= Scalar-potential conducting waveguide modes
{title2=$\gamma^2=\omega^2/c^2-k^2$}
For dependence $e^{i(kz-\omega t)}$, a <Helmholtz equation> $(\Delta_\perp+\gamma^2)\psi=0$ generates transverse-magnetic fields $e_\perp=\nabla_\perp\psi$, $e_z=-i\gamma^2\psi/k$, $b_\perp=\omega\,\widehat z\times\nabla_\perp\psi/(kc^2)$, $b_z=0$, for $k\ne0$. The conducting wall requires a <Dirichlet boundary condition> on $\psi$. A transverse-electric representation is $e_\perp=\widehat z\times\nabla_\perp\psi$, $e_z=0$, $b_\perp=-k\nabla_\perp\psi/\omega$, $b_z=i\gamma^2\psi/\omega$, with a <Neumann boundary condition>. These are families of modes, not a completeness assertion for multiply connected guides and their possible TEM fields.
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