Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM

Let the laboratory region $U\cong I\times \mathrm{\Omega }$, its frame basis $(e_1,e_2,e_3)$, its SI units and its fields $E,B$ be those of Laboratory geometry: a restricted inertial chart, its spatial slices and fixed-time fields, Framework-local SI units, the vacuum constants and the exact/measured status of their values and The electric and magnetic fields as fixed-time spatial vector fields. Fix $a,b>0$, write $x_t=(x_1,x_2)$, $e_z:=e_3$, and define ${\mathrm{\nabla }}_{t}f=({\mathrm{\partial }}_{1}f,{\mathrm{\partial }}_{2}f,0)$, ${\mathrm{\Delta }}_{t}f={\mathrm{\partial }}_1^{2}f+{\mathrm{\partial }}_2^{2}f$ and ${\mathrm{div}}_t{u}_t={\mathrm{\partial }}_1{u}_1+{\mathrm{\partial }}_2{u}_2$ for the stated regularity, using Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$, Divergence and curl of a $C^1$ vector field and The Laplacian of a $C^2$ function and of a $C^2$ vector field. (a) [Straight guide] A straight guide region is ${\mathrm{\Omega }}_A:=A\times \mathbb{R}$ where $A$ is either a bounded connected $C^1$ plane domain (Bounded C^k domains and boundary charts) or the explicit rectangular cross-section; the rectangle has corners and is treated by its open faces, not asserted to be a $C^1$ boundary domain, with longitudinal axis $e_z$; for a rectangular cross-section $A=R_{ab}:=(0,a)\times (0,b)$ the region is called the rectangular guide and its walls are the four open faces $\{x_1=0\},\{x_1=a\},\{x_2=0\},\{x_2=b\}$. A rectangular cavity region is ${\mathrm{\Omega }}_c:=(0,a)\times (0,b)\times (0,d)$ for fixed $d>0$, with the six open faces as its walls. The separate smooth-cavity spectral branch uses an explicitly declared bounded $C^{\mathrm{\infty }}$ domain in ${\mathbb{R}}^3$ with its smooth PEC boundary; it is not the rectangular reflection class. Spatial derivatives are taken at fixed $t$; the fields are defined on $\mathrm{\Omega }\times I$ with the regularity declared by each theorem. (b) [PEC boundary model] On every open wall (a wall minus its edges) the ideal perfectly conducting boundary model declares the two conditions $n\times E=0$ and $n\cdot B=0$, where $n$ is the outward unit normal of that wall ($-e_1,+e_1,-e_2,+e_2$ on the four rectangular walls, and $\pm e_j$ on the six cavity walls). Its physical reading uses the stationary interface conditions of Stationary macroscopic interface jump conditions for D, B, E and H with $E=B=0$ on the conductor side: tangential-$E$ and normal-$B$ continuity then give the two filling-side conditions. For that reading write $D_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}},H_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}}$ and $D_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}},H_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}$ for the filling- and conductor-side traces of electric displacement (in $\mathrm{C}{\mathrm{m}}^{-2}$) and magnetic intensity (in $\mathrm{A}{\mathrm{m}}^{-1}$). Since $n$ points from the filling into the conductor, the general sheet readings are ${\sigma }_f=n\cdot (D_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}-D_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}})$ and $K_f=n\times (H_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}-H_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}})$, in $\mathrm{C}{\mathrm{m}}^{-2}$ and $\mathrm{A}{\mathrm{m}}^{-1}$ respectively. Only with the additional conductor-side assumptions $D_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}=H_{\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{d}}=0$ do they reduce to ${\sigma }_f=-n\cdot D_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}}$ and $K_f=-n\times H_{\mathrm{f}\mathrm{i}\mathrm{l}\mathrm{l}}$; $E=B=0$ alone does not imply those additional assumptions. No sheet density or conductor constitutive relation is prescribed here. The supplier assumes ${\mathrm{A}\mathrm{C}}_{\omega }$, a compact embedded $C^2$ stationary hypersurface, joint one-sided spacetime $C^1$ extensions and distributional macroscopic equations with the stated thin sheet sources for its pointwise branch; it covers that smooth interface model, not a deduction of the $\sigma \to \mathrm{\infty }$ idealization nor the box edges. The guide/cavity conditions on the open walls are therefore declared as an idealization of this definition, and no pointwise statement on the edges is made. (c) [Zero-normal-B reflection class] For the rectangular box $Q={\prod }_{j=1}^3(0,L_j)$ with $L=(a,b,d)$, the reflection class is the class of locally integrable fields $E,B$ on $Q$ whose extensions to the rectangular torus ${\prod }_j\mathbb{R}\mathrm{/}(2L_j\mathbb{Z})$, defined by extending $E_1$ evenly across the $x_1$-walls and oddly across the $x_2$-, $x_3$-walls and cyclically for $E_2,E_3$, and extending $B_1$ oddly across $x_1$ and evenly across $x_2,x_3$ and cyclically for $B_2,B_3$, are $L^2$ fields, together with the distributional constraints that $\mathrm{div}E$ and $\mathrm{div}B$ vanish on the torus for the extensions. This is the declared weak PEC model used for the cavity initial-value problem; when the reflected extensions are continuous across the walls, their odd components vanish there, giving $n\times E=0$ and $n\cdot B=0$ on the open walls. Smoothness only inside $Q$ does not imply this extension continuity. Initial data not in this class (for example data with trapped normal $B$) are excluded, as are the general $H(\mathrm{curl})$ boundary classes, which are defined by the mathematical suppliers and are not identified here with the classical conditions. (d) [TE, TM, TEM] For a guide or cavity field $(E,B)$, the longitudinal components are $E_z:=E\cdot e_z$ and $B_z:=B\cdot e_z$; the field is called TE (transverse electric) if $E_z\equiv 0$, TM (transverse magnetic) if $B_z\equiv 0$, and TEM if both vanish, always relative to the named axis $e_z$. (e) [Separated phasor convention] For $\omega >0$ and $\beta \in \mathbb{C}$, a $\beta$-$\omega$ separated phasor field on the guide is a real field of the form $F(x,t)=\mathrm{Re}(\tilde{F}(x_1,x_2)e^{i(\beta x_3-\omega t)})$ with a complex amplitude $\tilde{F}$, componentwise; $\omega \mathrm{/}(2\pi )$ is the frequency and $\beta$ the longitudinal propagation constant; when $\beta$ is real, $\beta >0$ describes propagation in the $+e_z$ direction under this phase convention, while a complex $\beta$ gives the spatial factor $e^{-\mathrm{Im}\beta x_3}$ and may grow in the opposite direction; each consuming branch states its selection. (f) [Scope] The definition fixes the geometry, the PEC boundary model, the weak reflection class and the mode terminology; it asserts no field equation, no constitutive law and no existence or completeness of modes, Rectangular TE/TM candidates and the rectangular reflection class, declared smooth bounded PEC cavity spectral classes, and smooth straight-guide TEM potential branches (including the coaxial annulus) are within this geometry. No general non-rectangular TE/TM guide theory, circular-guide spectral classification, open-end or junction scattering, or arbitrary lossy model is supplied. The same stationary PEC geometry also hosts the explicitly declared constant scalar Ohmic cavity class and the full longitudinal finite-energy reflection class of the straight rectangular guide; constitutive and evolution claims belong to their consuming theorems, not this definition.

Conventions fixed by this definition. The definition records objects and conditions only: the guide and cavity regions, their open walls and outward normals, the ideal PEC boundary conditions with their interface reading and excluded idealization, the zero-normal-$B$ reflection class with its distributional divergence constraints, the axis-relative TE/TM/TEM terminology, and the separated phasor encoding of real fields. It asserts no field equation, no constitutive relation, no existence, uniqueness or completeness of modes, and no dissipation or energy claim; those belong to the consuming theorems, each of which restates the branch of the material model it uses. The rectangular cross-section is treated through its four open faces and is not asserted to be a $C^1$ boundary domain, and no pointwise condition on box edges is made.