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Announcements 2/11/11

Announcements 2/11/11. Prayer Exam 1 ends on Tuesday night Lab 3: Dispersion lab – computer simulations, find details on class website “Starts” tomorrow, due next Saturday… but we won’t talk about dispersion until Monday, so I recommend you do it after Monday’s lecture.

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Announcements 2/11/11

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  1. Announcements 2/11/11 • Prayer • Exam 1 ends on Tuesday night • Lab 3: Dispersion lab – computer simulations, find details on class website • “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: • cos(x) = 1 – x2/2! + x4/4! – x6/6! + … • sin(x) = x – x3/3! + x5/5! – x7/7! + … • ex = 1 + x + x2/2! + x3/3! + x4/4! + … • (1 + x)n = 1 + nx + …

  2. Reminder • What is w? • What is k?

  3. Reading Quiz • What’s the complex conjugate of:

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

  5. Complex Numbers, cont. • Adding • …on complex plane, graphically? • Multiplying • …on complex plane, graphically? • How many solutions are there to x2=1? • What are the solutions to x5=1? (xxxxx=1) • Subtracting and dividing • …on complex plane, graphically?

  6. Polar/rectangular conversion • Warning about rectangular-to-polar conversion: tan-1(-1/2) = ? • 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) • “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 2nd 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 – wt + f) • 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 x = 0 • Why are k and w the same for I and R? (both labeled k1 and w1) • “The Rules” (aka “boundary conditions”) • At boundary: f1 = f2 • At boundary: df1/dx = df2/dx Region 1: light string Region 2: heavier string transmitted wave in-going wave Goal: How much of wave is transmitted and reflected? (assume k’s and w’s are known) reflected wave

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