1. Announcements 10/7/11 Prayer We’re likely not going to finish dispersion today, so you might want to wait until after Monday before doing Lab 3 (Dispersion). I just got the exams from the Testing Center, TA & I will work on grading them today & this weekend. Non Sequitur

2. 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

3. 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

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

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

6. 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 = -0.25 x z = 0.75 x

7. 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….

8. Special Cases Do we ever have a phase shift in reflected or transmitted waves? 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….

9. 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

10. Quick Writing We saw that A 1 cos(kx +  1 ) + A 2 cos(kx +  2 ) gives you a cosine wave with the same k, and hence wavelength. If you add a third, fourth, fifth, etc., such cosine wave, you still get a simple cosine wave. See  How can you then add together cosine waves to get more complicated shapes? Or can you?

12. Wave packets HW 17-5

13. Wave packets, cont. Results: a. a.To localize a wave in space, you need lots of spatial frequencies (k values) b. b.To remove neighboring localized waves (i.e. to make it non-periodic), you need those frequencies to spaced close to each other. (infinitely close, really)

14. 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 transform” lectures