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:
_&perp#perp;·
_&perp#perp;& = & 0

_&perp#perp;×E_&perp#perp;& = & 0
from which we can immediately see that:
&nabla#nabla;_&perp#perp;^2
_&perp#perp;= 0
and that
_&perp#perp;= -&phis#phi;
for some suitable potential that satisfies
.
The solution looks like a propagating *electrostatic* wave. From
the wave equation we see that:
&mu#mu;&epsi#epsilon;&omega#omega;^2 = k^2
or
k = ±&omega#omega;&mu#mu;&epsi#epsilon;
which is just like a plane wave (which can propagate in either
direction, recall).

Again referring to our list of mutilated Maxwell equations above, we see
that:
ik
_&perp#perp;& = & -i&omega#omega;(
×
_&perp#perp;)

_&perp#perp;& = & -&omega#omega;&mu#mu;&epsi#epsilon;k (
×
_&perp#perp;)

_&perp#perp;& = & ±&mu#mu;&epsi#epsilon; (
×
_&perp#perp;)
or working the other way, that:
_&perp#perp;= ±&mu#mu;&epsi#epsilon; (
×
_&perp#perp;)
so we can easily find one from the other.

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.