Announcements 2/14/11 Prayer Exam 1: last day = tomorrow! We’re likely not going to finish dispersion today, so extra time on Lab 3: now due Tues Feb 22. Monday Feb 21 is President’s Day holiday. Tues Feb 22 is a virtual Monday

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

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

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.

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 HW 17-5

Wave packets, cont. 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)