1. Announcements 2/11/11 Prayer Exam 1 ends on Tuesday night Lab 3: Dispersion lab – computer simulations, find details on class website a. a.“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. 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 + …

2. Reminder What is  ? What is k?

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

4. Complex Numbers – A Summary What is “i”? What is “-i”? The complex plane Complex conjugate a. a.Graphically, complex conjugate = ? Polar vs. rectangular coordinates a. a.Angle notation, “A  ” Euler’s equation…proof that e i  = cos  + isin  a. a.  must be in radians b. b.Where is 10e i(  /6) located on complex plane? What is the square root of 1… 1 or -1?

5. 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? 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?

6. 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!!

7. 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. 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!!

8. Using complex numbers to solve equations 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) A few words about HW 16.5…

9. 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

10. 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