Rectangular electromagnetic waveguide

A rectangular electromagnetic waveguide has a vacuum interior bounded by four perfect conductor walls parallel to its axis. The harmonic fields obey zero tangential electric field and zero normal magnetic field at the walls. Maxwell equations determine transverse fields from longitudinal components; a purely longitudinal oscillating magnetic field by itself cannot propagate, since its divergence would be nonzero.

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.

For a transverse electric mode, $E_z=0$ and $B_z=B_0(x,y)e^{i(kz-\omega t)}$. Put $\Delta =(\omega /c{)}^2-k^2$. Maxwell equations give $B_x=ik{\partial }_x{B}_z/\Delta$, $B_y=ik{\partial }_y{B}_z/\Delta$, $E_x=i\omega {\partial }_y{B}_z/\Delta$, and $E_y=-i\omega {\partial }_x{B}_z/\Delta$. The wall conditions become Neumann boundary conditions for $B_0$, whose separated solutions are $\cos (m\pi x/a)\cos (n\pi y/b)$. The constant $(0,0)$ candidate is excluded by Faraday's equation because its corresponding electric field vanishes.