part and the 
part,
work on each separately, and recombine them. This can be cumbersome.
Jackson works out in detail the fields in a waveguide, then uses the
waveguide solutions to work out the fields in a cavity. His procedure is to
break up the cavity fields into the part and the part,
work on each separately, and recombine them. This can be cumbersome.
One can obtain formulas that work directly for fields proportional to any combination of sinusoidal or exponential functions. The result shown here is in cartesian coordinates, and thus it is especially suited for box-shaped cavities.
Start from Maxwell's equations written out explicitly:
Assuming only that the each field component satisfies 
we can obtain all the other field components from the
``superpotentials'' 
and 
To derive (7), for example, all we need is Eq. (1) and the z derivative of (5):
Eliminate 
from these two equations and replace 
with 
.
Jackson is rather vague on this point. For a cavity with perfectly
conducting walls and end walls perpendicular to the z axis:
on the side walls and 
on the end walls
on the side walls and 
on the end walls
To derive these boundary conditions, we note first that 
must be inversely proportional to the conductivity, otherwise the current
would be infinite. This implies that on the side walls 
and on the
end walls 
and their lateral derivatives also vanish. Then we
can use 
and 
in the divergence equation
to conclude that 
on the end walls
We can also use 
and 
in the equation 
from 
to conclude that 
on the end walls. In general
the normal component 
vanishes on a wall, and so do its lateral
derivatives.
Finally, we show that the normal derivative 
vanishes on the
side walls. Consider a wall element and orient the x axis normal to it.
Use 
from 
and
note that 
because it is a parallel component of 
and
that 
because it is a lateral derivative of the normal
component of 
It follows that 
. In general, 
on a wall.
