The transverse wave equation and boundary condition (dirichlet or
neumann) are an *eigenvalue problem*. We can see two things right
away. First of all:
&mu#mu;&epsi#epsilon;&omega#omega;^2 &ge#ge;k^2
or we no longer have a wave, we have an exponential function that cannot
be made to satisfy the boundary conditions on the entire surface.
Alternatively,
v_p^2 = &omega#omega;^2k^2 &ge#ge;1&mu#mu;&epsi#epsilon; = v^2
which has the lovely property (as a phase velocity) of being faster than
the speed of light in the medium!

To proceed further in our understanding, we need to look at an actual example - we'll find that only certain for will permit the boundary conditions to be solved, and we'll learn some important things about the propagating solutions at the same time.

Robert G. Brown 2017-07-11