Waveguide boundary conditions

I am calculating useful band width for the case of half circular waveguide with radius a, filed with plasma with $$\epsilon=1-\omega_p^2/\omega^2$$, booth for transferse electric (TE), and transferse magnetic (TE) mode. I know that the solution are Bessel's functions, but I am unable to determine boundary conditions for booth modes.

In our exercise, we determined z component of $$\textbf{E}$$ is zero ($$E_z=0$$) on border of waveguide for TM mode, and that the derivative of z component of $$\textbf{H}$$ is zero on the border $$(\frac{\partial}{\partial \perp}H_z=0)$$, where $$\partial \perp$$ is derivative on a path perpendicular to the border.

I don't know where this boundary conditions come from, and if they are general for all waveguides (empty and filed with dielectric) or how to determine boundary conditions for waveguides in general.