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,
Now comes the relatively tricky part. Recall the boundary conditions for
a perfect 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
We therefore have two possibilities for non-zero or that can act as source term in the mutilated Maxwell Equations.