Download lecture 16 - complex numbers

Announcements 10/5/11
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Prayer
Exam 1 ends tomorrow night
Lab 3: Dispersion lab – computer
simulations, see website
a. “Starts” Saturday, due next Saturday
Taylor’s Series review:
a. cos(x) = 1 – x2/2! + x4/4! – x6/6! + …
b. sin(x) = x – x3/3! + x5/5! – x7/7! + …
c. ex = 1 + x + x2/2! + x3/3! + x4/4! + …
d. (1 + x)n = 1 + nx + …
Guy & Rodd
Reading Quiz
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1  3i
What’s the complex conjugate of:
4  5i
1  3i
a.
4  5i
1  3i
4  5i
c. 1  3i
4  5i
1  3i
d.
4  5i
b.
Complex Numbers – Polar Coordinates
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Where is 10ei(p/6) located on complex plane?
Proof that it is really the same as 1030
Complex Numbers, cont.
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Adding
a. …on complex plane, graphically?
Multiplying
a. …on complex plane, graphically?
b. How many solutions are there to x2=1? x2=-1?
c. What are the solutions to x5=1? (xxxxx=1)
Subtracting and dividing
a. …on complex plane, graphically?
Polar/rectangular conversion
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Warning about rectangular-to-polar conversion:
tan-1(-1/2) = ?
a. Do you mean to find the angle for (2,-1) or
(-2,1)?
Always draw a picture!!
Using complex numbers to add sines/cosines
Fact: when you add two sines or cosines having
the same frequency, you get a sine wave with the
same frequency!
a. “Proof” with Mathematica
 Worked problem: how do you find mathematically
what the amplitude and phase are?
 Summary of method:
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Just like adding vectors!!
Hw 16.5: Solving Newton’s 2nd Law
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Simple Harmonic Oscillator
(ex.: Newton 2nd Law for mass on spring)
2
d x
k
 x
2
m
dt
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Guess a solution like
x(t )  Ae
it  
what it means, really: x(t )  A cos(t   )
and take Re{ … } of each side
(“Re” = “real part”)
Complex numbers & traveling waves
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Traveling wave: A cos(kx – t + )
Write as:
f (t )  Ae
i kx t  
i i kx t 
Often: f (t )  Ae e
…or
f (t )  Ae
i kx t 
– where “A-tilde” = a complex number
 the amplitude of which represents the amplitude of
the wave
 the phase of which represents the phase of the wave
– often the tilde is even left off
Thought Question
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Which of these are the same?
(1) A cos(kx – t)
(2) A cos(kx + t)
(3) A cos(–kx – t)
a.
b.
c.
d.
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(1) and (2)
(1) and (3)
(2) and (3)
(1), (2), and (3)
Which should we use for a left-moving wave: (2) or (3)?
a. Convention: Usually use #3, Aei(-kx-t)
b. Reasons: (1) All terms will have same e-it factor. (2)
The sign of the number multiplying x then indicates
the direction the wave is traveling.
k  k iˆ
Reflection/transmission at boundaries: The setup
x=0
Region 1: light string
in-going wave
reflected wave
i ( k1x1t )
transmitted wave
AI e
i (  k1x1t )
AR e
i ( k1x1t )
f1  AI e
Region 2: heavier string
Goal: How much of wave is
transmitted and reflected?
(assume k’s and ’s are known)
i ( k1x1t )
 ARe
f1  AI cos(k1x  1t  I )  AR cos(k1x  1t  R )
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AT ei ( k2 x2t )
i ( k2 x2t )
f2  AT e
f 2  AT cos(k2 x  2t  T )
Why are k and  the same for I and R? (both labeled k1 and 1)
“The Rules” (aka “boundary conditions”)
a. At boundary: f1 = f2
b. At boundary: df1/dx = df2/dx