Field Interactions and Power Conservation at Lossy Interfaces Under Normal Incidence

The conservation of local complex power density and global complex power flow at interfaces between lossy media is not inherently guaranteed within classical plane-wave treatments, which often neglect the contribution of cross-field interaction terms. An analytical investigation is conducted for a normally incident time-harmonic plane wave impinging on a planar half-space interface, wherein the omission of such interaction terms is shown to preserve conservation in the lossless case but to induce significant violations in lossy configurations. The precise parametric conditions under which these violations occur are derived. A complete formulation is then constructed, incorporating the requisite cross terms to restore both local and global conservation laws. Verification is achieved through consistency between global complex power flow—obtained via surface integrals of the corrected local power density—and volumetric field–material interaction integrals. A parametric analysis over a broad frequency range (300 MHz–300 GHz) and varying permittivity and conductivity reveals that the error resulting from neglecting interaction terms is highly localized, typically confined within a quarter-wavelength of the interface. The results delineate the exact scenarios necessitating inclusion of interaction terms and establish a rigorously balanced power framework for modeling wave behavior at lossy media boundaries.