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.