Complete longitudinal Fourier energy evolution in a rectangular PEC guide

[Complete mixed transform] With the characters of [F2], first form for each component the transverse coefficients $F_{mn}(z)={\int }_{T_{ab}^2}F(x_t,z)\overline{{\varphi }_{mn}(x_t)}d{x}_t$ and then ${\widehat{F}}_{mn}(\xi )={\mathcal{F}}_2{F}_{mn}(\xi )$ for the exact $2\pi$-convention transform of [F3]; write $\beta =2\pi \xi$. Then the map $F\mapsto ({\widehat{F}}_{mn}{)}_{(m,n)\in {\mathbb{Z}}^2}$ is a unitary isomorphism of $L^2(T_{ab}^2\times \mathbb{R})$ onto ${\mathrm{\ell }}^2({\mathbb{Z}}^2;L^2(\mathbb{R}))$ and, for the reflection fields of the Statement, $\mathrm{\parallel }F{\mathrm{\parallel }}_{L^2(Q)}^2=\frac{1}{4}{\sum }_{m,n}{\int }_{\mathbb{R}}\mathrm{\mid }{\widehat{F}}_{mn}(\xi ){\mathrm{\mid }}^{2}d\xi$. Indeed the characters are orthonormal on the fundamental rectangle and, being products of the complete one-dimensional character systems, their finite span is dense in the transverse $L^2$ of each torus fibre by [F2]; Fubini–Tonelli therefore gives, for almost every $z$, the transverse expansion together with ${\int }_{T_{ab}^2}\mathrm{\mid }F(x_t,z){\mathrm{\mid }}^{2}d{x}_t={\sum }_{m,n}\mathrm{\mid }{F}_{mn}(z){\mathrm{\mid }}^2$ and, after integration in $z$, $\mathrm{\parallel }F{\mathrm{\parallel }}_{L^2(T_{ab}^2\times \mathbb{R})}^2={\sum }_{m,n}\mathrm{\parallel }{F}_{mn}{\mathrm{\parallel }}_{L^2(\mathbb{R})}^2$. Applying the unitary Plancherel transform [F3] to each of the countably many scalar functions $F_{mn}\in L^2(\mathbb{R})$ gives the coefficient map and the isometry on the torus product, with inverse from the $L^2$ inversion theorem of [F3]; the reflection extension doubles each transverse period, so the torus fundamental rectangle has $4ab$-area and the physical norm is one quarter of the extended norm, which is the displayed factor $\frac{1}{4}$. The parity relations follow from ${\varphi }_{-m,-n}=\overline{{\varphi }_{mn}}$ and the transverse parities by the coordinate changes $x_j\mapsto -x_j$ on the torus, and the reality relation ${\widehat{F}}_{-m,-n}(-\xi )=\overline{{\widehat{F}}_{mn}(\xi )}$ holds first for real integrable compactly supported longitudinal data by the defining integral and passes to $L^2$ by density and continuity of the transform [F3]. No pointwise reconstruction is asserted.

[Divergence constraints in coefficient form] For $K:=(\pi m\mathrm{/}a,\pi n\mathrm{/}b,2\pi \xi )$ the distributionally zero divergence of the extended field is equivalent to $K\cdot \widehat{E}=K\cdot \widehat{B}=0$ for almost every $\xi$, and the finite-energy reflection class is exactly the class of coefficient families satisfying these almost-everywhere constraints together with the parities and $L^2$ summability. The derivative correspondence is proved on the dense test class of finite transverse-character sums times longitudinal Schwartz functions: for such a test $\psi$, finite transverse integration by parts and the exact identity ${\mathcal{F}}_2({\mathrm{\partial }}_z\psi )=2\pi i\xi \widehat{\psi }$ of [F3] turn the pairing $\int F\cdot \mathrm{\nabla }\psi$ into the sum of the coefficient pairings against $iK\cdot \widehat{F}$, and the compactly supported smooth tests used in the definition are obtained by multiplying with cutoffs ${\chi }_R$ of [F3] and passing to the limit, since the tests and their first derivatives converge in $L^2$ and all pairings are bounded by Cauchy–Schwarz [F6]; conversely a test with $K\cdot \widehat{F}=0$ almost everywhere has zero weak divergence pairing, and the general compact smooth tests are reached through the transverse $L^2$ expansions of their derivatives and the Schwartz transform isomorphism [F3]. Hence the exact constrained class corresponds to the stated almost-everywhere coefficient constraints, not merely to a scalar basis without Gauss equations; for fields smooth in the interior the divergences vanish pointwise, and if their reflected extensions are continuous across the walls then the four-wall PEC conditions hold by Rectangular PEC guide and cavity geometry, the zero-normal-B reflection class, and axis-relative TE/TM/TEM(c).

[Unique energy evolution in the class] Adopt the constant-Ohmic first-order system of [F1] distributionally in the reflection class of step 2.1. For every initial pair in the class there is a unique $C(\mathbb{R};L^2(Q))$ solution in the same class, its mixed coefficients satisfy, for every $(m,n)$ and almost every $\xi$, ${\mathrm{\partial }}_t\widehat{E}=i{v}^{2}K\times \widehat{B}-\gamma \widehat{E}$ and ${\mathrm{\partial }}_t\widehat{B}=-iK\times \widehat{E}$, and finite transverse sums with bounded longitudinal frequency truncations reconstruct the solution in energy norm uniformly on compact time intervals and are dense in the class at every time, retaining the divergence constraints and parities without dividing by $K$ or $\mathrm{\mid }K\mathrm{\mid }$. Existence: for each $(m,n,\xi )$ the real and imaginary parts of the displayed system form a real finite-dimensional constant-coefficient linear system on all of $\mathbb{R}$, hence a globally Lipschitz system with a unique global solution through the initial coefficients by [F5]; the solution matrix depends continuously on $\xi$ uniformly on compact parameter and time sets by the continuous-dependence theorem [F5] (applied on a common bounded state cylinder, which the uniform energy bound below supplies), so multiplying the measurable initial coefficient vector by the continuous fundamental matrix gives measurable coefficient trajectories [F6]; no uncountable choice of solutions is made, and the construction uses only the countable families of coefficients. Energy: with $q:=\epsilon \mathrm{\mid }\widehat{E}{\mathrm{\mid }}^2+{\mu }^{-1}\mathrm{\mid }\widehat{B}{\mathrm{\mid }}^2$, conjugation and the triple-product cancellation of the displayed system give $q^{\mathrm{\prime }}=-2\sigma \mathrm{\mid }\widehat{E}{\mathrm{\mid }}^2$, whence $q(t)\le e^{2\gamma T}q(0)$ for $\mathrm{\mid }t\mathrm{\mid }\le T$ and forward contractivity at $\sigma \ge 0$; the constraints $K\cdot \widehat{E}=K\cdot \widehat{B}=0$ and the parity and reality relations of step 1.1 are preserved because the right-hand side is linear, $K\cdot (K\times \widehat{B})=0$ and the parities are respected by the cross product with $K$. Cutoffs $\mathrm{\mid }m\mathrm{\mid },\mathrm{\mid }n\mathrm{\mid }\le N$, $\mathrm{\mid }\xi \mathrm{\mid }\le N$ preserve every relation; their initial energy tails tend to zero by Tonelli and dominated convergence [F4], and the uniform bound makes the time-dependent tails tend to zero uniformly on compact times; each truncated solution is strongly differentiable in $L^2$ with bounded multiplier $K$ on the cutoff, and its linear equations follow from the test derivative identity of step 2.1, so the compact-time $L^2$ limits are continuous, have the prescribed data and satisfy the equations against compact smooth tests. Uniqueness: for any weak $C_t{L}^2$ reflection solution and bounded spectral cutoff $N$, project to the cutoff coefficients; the test identity of step 2.1 turns the evolution into $f^{\mathrm{\prime }}=A_{N}f$ distributionally in time in the cutoff Hilbert space with bounded constant matrix $A_N$, and the finite transverse tests and smooth frequency tests supported in the cutoff interval, whose inverse transforms are Schwartz, are dense in the cutoff space, so the identity extends to every cutoff vector by the bounded multiplier and Cauchy–Schwarz [F6]. Subtracting the continuous Hilbert-valued primitive of $A_{N}f$, constructed as the norm limit of finite Riemann sums (which is independent of the tags by the uniform oscillation bound on a compact interval, exists by completeness of $L^2$, has the correct derivative by the average-increment bound, and is the vector version of the primitive argument of [F4]), pairing with every fixed cutoff vector and applying the scalar zero-derivative distribution theorem [F4] identifies the difference with its constant initial value; the difference $d$ of two solutions with zero data satisfies $d(t)={\int }_0^t{A}_{N}d(s)ds$, so on an interval of length $h$ with $h\mathrm{\parallel }{A}_N\mathrm{\parallel }<1$ the supremum norm forces $d=0$, and finitely iterating these intervals in either direction on any compact time interval gives $d=0$; letting the cutoffs exhaust the representation gives uniqueness of the whole field. The zero coefficient $K=0$ is treated directly by the same equations with $K=0$, without division by $K$ or $\mathrm{\mid }K\mathrm{\mid }$.

[Energy and dispersion] The total stored energy of the whole infinite guide is finite and equals $\mathcal{U}(t)=\frac{1}{8}{\sum }_{m,n}{\int }_{\mathbb{R}}(\epsilon \mathrm{\mid }\widehat{E}{\mathrm{\mid }}^2+{\mu }^{-1}\mathrm{\mid }\widehat{B}{\mathrm{\mid }}^2)d\xi$, and for every $t_0<t_1$ one has $\mathcal{U}(t_1)+\sigma {\int }_{t_0}^{t_1}\mathrm{\parallel }E(t){\mathrm{\parallel }}_{L^2(Q)}^{2}dt=\mathcal{U}(t_0)$ with ${\mathcal{U}}^{\mathrm{\prime }}(t)=-\sigma \mathrm{\parallel }E(t){\mathrm{\parallel }}_2^2$. Indeed, summing the finite/cutoff modal identities $q^{\mathrm{\prime }}=-2\sigma \mathrm{\mid }\widehat{E}{\mathrm{\mid }}^2$ of step 3.1 over the bounded cutoffs, integrating in $\xi$ and summing the finite transverse indices gives the truncated balance, and dominated convergence with the compact-time tail bound of step 3.1 passes it to the limit; the physical energy carries the factor $\frac{1}{2}\cdot \frac{1}{4}=\frac{1}{8}$ from step 1.1, and the dissipation term is continuous in time, so ordinary FTC [F4] gives the derivative form. At $\sigma =0$ the energy is constant, and on the constrained coefficient subspace differentiating the first-order equation once and using $K\times (K\times e)=K(K\cdot e)-\mathrm{\mid }K{\mathrm{\mid }}^{2}e$ together with $K\cdot \widehat{E}=0$ gives ${\widehat{E}}^{\mathrm{\prime }\mathrm{\prime }}+v^2\mathrm{\mid }K{\mathrm{\mid }}^2\widehat{E}=0$. Substitution of a harmonic coefficient $e{e}^{-i\omega t}$ gives $(v^2\mathrm{\mid }K{\mathrm{\mid }}^2-{\omega }^2)e=0$, so for $e\mathrm{\ne }0$ and $\omega >0$ the stated dispersion relation follows with $\beta =2\pi \xi$; for $e=0$ it supplies no condition on $\omega$; at $K=0$ the direct equations ${\widehat{E}}^{\mathrm{\prime }}=-\gamma \widehat{E}$, ${\widehat{B}}^{\mathrm{\prime }}=0$ are retained without dividing by $K$. For $\sigma >0$ the electric equation is ${\widehat{E}}^{\mathrm{\prime }\mathrm{\prime }}+\gamma {\widehat{E}}^{\mathrm{\prime }}+v^2\mathrm{\mid }K{\mathrm{\mid }}^2\widehat{E}=0$ at $\lambda =\mathrm{\mid }K{\mathrm{\mid }}^2\ge 0$ with the distinct/repeated-root alternatives of the finite-parameter classification [F5], including the zero-$K$ case; the zero-profile restrictions and individual cutoffs of the separated TE/TM candidates are unaffected, and no TE${}_{00}$ mode is created by a scalar constant coefficient.

[Scope] Clause (a) is steps 1.1 and 2.1, clause (b) is step 3.1, and clause (c) is step 4.1. The theorem supplies the complete longitudinal $L^2$ expansion of the declared straight rectangular guide with its lossless and constant-Ohmic energy evolutions and makes no completeness claim for arbitrary non-rectangular, dispersive, nonlocal, nonlinear, rough-wall or open-end/junction problems; the weak energy result is proved by linear mixed-transform evolution and modal limits, not by applying a nonlinear smooth Poynting identity to rough data, and no signal-speed or empirical inference is made.