Phase velocity

Beats

The previous definition of phase velocity has been demonstrated for an isolated wave. However, such a definition can be extended to a beat of waves, or to a signal composed of multiple waves. For this it is necessary to mathematically write the beat or signal as a low frequency envelope multiplying a carrier. Thus the phase velocity of the carrier determines the phase velocity of the wave set.

Dispersion

In the context of electromagnetics and optics, the frequency is some function 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_p is known as the refractive index, n = c / v_p = ck / ω.

In this way, we can obtain another form for group velocity for electromagnetics. Writing n = n(ω), a quick way to derive this form is to observe $${\displaystyle k=\frac{1}{c}\omega n(\omega )⟹dk=\frac{1}{c}\left(n(\omega )+\omega \frac{\mathrm{\partial }}{\mathrm{\partial }\omega }n(\omega )\right)d\omega .}$$ We can then rearrange the above to obtain $${\displaystyle v_g=\frac{\mathrm{\partial }w}{\mathrm{\partial }k}=\frac{c}{n+\omega \frac{\mathrm{\partial }n}{\mathrm{\partial }\omega }}.}$$

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