Phase velocity v­_p, and group velocity v_g in a waveguide are related to the velocity of light c as:

Phase velocity:

It is the rate at which the phase of the wave propagates in space. Mathematically:

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

ω_c = Cut off frequency

ω = Frequency of operation

Group velocity:

It is the velocity at which the overall envelope of the wave travels.

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

Multiplying both velocities, we get:

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

\({V_p}{V_g} = {c^2}\)