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Time Evolution of a Square Pulse - Fourier

  1. Oct 21, 2008 #1
    Hi There,
    A square pulse of light has the form f(x) = A exp(ik0x) 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.
     
  2. jcsd
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