Phase velocity

The group velocity of a collection of waves is defined as

${\displaystyle v_{g}={\frac {\partial \omega }{\partial k}}.}$ 

When multiple sinusoidal waves are propagating together, the resultant superposition of the waves can result in an "envelope" wave as well as a "carrier" wave that lies inside the envelope. This commonly appears in wireless communication when modulation (a change in amplitude and/or phase) is employed to send data. To gain some intuition for this definition, we consider a superposition of (cosine) waves f(x, t) with their respective angular frequencies and wavevectors.

${\displaystyle {\begin{aligned}f(x,t)&=\cos(k_{1}x-\omega _{1}t)+\cos(k_{2}x-\omega _{2}t)\\&=2\cos \left({\frac {(k_{2}-k_{1})x-(\omega _{2}-\omega _{1})t}{2}}\right)\cos \left({\frac {(k_{2}+k_{1})x-(\omega _{2}+\omega _{1})t}{2}}\right)\\&=2f_{1}(x,t)f_{2}(x,t).\end{aligned}}}$ 

So, we have a product of two waves: an envelope wave formed by f₁ and a carrier wave formed by f₂ . We call the velocity of the envelope wave the group velocity. We see that the phase velocityof f₁ is

${\displaystyle {\frac {\omega _{2}-\omega _{1}}{k_{2}-k_{1}}}.}$ 

In the continuous differential case, this becomes the definition of the group velocity.

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

${\displaystyle k={\frac {1}{c}}\omega n(\omega )\implies dk={\frac {1}{c}}\left(n(\omega )+\omega {\frac {\partial }{\partial \omega }}n(\omega )\right)d\omega .}$ 

We can then rearrange the above to obtain

${\displaystyle v_{g}={\frac {\partial w}{\partial k}}={\frac {c}{n+\omega {\frac {\partial n}{\partial \omega }}}}.}$ 

From this formula, we see that the group velocity is equal to the phase velocity only when the refractive index is independent of frequency ${\textstyle \partial n/\partial \omega =0}$ . 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.