Waveguide phase and group velocities
= Waveguide phase and group velocities
{title2=$v_{\mathrm{ph}}v_g=c^2$}
Above a mode's <waveguide cutoff frequency> $\omega_c$, its <phase velocity> is $c/[1-(\omega_c/\omega)^2]^{1/2}$ and its <group velocity> is $c[1-(\omega_c/\omega)^2]^{1/2}$. Differentiate $\omega^2=c^2k^2+\omega_c^2$ to obtain the latter. Phase travels faster than $c$ while the wave packet travels slower, so the phase speed does not imply superluminal information transport.