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Case 1: Constant v

Explore wave equation solutions using simple functions, and understand the effects of phase velocity, group velocity, and relationships between variables like k, w, l, and T. Learn about adding multiple waves and their properties.

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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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