announcements 10/5/11 prayer exam 1 ends tomorrow night lab 3: dispersion lab – computer...

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