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I don't understand how the spectral phase (the phase in the frequency domain) describes dispersion.

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For example, if you measure the amplitude of a wave in point A you can find that the signal there has the form:

$s_a(t) = sin(\omega t)+sin(2\omega t)+sin(3\omega t)$

If you do the same measurement in point B you can get:

$s_b(t) = sin(\omega t)+sin(2\omega t+2\pi/8)+sin(3\omega t+3\pi/8)$

The last two sinus are shifted with $2\pi/8$ and $3\pi/8$. This shift with various phases is called dispersion.

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