Now we can start looking at waveforms in various cavities. Suppose we let . Then the wave in the cavity is a pure transverse electromagnetic (TEM) wave just like a plane wave, except that it has to satisfy the boundary conditions of a perfect conductor at the cavity boundary!
Note from the equations above that:
Again referring to our list of mutilated Maxwell equations above, we see
TEM waves cannot be sustained in a cylinder because the surrounding (perfect, recall) conductor is equipotential. Therefore is zero as is . However, they are the dominant way energy is transmitted down a coaxial cable, where a potential difference is maintained between the central conductor and the coaxial sheathe. In this case the fields are very simple, as the is purely radial and the field circles the conductor (so the energy goes which way?) with no components.
Finally, note that all frequencies are permitted for a TEM wave. It is not ``quantized'' by the appearance of eigenvalues due to a constraining boundary value problem.