Let consider the simple problem of :
1D electromagnetic cavity with perfect conducting walls filled with a conducting medium
Let the length of the cavity be L, the conductivity be $\sigma$ and dielectric be $\epsilon$.
General solution
From Maxwell equations the solution of the wave equation inside this medium is :
$E_z=Ae^{-jk_{m}y}+Be^{+jk_{m}y} $
With $k_{m}$ given by the complex dispersion relation : $k_{m}=\frac{\omega}{c}\bar{n_{m}}=\beta-j\alpha $
Whose complex refractive index(which can easily be expanded) is : $\bar{n_{m}}=\sqrt{1+j\frac{\sigma}{\epsilon\omega}}$
Continuity at the PEC boundary $E_z(0,L)=0$
We consequently have the 2 sets of equations :
$A+B=0$
$Ae^{-j\bar{k_{m}L}}+Be^{+j\bar{k_{m}L}}=0 $
Since $k_m$ is complex how do you manage to find the resonant frequencies along with their corresponding mode shapes???
I have tried manipulation but when you look at the amplitude of the complex sinh on a graph, it only passes by 0 once. What am I missing??