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Case 1: Constant v

Case 1: Constant v. d 2 y/dt 2 – v 2 d 2 y/dx 2 = 0 Is f(x – v t) a solution for any f? What does a simple solution look like?. y = exp[i (k x – w t)] {can also use cos( ) or sin( )}. What is the relation between k & w ?. w 2 = v 2 k 2 a w = v k or w = – v k.

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Case 1: Constant v

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  1. Case 1: Constant v d2y/dt2 – v2 d2y/dx2 = 0 Is f(x – v t) a solution for any f? What does a simple solution look like? y = exp[i (k x – w t)] {can also use cos( ) or sin( )} What is the relation between k & w? w2 = v2 k2 aw = v k or w = – v k What is the relationship between k and l and w and T? k = 2 p / l and w = 2 p / T

  2. cos(kx – wt) (phase velocity)

  3. Case 2: Adding 2 waves (Movies) [used cos(a+b) = cos(a)cos(b) – sin(a)sin(b)] Phase velocity arises from keeping F constant. Group velocity arises from keeping DF constant. Velocities are same only when w is proportional to k.

  4. Group velocity = Phase velocity

  5. Group velocity > Phase velocity

  6. Group velocity < Phase velocity

  7. Case 2: Adding 2 waves (Movies) What is relation between Dx & Dk? Dx Dk = 2 p General a Dx Dk ~ 1 The wave has width in time. What is relation between Dt & Dw? Dt Dw = 2 p General a Dt Dw ~ 1

  8. Adding Many Waves Make a wave at (t=0) by adding {cos[k x] + cos[(k+Dk)x] + cos[(k-Dk)x] + cos[(k+2Dk)x] + cos[(k-2Dk)x] + …}/N What will this look like as number of terms increase? Specific calculation with k = 10 p and Dk = p/4 Not important but can get simple expression for sumSum = cos[kx] sin[(2N+1) Dk x/2]/{(2N+1) sin[Dk x/2]}

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