Time Evolution of a Square Pulse - Fourier

[skynelson]

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.

 

[DrClaude]

The approach is basically correct. However, I am not sure this can be accomplished easily. I suggest instead of discretizing space and using a FFT to go to reciprocal space, where the time-dependent phase factor can be applied for the desired time, and doing an inverse FFT to recover the pulse shape. This can then be repeated for different times to visualize the evolution.