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Halos are not exclusive to ice crystals, and rainbow colored halos are commonly reported, they usually indicate some issue with eye's focusing. One possibility is a growing cataract:"Seeing rainbows or halos around light indicates a problem with how light is filtering into the eye. Light is made up of different colours but the rays are normally focused on a ...

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For this solution (which was not mentioned) is is assumed that the glass is isotropic. Then according to (for example) Sutherland we have the three room dimensions $x$, $y$ and $z$ for the incoming electrical field. Now due to isotropy the 81-element-matrix only has 21 non-zero elements with $$\chi_{XXXX}=\chi_{YYYY}=\chi_{ZZZZ}$$ ...

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You have a good question. The only answer that I can come up with at the moment is not very satisfying. I'll keep your question in the back of my head for a few days and update my answer if I come up with something. In the mean time, you can use dimensional analysis! You are interested in the difference in between the phase and group velocities, ...

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Note that $k>0$ so in shallow water $\omega \approx \sqrt{gh}k$. These are pure surface waves whose amplitudes are much smaller than the depth, and is also assumed that no other waves propagate, thus reflections from the bottom are completely ignored.

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So to your first question, when you calculate the time-evolution of a wavepacket, you first transform your wavepacket representation from a positional basis to a momentum basis. $$|\psi\rangle = \sum_k c_{x,p}|p\rangle$$ This moment basis is useful as they are eigenstates to your hamiltonian and satisfy $$H |p\rangle = E_p |p\rangle$$ This in turn allows ...

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