announcements 10/5/11 prayer exam 1 ends tomorrow night lab 3: dispersion lab – computer...
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![Page 1: Announcements 10/5/11 Prayer Exam 1 ends tomorrow night Lab 3: Dispersion lab – computer simulations, see website a. a.“Starts” Saturday, due next Saturday](https://reader036.vdocuments.site/reader036/viewer/2022062516/56649d355503460f94a0d451/html5/thumbnails/1.jpg)
Announcements 10/5/11 Prayer Exam 1 ends tomorrow night Lab 3: Dispersion lab – computer
simulations, see websitea. “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
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Reading Quiz What’s the complex conjugate of:
a.
b.
c.
d.
1 3
4 5
i
i
1 3
4 5
i
i
1 3
4 5
i
i
1 3
4 5
i
i
1 3
4 5
i
i
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Complex Numbers – Polar Coordinates Where is 10ei(/6) located on complex plane? Proof that it is really the same as 1030
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Complex Numbers, cont. 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?
(xxxxx=1) Subtracting and dividing
a. …on complex plane, graphically?
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Polar/rectangular conversion 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!!
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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:
Just like adding vectors!!
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Hw 16.5: Solving Newton’s 2nd Law Simple Harmonic Oscillator
(ex.: Newton 2nd Law for mass on spring)
Guess a solution like
what it means, really: and take Re{ … } of each side
(“Re” = “real part”)
2
2
d x kx
mdt
( ) i tx t Ae
( ) cos( )x t A t
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Complex numbers & traveling waves Traveling wave: A cos(kx – t + )
Write as:
Often:
…or – 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
( ) i kx tf t Ae
( ) i kx tif t Ae e ( ) i kx tf t Ae
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Thought Question Which of these are the same?
(1) A cos(kx – t)(2) A cos(kx + t)(3) A cos(–kx – t)
a. (1) and (2)b. (1) and (3)c. (2) and (3)d. (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-it factor. (2) The sign of the number multiplying x then indicates the direction the wave is traveling.
ˆk k i
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Reflection/transmission at boundaries: The setup
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
Region 1: light string Region 2: heavier string
in-going wave transmitted wave
reflected wave
1 1( )i k x tIA e
1 1( )i k x tRA e
2 2( )i k x tTA e
1 1 1 1( ) ( )1
i k x t i k x tI Rf A e A e 2 2( )
2i k x t
Tf A e
Goal: How much of wave is transmitted and reflected? (assume k’s and ’s are known)
x = 0
1 1 1 1 1cos( ) cos( )I I R Rf A k x t A k x t 2 2 2cos( )T Tf A k x t