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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.

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They come from the fact that you consider your wall as perfect electric conductor(PEC) inside which the E field has to be zero. This is a property of PEC materials which are an ideal case of metals.

In general if the surrounding medium is air as for fiber optic the conditions have to be the standard continuity relations of the E and H fields at a boundary between 2 media.

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