan  2  tan 1   2 y
 2



Dx
 t
T2
T1
T2
2
1
T1
x x+Dx
 y

T  x



What is tan2 in picture?
 tan2 = opp/adj = rise/run
= slope!
(at x + Dx)
y 


2

x
 y
x Dx
x 
 2
 t
Dx


T 2 y 2 y
 2
2
 x
t
v
T

Demo: wave speed vs tension
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I measured mass of spring to be 0.28 kg.
Can we predict how fast wave will travel?
Reading Quiz
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A wave pulse traveling on a string meets an
interface, where the medium abruptly switches to
a thicker string. What happens?
a. The pulse continues on, but flipped over
b. The pulse continues on, not flipped over
c. The pulse reflects, flipped over
d. The pulse reflects, not flipped over
e. The pulse partially reflects and partially
transmits
Advertisement: We’ll figure out the equations for
reflection and transmission on Monday
Power: energy transfer
1
P  w 2 A2v
2
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(Proved in book; most important is P ~ A2)
The Wave Equation: Linear
2
2 f

f
2
v
2
t
x 2
Any function that has
“x-vt” will work!
Why is it called the linear wave equation?
a. Because we don’t have nonlinear terms like f2, x2, xf,
ex, etc., in the equation itself.
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Properties of linear differential equations:
a. If f1 is a solution, then so is C  f1
b. If f1 and f2 are solutions, then so is (f1 + f2)
Consider:
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f  A1 cos(k1 ( x  vt )  1 ) A2 cos(k2 ( x  vt )  2 )
Thought Question
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What happens when two wave pulses on a linear medium
run into each other head on?
a. They reflect off of each other and go back the way
they came.
b. Part of each wave is reflected and part transmitted.
c. They pass right through each other.
Demo: Shive wave machine interference
Web demo again
Complex Numbers – A Summary
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What is the square
What is “i”?
root of 1… 1 or -1?
What is “-i”?
The complex plane
Complex conjugate
a. Graphically, complex conjugate = ?
Polar vs. rectangular coordinates
a. Angle notation, “A”
Euler’s equation…proof that ei = cos + isin
a.  must be in radians
b. Where is 10ei(p/6) located on complex plane?
Reminder
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What is w?
What is k?
How do they relate to the velocity?
Relationship between w and T
Relationship between k and l
Consistency check: w/k = ?