Editing Phase velocity (section)

== Refractive index ==

In the context of electromagnetics and optics, the frequency is some function {{math|''ω''(''k'')}} of the wave number, so in general, the phase velocity and the group velocity depend on specific medium and frequency. The ratio between the speed of light ''c'' and the phase velocity ''v''<sub>''p''</sub> is known as the [[refractive index]], {{math|''n'' {{=}} ''c'' / ''v''<sub>''p''</sub> {{=}} ''ck'' / ''ω''}}.

In this way, we can obtain another form for group velocity for electromagnetics. Writing {{math| ''n'' {{=}} ''n''(ω)}}, a quick way to derive this form is to observe
:<math> k = \frac{1}{c}\omega n(\omega) \implies dk = \frac{1}{c}\left(n(\omega) + \omega \frac{\partial}{\partial \omega}n(\omega)\right)d\omega.</math>
We can then rearrange the above to obtain 
:<math> v_g = \frac{\partial w}{\partial k} = \frac{c}{n+\omega\frac{\partial n}{\partial \omega}}.</math>

From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is independent of frequency <math display=inline>\partial n / \partial\omega = 0</math>. When this occurs, the medium is called non-dispersive, as opposed to [[dispersion (optics)|dispersive]], where various properties of the medium depend on the frequency {{mvar|ω}}. The relation <math>\omega(k)</math> is known as the [[dispersion relation]] of the medium.