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handout06

# handout06 - PHYSICS 218 SOLUTION TO HANDOUT 6 Created...

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PHYSICS 218 SOLUTION TO HANDOUT 6 Created: September 25, 2004 8:55pm Last updated: September 26, 2004 8:14am Gaussian Wave Packet Consider a transverse wave propagating along an infinite one-dimensional string which satisfies the wave equation ∂η ∂x 2 - 1 c 2 ∂η ∂t 2 , where η ( x, t ) is the displacement from the string’s equilibrium position. The initial profile of the wave is given by η ( x, t = 0) = A e - x 2 , where A is a constant independent of time. Waves of this form are called Gaussian. Let the initial velocity profile be given by ∂t η ( x, t = 0) = A c x e - x 2 , where c is the wave speed. 1. Let the uniform in the string be τ 0 , what is the linear mass density λ 0 of the string? Solution: This is to refresh your memory on wave equation for one-dimensional string. Recall that c = τ 0 0 . So λ 0 = τ 0 /c 2 . 2. Let’s assume that the wave can be expressed in terms of two travelling waves, namely η ( x, t ) = f 1 ( x - c t ) + f 2 ( x + c t ) . Let the right-travelling wave f 1 have boundary condition f 1 ( x = 0) = 3 4 A . Find the equation

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