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1-4 Superposition soln - Solution Physics 214 Problem 4...

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Physics 214 Problem 4 Week 1 Superposition and the Wave Equation The wave equation for electromagnetic waves is given by: d 2 y dx 2 = 1 v 2 d 2 y dt 2 with ν = c, the speed of light a) What relation between ω 1 , k 1 and velocity v is needed to make y 1 = A 1 cos( k 1 x ω 1 t ) a solution to this equation? (In fact, any function of the form y = f( kx ω t + φ ) is a solution if that relation holds, although we would not usually use the ω , k notation for other functional forms.). b) Show that the superposition of two solutions, y 3 (x,t) = y 1 (x,t) + y 2 (x,t) is also a solution to this wave equation if each of y 1 (x,t) and y 2 (x,t) is separately a solution. If that seems to abstract, try a specific case: y 3 (x,t) = A 1 cos( k 1 x ω 1 t ) + A 2 cos( k 2 x ω 2 t+ φ ), where the ω ’s and k ’s satisfy the necessary condition from part (a). ( ) ( ) ( ) ( ) ( ) ( ) 2 2 2 2 1 2 1 2 2 2 3 1 1 2 2 2 2 2 2 2 2 2 2 1 2 1 2 2 2 3 1 1 2 2 2 2 2 2 2 3 1 1 2 2 2 2 2 2 3 1 1 1 1 2 2 2 2 2 2 2 3 1 1 1 1 2 iff /k /k .
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