Simple question: How does the following wave look like? $$ U(x,y,z,t) = U_0e^{-\frac{x^2+y^2}{2\sigma^2}}e^{ikz}e^{-i\omega t} $$ Is it a plane wave? How does this propagate in space? What is the wavenumber vector($\vec{k}$)? Do the spatial frequencies $k_x,k_y$ of the gaussian profile contribute to $\vec{k}$?
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1$\begingroup$ This is not even an exact solution of the wave equation, I am afraid. $\endgroup$– CuriousOneCommented Jan 31, 2016 at 19:29
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Assuming (sigma)^2 is >>wavelength, then is is a superposition of several plane waves. Essentially the spatial frequency spectrum is the fourier transform of the gaussian term. So, the spectrum would be centered at "k," but with a width ~sigma.
The smaller sigma, the larger the spread in wave vectors. Similarly if sigma were very large, such that the amplitude of U were nearly uniform, then the range of wave vectors will be small, and very nearly a single plane wave.
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$\begingroup$ Per values of $z$, what does it look like? $\endgroup$– JonTrav1Commented Jan 31, 2016 at 19:31
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$\begingroup$ It doesn't matter how large one makes $\sigma$, the wave packet will always spread out, so even in approximation it's only good in a small volume. I have a hunch that it's actually only a good approximation for a volume smaller than $\sigma$, which kind of defeats the purpose somewhat. At the very least one should model the phase modulation as a function of cylinder radius, I think. $\endgroup$ Commented Jan 31, 2016 at 19:46