Announcements 2/11/11 Prayer Exam 1 ends on Tuesday night Lab 3: Dispersion lab – computer

simulations, find details on class websitea. “Starts” tomorrow, due next Saturday…

but we won’t talk about dispersion until Monday, so I recommend you do it after Monday’s lecture.

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 + …

Reminder What is ? What is k?

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 – A Summary 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 +

isina. must be in radiansb. Where is 10ei(/6) located on complex

plane?

What is the square root of 1… 1 or -1?

Complex Numbers, cont. Adding

a. …on complex plane, graphically? Multiplying

a. …on complex plane, graphically?b. How many solutions are there to 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 (with possibly different amplitudes and phases), you get a sine wave with the same frequency! (but a still-different amplitude and phase)

a. “Proof” with Mathematica… (class make up numbers)

Worked problem: how do you find mathematically what the amplitude and phase are?

Summary of method:Just like adding vectors!!

Using complex numbers to solve equations

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)

2

2

d x kx

mdt

( ) i tx t Ae

( ) cos( )x t A t

A few words about HW 16.5…

Complex numbers & traveling waves Traveling wave: A cos(kx – t + )

Write as:

Often:

…or – where “A-tilde” = a complex number, 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

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