In a dispersive medium, the group velocity is:

In a dispersive medium, the group velocity is less than the phase velocity only.

Derivation:

Phase velocity is defined as:

\({V_p} = \frac{\omega }{\beta }\)

β is the phase constant defined as:

\(\beta = \sqrt {{\omega ^2}\mu \epsilon - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} \)

\({V_p} = \frac{\omega }{{\sqrt {{\omega ^2}\mu \epsilon - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} }}\)

\({V_p} = \frac{1}{{\sqrt {\mu \epsilon - {{\left( {\frac{{m\pi }}{{a\omega }}} \right)}^2}} }}\)

\({V_p} = \frac{{\frac{1}{{\sqrt {\mu \epsilon } }}}}{{\sqrt {1 - {{\left( {\frac{{m\pi }}{{a\omega \sqrt {\mu \epsilon} }}} \right)}^2}} }}\)

Using \(c = \frac{1}{{\sqrt {\mu C} }}\) where c =  speed of light, the above expression becomes:

\({V_p} = \frac{c}{{\sqrt {1 - {{\left( {\frac{{m\pi C}}{{a\omega }}} \right)}^2}} }};\)

Also \({\omega _c} = \frac{{m\pi c}}{a}\)

\({V_p} = \frac{C}{{\sqrt {1 - {{\left( {\frac{{{\omega _c}}}{\omega }} \right)}^2}} }}\)

Using \(\sin \theta = \frac{{{\omega _c}}}{\omega }\), we get:

\({V_p} = \frac{c}{{\sqrt {1 - {{\sin }^2}\theta } }}\)

\({V_p} = \frac{c}{{\cos \theta }};\)

Since -1 ≤ cos θ ≤ 1

∴ Vp > c

Group velocity is given by:

\({V_g} = \frac{{d\omega }}{{d\beta }}\)

\(\beta = \sqrt {{\omega ^2}\mu \epsilon - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} \)

\(\frac{{d\beta }}{{d\omega }} = \frac{{2\omega \mu\epsilon }}{{2\sqrt {{\omega ^2}\mu\epsilon - {{\left( {\frac{{m\pi }}{a}} \right)}^2}} }}\)

\(\frac{{d\beta }}{{d\omega }} = \frac{{\sqrt {\mu\epsilon } }}{{\sqrt {1 - {{\left( {\frac{{m\pi }}{{a\omega \sqrt {\mu\epsilon } }}} \right)}^2}} }}\)

\(\frac{{d\beta }}{{d\omega }} = \frac{1}{{C\sqrt {1 - {{\left( {\frac{{{\omega _C}}}{\omega }} \right)}^2}} }}\)

\({V_g} = c\sqrt {1 - {{\left( {\frac{{{\omega _C}}}{\omega }} \right)}^2}} \)

Vg = c cos θ

Vg < c

Conclusion: 

The phase velocity is always greater than the speed of light and group velocity is always less than the speed of light. Hence, the group velocity is less than the phase velocity 

Extra Information:

For a Non-dispersive medium:

A one-dimensional wave defined as:

U(x, t) = A0 sin (ωt – kx + ϕ) has a phase angle (θ) of ωt – kx + ϕ

In general, the phase is constant,

i.e. \(\frac{{d\theta }}{{dt}} = \frac{\omega }{k} = {v_p}\;\left( {phase\;velocity} \right)\)

Group velocity is defined as:

 \({V_{group}} = \frac{{{\omega _2} - {\omega _1}}}{{{k_2} - {k_1}}} = \frac{{d\omega }}{{dk}}\).

Dispersion is when the distinct phase velocities of the components of the envelope cause the wave packet to “Spread out” over time.

When there is no dispersion derivative term is 0 and

 Vp = Vg