Announcements 10/1/12
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Prayer
Exam going on, ends Thurs night
No office hours for me today, sorry. (Clement still has his.)
The four Taylors series that all physics majors need to know
Complex numbers next time. The Colton “Complex numbers
summary” handout on class website should be helpful.
The Far Side
From warmup
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Extra time on?
a. partial derivatives:
– What do the funny little backward sixes mean?
– I don't understand partial derivations at all so that all
just confused me.
– what do we do for the partial derivatives in section
16.6?!?!
– What is that symbol that looks like an upside down e
in section 16.6?
b. Can we have a demo on the wave transmittance
between a heavier string to a lighter one?
Other comments?
a. rate my professor you put a generous curve on the
tests. Will you?
Review: k and w
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Reminder: 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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v = w/k
v = lf
k = “wavenumber”; w = “angular frequency”
k = 2p/l
w = 2p/T
Clicker question:
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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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Rubber tubing
From warmup: String attached to a wall. Which of the following will
decrease the time required for the the pulse to reach the wall? Mark all
that apply.
Moving your hand up and down more quickly but by the same amount
Moving your hand up and down more slowly but by the same amount
Moving your hand the same speed but farther up and down
Moving your hand the same speed but a shorter distance up and down
Using a heavier string of the same length under the same tension
 Using a lighter string of the same length under the same tension
 Using the same string of the same length but under more tension
Using the same string of the same length but under less tension
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Shive wave machine
Web demo:
http://www.colorado.edu/physics/phet/simulations/stringwave/stringWa
ve.swf
The (Linear 1D) Wave Equation
2s
2s
C 2
2
t
x
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C = v2
What’s that
funny symbol?
Why is it called the wave equation?
a. Because traveling waves are solutions of the equation!
s  A cos( x  vt )
Any function that has
“x-vt” will work! …or “x+vt”
s
  A sin( x  vt )
x
s
  A sin( x  vt )  v 
t
  Av sin( x  vt )
2s
  A cos( x  vt )
2
x
2s
  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
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Demo: wave speed vs tension
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Can we predict how fast wave will travel on
spring (or slinky)?
We need to measure some things:
a. mass of spring
b. length of spring
c. tension of spring
Do the experiment! Time the wave!
Clicker question:
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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 in the class after next
Power: energy transfer
1
P  w 2 A2v
2
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What does everything stand for?
Proved in book; most important thing for us now is P ~ A2
From warmup: Explain how each of the following (keeping
everything else the same) would impact the energy transfer rate of
a wave on a string:
a. reducing the mass density of the string
– reduces P linearly, assuming that the velocity is unchanged
b. doubling the wavelength
– irrelevant (if no dispersion)
c. doubling the tension
– increases v and therefore P (by sqrt(2))
d. doubling the amplitude
– increases by 4
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 a medium with v = 3 m/s:
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f  cos(2( x  3t )  5.5)  cos(9( x  3t )  0.3)
It’s a perfectly acceptable wave!
Clicker 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?