Scalar-potential conducting waveguide modes (${\gamma }^2={\omega }^2/c^2-k^2$)

For dependence $e^{i(kz-\omega t)}$, a Helmholtz equation $({\Delta }_{\perp }+{\gamma }^2)\psi =0$ generates transverse-magnetic fields $e_{\perp }={\nabla }_{\perp }\psi$, $e_z=-i{\gamma }^2\psi /k$, $b_{\perp }=\omega \hat{z}\times {\nabla }_{\perp }\psi /(k{c}^2)$, $b_z=0$, for $k\ne 0$. The conducting wall requires a Dirichlet boundary condition on $\psi$. A transverse-electric representation is $e_{\perp }=\hat{z}\times {\nabla }_{\perp }\psi$, $e_z=0$, $b_{\perp }=-k{\nabla }_{\perp }\psi /\omega$, $b_z=i{\gamma }^2\psi /\omega$, with a Neumann boundary condition. These are families of modes, not a completeness assertion for multiply connected guides and their possible TEM fields.