Waveguide cutoff frequency (${\omega }_{mn}=c\pi \sqrt{m^2/a^2+n^2/b^2}$)

For a transverse electric mode of a rectangular waveguide, the dispersion relation is ${\omega }^2=c^2(k^2+{\kappa }_{mn}^2)$, with ${\kappa }_{mn}^2=(m\pi /a{)}^2+(n\pi /b{)}^2$. Below ${\omega }_{mn}=c{\kappa }_{mn}$, the longitudinal wave number is imaginary and the mode is evanescent. If $a\ge b$, the lowest propagating cutoff is ${\omega }_{10}=\pi c/a$, or ordinary frequency $c/(2a)$.

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 ${\omega }^2=c^2{k}^2+{\omega }_c^2$ 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.