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A square pulse of light has the form f(x) = A exp(ik_{0}x) for |x| < L/2

and f(x) = 0 everywhere else.

I want to know how that pulse evolves overtime. I want to graph it in Mathematica.

I did a fourier transform to find the wave number spectrum, with the general form:

F(k) = Sin((k - k0) L/2) / (k - k0)

this is a sinusoid that decays inversely, centered on k0. I figure that this wave packet will disperse over time, due to the different wave number components.

my question is "how do I calculate the time evolution of this pulse?

my attempted answer (thanks to Lewis A. Riley): calculate the inverse transformation, with the addition of a factor to account for the time progression of the wave:

y(x,t) = Integral[ { sin((k-k0) L/2) / (k-k0) } * exp (ik {x - vt}) ] dk

this is not a trivial calculation (I'm using Mathematica) and it returns an answer in the form of the exponential integral. I'm not sure how to make sense of this. I expect I should be able to graph a real function here, since we are talking about the evolution of real waves. I don't know how to do that.

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# Time Evolution of a Square Pulse - Fourier

Can you offer guidance or do you also need help?

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