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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
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
Complex Numbers – Polar Coordinates Where is 10ei(/6) located on complex plane? Proof that it is really the same as 1030
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?
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!!
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!!
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
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
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
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