Geometric Wave Engineering: Mode-Resolved Poynting-Flow Redistribution in an Axisymmetric Pseudohyperboloidal PEC Cavity — A Preliminary Full-Vector Cylindrical-FDTD Comparison at PHB-Selected Frequencies

Geometric Wave Engineering investigates how the geometry of a physical boundary can reorganise electromagnetic fields and energy-flow channels. This preliminary study compares an author-defined open pseudohyperboloidal, or PHB, perfect-electric-conductor cavity with a matched straight-generatrix reference using six full-vector cylindrical finite-difference time-domain calculations. Three prescribed excitation cases are considered for azimuthal orders zero, one and two, each at a corresponding frequency selected from the PHB modal search. Within each case, the same PHB-selected frequency is applied to both geometries. Because the frequency changes between the three azimuthal sectors, the cross-sector comparison combines the effects of azimuthal order and frequency and is not a pure sweep of azimuthal order alone. The straight-generatrix reference also does not isolate negative Gaussian curvature from simultaneous changes in surface normals, optical paths, cavity volume and spectral response. The results therefore establish geometry-associated differences between the complete PHB and straight-generatrix configurations rather than a causal effect of curvature alone. For azimuthal order zero at frequency f0, the PHB calculation exhibits 2.07 times the total safe-vacuum energy of the reference, while the raw annular-monitor flux is reduced to 0.436 of the reference value. For azimuthal order one at frequency f1, the fixed R3-side analysis window shows a pronounced local axial-component sign inversion. The axial sign index is positive 0.983 for PHB and negative 0.907 for the straight reference. In the same excitation case, the raw M1 annular flux is 2.84 times as large in PHB, despite its total energy being only 29.9 percent of the reference value. For azimuthal order two at frequency f2, the broad internal annular region shows modest enrichment, whereas the pre-exit component magnitudes are predominantly radial. The downstream-plane flux is 1.90 times the reference value, but 93.97 percent of the closed-boundary PHB outflow remains radial. The enhanced downstream signal therefore represents a secondary sampled channel rather than a collimated far-field beam. Accepted source power was not recorded. Consequently, the reported energy and flux ratios are unnormalised forced-response quantities and not conversion efficiencies. All signatures were obtained at resolution 96 using one source ensemble, one random seed and one PML configuration. The defensible result is a preliminary numerical observation of mode-resolved forced-response redistribution associated with the complete PHB generatrix under the specified protocol. Convergence studies, source-power normalisation, frequency sweeps, independent modal optimisation, PML testing, near-to-far-field transformation and experimental verification remain necessary.