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