Note well that we have written the mutilated Maxwell Equations so that the components are all on the right hand side. If they are known functions, and if the only dependence is the complex exponential (so we can do all the derivatives and just bring down a ) then the transverse components and are determined!
In fact (for propagation in the
direction,
):
ik
_&perp#perp;+ i&omega#omega;(
×
_&perp#perp;) & = & _&perp#perp;E_z
ik(
×
_&perp#perp;) +
i&omega#omega;
×(
×
_&perp#perp;) & = &
×_&perp#perp;E_z
ik(
×
_&perp#perp;) & = &
i&omega#omega;
_&perp#perp;+
×_&perp#perp;E_z
·(
_&perp#perp;×
_&perp#perp;) & = &
i&omega#omega;&mu#mu;&epsi#epsilon;E_z
and
ik
_&perp#perp; i&omega#omega;&mu#mu;&epsi#epsilon;(
×
_&perp#perp;) & = & _&perp#perp;
B_z
ik
_&perp#perp; _&perp#perp;B_z & = & i&omega#omega;&mu#mu;&epsi#epsilon;
(
×
_&perp#perp;)
ik^2&omega#omega;&mu#mu;&epsi#epsilon;
_&perp#perp;
k&omega#omega;&mu#mu;&epsi#epsilon;_&perp#perp;B_z & = &
ik (
×
_&perp#perp;)
ik^2&omega#omega;&mu#mu;&epsi#epsilon;
_&perp#perp;
k&omega#omega;&mu#mu;&epsi#epsilon;_&perp#perp;B_z & = &
i&omega#omega;
_&perp#perp;+
×_&perp#perp;E_z
or
_&perp#perp;& = & i&mu#mu;&epsi#epsilon;&omega#omega;^2  k^2( k_&perp#perp;
B_z + &mu#mu;&epsi#epsilon;&omega#omega;(
×_&perp#perp;E_z))
_&perp#perp;& = & i&mu#mu;&epsi#epsilon;&omega#omega;^2  k^2( k_&perp#perp;
E_z  &omega#omega;(
×_&perp#perp;B_z))
(where we started with the second equation and eliminated
to get the second equation just like the
first).
Now comes the relatively tricky part. Recall the boundary conditions for
a perfect conductor:
×(

_c) =
×
& = & 0
·(

_c) =
·
& = & 0
×
& = &
·
& = & &Sigma#Sigma;
They tell us basically that
(
) is strictly
perpendicular to the surface and that
(
) is strictly
parallel to the surface of the conductor at the surface of the
conductor.
This means that it is not necessary for or both to vanish everywhere inside the dielectric (although both can, of course, and result in a TEM wave or no wave at all). All that is strictly required by the boundary conditions is for E_z_S = 0 on the conducting surface (it can only have a normal component so the component must vanish). The condition on is even weaker. It must lie parallel to the surface and be continuous across the surface (where can discontinuously change because of ). That is: B_zn_S = 0
We therefore have two possibilities for nonzero or that can act as source term in the mutilated Maxwell Equations.