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 Taylor’s Series review: a. a.cos(x) = 1 – x 2 /2! + x 4 /4! – x 6 /6! + … b. b.sin(x) = x – x 3 /3! + x 5 /5! – x 7 /7! + … c. c.e x = 1 + x + x 2 /2! + x 3 /3! + x 4 /4! + … d. d.(1 + x) n = 1 + nx + … Guy & Rodd

Reading Quiz What’s the complex conjugate of: a. a. b. b. c. c. d. d.

Complex Numbers – Polar Coordinates Where is 10e i(  /6) located on complex plane? Proof that it is really the same as 10  30 

Complex Numbers, cont. Adding a. a.…on complex plane, graphically? Multiplying a. a.…on complex plane, graphically? b. b.How many solutions are there to x 2 =1? x 2 =-1? c. c.What are the solutions to x 5 =1? (x  x  x  x  x=1) Subtracting and dividing a. a.…on complex plane, graphically?

Polar/rectangular conversion Warning about rectangular-to-polar conversion: tan -1 (-1/2) = ? a. 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. 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 2 nd Law Simple Harmonic Oscillator (ex.: Newton 2 nd Law for mass on spring) Guess a solution like what it means, really: and take Re{ … } of each side (“Re” = “real part”)

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

Thought Question Which of these are the same? (1) A cos(kx –  t) (2) A cos(kx +  t) (3) A cos(–kx –  t) a. a.(1) and (2) b. b.(1) and (3) c. c.(2) and (3) d. d.(1), (2), and (3) Which should we use for a left-moving wave: (2) or (3)? a. a.Convention: Usually use #3, Ae i(-kx-  t) b. 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.

Reflection/transmission at boundaries: The setup Why are k and  the same for I and R? (both labeled k 1 and  1 ) “The Rules” (aka “boundary conditions”) a. a.At boundary: f 1 = f 2 b. b.At boundary: df 1 /dx = df 2 /dx Region 1: light stringRegion 2: heavier string in-going wave transmitted wave reflected wave Goal: How much of wave is transmitted and reflected? (assume k’s and  ’s are known) x = 0