circular disk cavity
From equation (7) of my post [Modes of Slab Waveguide] the scalar mode equation can be expressed by
$$
\begin{align}
\nabla^2 \psi(x,y) + n^2 k^2 \psi(x,y) = 0 \tag{1}
\end{align}
$$
by assuming $k_z \approx 0$, that propagation in the z-direction is negligible.
now with $\psi(r,\phi) = \Psi(r)e^{im\phi}$ the continuous rotational symmetry