1 / 10

Announcements 10/5/11

Announcements 10/5/11. Prayer Exam 1 ends tomorrow night Lab 3: Dispersion lab – computer simulations, see website “Starts” Saturday, due next Saturday Taylor’s Series review: cos(x) = 1 – x 2 /2! + x 4 /4! – x 6 /6! + … sin(x) = x – x 3 /3! + x 5 /5! – x 7 /7! + …

hillary-goff
Download Presentation

Announcements 10/5/11

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. Announcements 10/5/11 • Prayer • Exam 1 ends tomorrow night • Lab 3: Dispersion lab – computer simulations, see website • “Starts” Saturday, due next Saturday • 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 + … Guy & Rodd

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

  3. Complex Numbers – Polar Coordinates • Where is 10ei(p/6) located on complex plane? • Proof that it is really the same as 1030

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

  5. 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!!

  6. 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! • “Proof” with Mathematica • Worked problem: how do you find mathematically what the amplitude and phase are? • Summary of method: Just like adding vectors!!

  7. 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”)

  8. Complex numbers & traveling waves • Traveling wave: A cos(kx – wt + f) • 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

  9. Thought Question • Which of these are the same? (1) A cos(kx – wt) (2) A cos(kx + wt) (3) A cos(–kx – wt) • (1) and (2) • (1) and (3) • (2) and (3) • (1), (2), and (3) • Which should we use for a left-moving wave: (2) or (3)? • Convention: Usually use #3, Aei(-kx-wt) • Reasons: (1) All terms will have same e-iwt factor. (2) The sign of the number multiplying x then indicates the direction the wave is traveling.

  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

More Related