Announcements 10/4/10
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Prayer
Exam goes until Saturday
a. Correction to syllabus: on Saturdays, the
Testing Center gives out last exam at 3 pm,
closes at 4 pm.
Complex numbers next time. The Colton
“Complex numbers summary” should be helpful.
(We’ll go over today, if enough time.)
Homework survey
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:
s  A cos(kx  wt   )
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k = “wavevector”; w = “angular frequency”
w=…
Quick Writing
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If I have a longitudinal wave passing down my
slinky, described by the function s(x,t), what
does s represent, what does x represent, and
what does t represent?
Other good questions: For a fixed time, what
would s(x) represent? For a fixed x, what would
s(t) represent?
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
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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 think mass of spring is 0.276 kg. (Check
with balance.)
Can we predict how fast wave will travel?
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
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 Friday
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. 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 “i”?
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