Scalar-Longitudinal Modes of Extended Electrodynamics in Perfectly Conducting Waveguides and Divergence Cleaning Interpretation

Extended electrodynamics keeps the scalar combination $C=\mathrm{\nabla }\cdot \mathbf{A}+{\mu }_0{\epsilon }_0\mathrm{\partial }\mathrm{\Phi }\mathrm{/}\mathrm{\partial }t$ as a dynamical field rather than imposing the Lorenz gauge condition, and this scalar adds a scalar-longitudinal (SL) family to the TE, TM and TEM modes. With $C=0$ on the wall of a closed perfectly conducting waveguide, the classical modes are unchanged. The SL modes have $\mathbf{B}=0$ and an electric field along $\mathrm{\nabla }C$, and their cutoffs are the Dirichlet eigenvalues of the cross-section, the same as the TM cutoffs, with no coaxial TEM counterpart. Charge-conserving sources with zero scalar data do not excite them. Matching at a conductor described by its constitutive laws gives $\mathrm{\partial }C\mathrm{/}\mathrm{\partial }n=0$ instead and couples the scalar-bearing modes to the Maxwell field. The results belong to the $C=0$ problem, the condition that separates the sectors and the one divergence cleaning uses, with benchmarks for the rectangular, circular cylinder, coaxial and L-shaped guides.