Announcements 10/8/10 Prayer Exam: last day = tomorrow! a. a.Correction to syllabus: on Saturdays, the Testing Center gives out last exam at 3 pm, closes at 4 pm. Homework problem changes: some extra credit, one moved. See . Lab 3 starting tomorrow: it’s a computer simulation. See class website. a. a.If we don’t finish the relevant discussion today (which seems likely), you probably should wait until after Monday’s class before starting the lab. Quick writing assignment while you wait: Ralph is still not quite grasping this… he asks you, “How are complex exponentials related to waves on a string?” What should you tell him to help him understand? (Please actually write down your answer.)

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.

Reading Quiz Which of the following was not a major topic of the reading assignment? a. a.Dispersion b. b.Fourier transforms c. c.Reflection d. d.Transmission

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

Boundaries: The math x = 0 and Goal: How much of wave is transmitted and reflected?

Boundaries: The math x = 0 Goal: How much of wave is transmitted and reflected?

Boundaries: The math Like: and How do you solve? x = 0 Goal: How much of wave is transmitted and reflected? 2 equations, 3 unknowns?? Can’t get x, y, or z, but can get ratios! y = x z = 0.75 x

Boundaries: The results Recall v =  /k, and  is the same for region 1 and region 2. So k ~ 1/v Can write results like this: x = 0 Goal: How much of wave is transmitted and reflected? “reflection coefficient” “transmission coefficient” The results….

Special Cases Do we ever have a phase shift? a. a.If so, when? And what is it? What if v 2 = 0? a. a.When would that occur? What if v 2 = v 1 ? a. a.When would that occur? x = 0 The results….

Power Recall: (A = amplitude) Region 1:  and v are same … so P ~ A 2 Region 2:  and v are different… more complicated …but energy is conserved, so easy way is: x = 0 r,t = ratio of amplitudes R,T = ratio of power/energy

Dispersion A dispersive medium: velocity is different for different frequencies a. a.Any real-world examples? Why do we care? a. a.Real waves are often not shaped like sine waves. – – Non sine-wave shapes are made up of combinations of sine waves at different frequencies. b. b.Real waves are not infinite in space or in time. – – Finite waves are also made up of combinations of sine waves at different frequencies. Focus on (b) for now… (a) is the main topic of the “Fourier” lectures of next week.

Wave packets Adding cosines together with Mathematica, “sum of cosines.nb” fall10/lectures/lecture%2017%20-%20sum%20of%20cosines.nb What did we learn? a. a.To localize a wave in space, you need lots of frequencies b. b.To remove neighboring localized waves, you need those frequencies to spaced close to each other. (infinitely close, really)

Back to Dispersion What happens if a wave pulse is sent through a dispersive medium? Nondispersive? Dispersive wave example: a. a.s(x,t) = cos(x-4t) + cos(2 (x-5t)) – – What is “v”? – – What is v for  =4? What is v for  =10? What does that wave look like as time progresses? (Mathematica “dispersion of two cosines.nb”; on website. And next slide.)

Mathematica 0.7 seconds1.3 seconds 0.1 seconds

Femtosecond Laser Pulse E t=0 =sin(10 x)*exp(-x^2) Power Spectrum Credit: Dr. Durfee Initial shape of wave How much energy is contained in each frequency component (  = vk) Note: frequencies are infinitely close together

Propagation Of Light Pulse E(x,t) Power Spectrum Credit: Dr. Durfee Wave moving in time How much energy is contained in each frequency component

Tracking a Moving Pulse E(x+vt,t) Power Spectrum Credit: Dr. Durfee Graph “window” is moving along with speed v How much energy is contained in each frequency component

Laser Pulse in Dispersive Medium E t=0 = sin(10 x)*exp(-x^2) Power Spectrum Credit: Dr. Durfee How much energy is contained in each frequency component Initial shape of wave Not all frequency components travel at same speed

Time Evolution of Dispersive Pulse Credit: Dr. Durfee Wave moving in time Peak moves at about 13 m/s (on my office computer) How much energy is contained in each frequency component

Tracking a Dispersive Pulse E(x+v g t,t) Credit: Dr. Durfee Graph window moving along with peak, ~13 m/s