Summary of TE/TM waves

The transverse wave equation and boundary condition (dirichlet or neumann) are an eigenvalue problem. We can see two things right away. First of all:

$$\mu\epsilon\omega^2 \ge k^2$$ (10.70)

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,

$$v_p^2 = \frac{\omega^2}{k^2} \ge \frac{1}{\mu\epsilon} = v^2$$ (10.71)

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 $k_n = k_0 n$ for $n = 1,2,3...n_{\rm cutoff}$ will permit the boundary conditions to be solved, and we'll learn some important things about the propagating solutions at the same time.