Evanescent transverse electric mode in a rectangular waveguide (${\kappa }^2=(\pi /a{)}^2-{ϵ}_r{\omega }^2/c^2$)

Below the lowest waveguide cutoff frequency, the axial wave number of the ${\mathrm{TE}}_{10}$ mode is $i\kappa$, with $\kappa >0$. For a real peak scale $E_0$ and time convention $e^{-i\omega t}$, $E_y=E_0\sin (\pi x/a)e^{-\kappa z}$, $B_x=-i\kappa E_y/\omega$ and $B_z=-i{E}_0(\pi /a)\cos (\pi x/a)e^{-\kappa z}/\omega$. The electric field and magnetic field are in quadrature. A single decaying mode in a lossless semi-infinite guide consequently has zero time-averaged Poynting vector, although its instantaneous field stores energy.