> TO BE READ: rough AI draft pending detailed human review. # Appendix A2. Wave Equation on a Toroidal Domain Chapter 8 writes the wave equation on a toroidal closure in the separated form $$ \partial_t^2 f = c^2\left( \partial_s^2 f + \frac{1}{r^2}\partial_\theta^2 f \right), $$ with curvature corrections omitted. This appendix derives the full Laplacian on a torus, identifies the omitted terms, and shows that the integer mode counting is exact while the mode frequencies receive aspect-dependent corrections. ## A2.1 Toroidal Coordinates Parametrize the torus by three coordinates $(\phi, \theta, \rho)$: - $\phi \in [0, 2\pi)$: angle around the major cycle (centerline), - $\theta \in [0, 2\pi)$: angle around the minor cycle (cross-section), - $\rho \in [0, r]$: radial distance from the tube center. A point in space is $$ \mathbf{x}(\phi, \theta, \rho) = \bigl((R + \rho\cos\theta)\cos\phi,\; (R + \rho\cos\theta)\sin\phi,\; \rho\sin\theta\bigr). $$ Define $$ \eta(\theta, \rho) := R + \rho\cos\theta. $$ This is the distance from the symmetry axis to the point $(\phi,\theta,\rho)$. On the tube surface ($\rho = r$), $\eta$ varies from $R - r$ on the inner equator ($\theta = \pi$) to $R + r$ on the outer equator ($\theta = 0$). ## A2.2 Metric and Scale Factors The scale factors of the toroidal coordinate system are $$ h_\phi = \eta = R + \rho\cos\theta, \qquad h_\theta = \rho, \qquad h_\rho = 1. $$ The volume element is $$ dV = h_\phi\,h_\theta\,h_\rho\;d\phi\,d\theta\,d\rho = \eta\,\rho\;d\phi\,d\theta\,d\rho. $$ ## A2.3 The Laplacian The Laplacian in orthogonal curvilinear coordinates is $$ \nabla^2 f = \frac{1}{h_\phi h_\theta h_\rho} \left[ \frac{\partial}{\partial\phi}\!\left(\frac{h_\theta h_\rho}{h_\phi}\frac{\partial f}{\partial\phi}\right) + \frac{\partial}{\partial\theta}\!\left(\frac{h_\phi h_\rho}{h_\theta}\frac{\partial f}{\partial\theta}\right) + \frac{\partial}{\partial\rho}\!\left(\frac{h_\phi h_\theta}{h_\rho}\frac{\partial f}{\partial\rho}\right) \right]. $$ Substituting the scale factors: $$ \nabla^2 f = \frac{1}{\eta\rho} \left[ \frac{\partial}{\partial\phi}\!\left(\frac{\rho}{\eta}\frac{\partial f}{\partial\phi}\right) + \frac{\partial}{\partial\theta}\!\left(\eta\frac{\partial f}{\partial\theta}\right) + \rho\frac{\partial}{\partial\rho}\!\left(\eta\frac{\partial f}{\partial\rho}\right) \right]. $$ ### Surface Laplacian For a field confined to the tube surface ($\rho = r$, no radial dependence), the radial term is dropped and the surface Laplacian is $$ \nabla^2_{\mathrm{surf}} f = \frac{1}{\eta\,r} \left[ \frac{\partial}{\partial\phi}\!\left(\frac{r}{\eta}\frac{\partial f}{\partial\phi}\right) + \frac{\partial}{\partial\theta}\!\left(\eta\frac{\partial f}{\partial\theta}\right) \right], $$ where now $\eta = R + r\cos\theta$. Expanding the derivatives: $$ \nabla^2_{\mathrm{surf}} f = \frac{1}{\eta^2}\frac{\partial^2 f}{\partial\phi^2} + \frac{1}{r^2}\frac{\partial^2 f}{\partial\theta^2} - \frac{\sin\theta}{r\,\eta}\frac{\partial f}{\partial\theta}. $$ The three terms have clear geometric meanings: 1. $\dfrac{1}{\eta^2}\dfrac{\partial^2 f}{\partial\phi^2}$: variation around the major cycle, weighted by the local circumferential distance $\eta$ from the symmetry axis. 2. $\dfrac{1}{r^2}\dfrac{\partial^2 f}{\partial\theta^2}$: variation around the minor cycle, as in the flat approximation. 3. $-\dfrac{\sin\theta}{r\,\eta}\dfrac{\partial f}{\partial\theta}$: the curvature correction. It couples the minor-cycle derivative to the position on the cross-section. ## A2.4 What Chapter 8 Drops Chapter 8 replaces $\eta = R + r\cos\theta$ by its mean value $R$, giving $$ \nabla^2_{\mathrm{flat}} f \approx \frac{1}{R^2}\frac{\partial^2 f}{\partial\phi^2} + \frac{1}{r^2}\frac{\partial^2 f}{\partial\theta^2}. $$ This amounts to two approximations: **Approximation 1.** In the major-cycle term, $$ \frac{1}{\eta^2} \to \frac{1}{R^2}. $$ The fractional error is $$ \frac{1}{\eta^2} - \frac{1}{R^2} = \frac{1}{R^2}\left(\frac{1}{(1+\epsilon\cos\theta)^2} - 1\right), \qquad \epsilon := \frac{r}{R}. $$ For $\epsilon \ll 1$ (thin torus), the correction is $O(\epsilon)$. For $\epsilon \sim 1$ (fat torus, as Appendix A0 shows is the self-consistent regime), the correction is $O(1)$ and cannot be neglected. **Approximation 2.** The curvature correction term $$ -\frac{\sin\theta}{r\,\eta}\frac{\partial f}{\partial\theta} $$ is dropped entirely. This term is first order in the minor-cycle derivative and couples the angular position on the cross-section to the transport around the torus. It vanishes identically at $\theta = 0$ (outer equator) and $\theta = \pi$ (inner equator), and is maximal at $\theta = \pi/2$ and $3\pi/2$. ## A2.5 Full Surface Wave Equation The full scalar wave equation on the toroidal surface is $$ \partial_t^2 f = c^2\left( \frac{1}{\eta^2}\frac{\partial^2 f}{\partial\phi^2} + \frac{1}{r^2}\frac{\partial^2 f}{\partial\theta^2} - \frac{\sin\theta}{r\,\eta}\frac{\partial f}{\partial\theta} \right), $$ with $\eta = R + r\cos\theta$. ## A2.6 Mode Structure with Curvature Corrections Seek separated solutions of the form $$ f(\phi,\theta,t) = e^{im\phi}\,\Theta(\theta)\,e^{-i\omega t}. $$ The $\phi$-periodicity enforces $m \in \mathbb{Z}$, exactly as in Chapter 8. This integer condition is independent of the curvature corrections. Substituting into the full surface wave equation gives the ordinary differential equation for $\Theta(\theta)$: $$ \frac{\omega^2}{c^2}\,\Theta = -\frac{m^2}{\eta^2}\,\Theta + \frac{1}{r^2}\,\Theta'' - \frac{\sin\theta}{r\,\eta}\,\Theta', $$ where primes denote $d/d\theta$ and $\eta = R + r\cos\theta$. Rearranging: $$ \Theta'' - \frac{r\sin\theta}{\eta}\,\Theta' + r^2\left(\frac{\omega^2}{c^2} + \frac{m^2}{\eta^2}\right)\Theta = 0. $$ This is a Hill-type equation: a second-order ODE with periodic coefficients (period $2\pi$ in $\theta$). The $\theta$-periodicity of $\Theta$ enforces a discrete spectrum of allowed $\omega$ values for each $m$. In the flat approximation ($\eta \to R$, curvature term dropped), this reduces to $$ \Theta'' + \left(\frac{r^2\omega^2}{c^2} + \frac{m^2 r^2}{R^2}\right)\Theta = 0, $$ which has the constant-coefficient solutions $\Theta = e^{in\theta}$ with $n \in \mathbb{Z}$ and $$ \omega^2 = c^2\left(\frac{m^2}{R^2} + \frac{n^2}{r^2}\right), $$ recovering Chapter 8. ### Structure of the corrections Write $\epsilon = r/R$ and expand the Hill equation to first order in $\epsilon$. The coefficients become $$ \frac{r\sin\theta}{\eta} = \epsilon\sin\theta\,(1 + O(\epsilon)), $$ $$ \frac{m^2}{\eta^2} = \frac{m^2}{R^2}(1 - 2\epsilon\cos\theta + O(\epsilon^2)). $$ At first order, the curvature correction couples the $n$-th Fourier mode of $\Theta$ to its neighbors $n \pm 1$. The leading effect is a frequency shift proportional to $\epsilon$: $$ \omega_{mn}^2 = c^2\left(\frac{m^2}{R^2}+\frac{n^2}{r^2}\right) \left(1 + O(\epsilon)\right). $$ More precisely, the first-order correction to $\omega_{mn}^2$ from standard perturbation theory of Hill's equation is $$ \delta\omega_{mn}^2 = -\frac{c^2 m^2}{R^2}\,\epsilon\,\langle\cos\theta\rangle_{n} + O(\epsilon^2), $$ where $\langle\cos\theta\rangle_n$ is the diagonal matrix element of $\cos\theta$ in the $n$-th Fourier mode. Since $\cos\theta$ connects $n$ to $n \pm 1$, the diagonal element vanishes: $$ \langle n|\cos\theta|n\rangle = \frac{1}{2\pi}\int_0^{2\pi} e^{-in\theta}\cos\theta\,e^{in\theta}\,d\theta = 0. $$ Therefore the first-order frequency correction vanishes: $$ \delta\omega_{mn}^2 = O(\epsilon^2). $$ The leading correction to the mode frequencies is second order in $\epsilon$. ## A2.7 What the Curvature Terms Do and Do Not Change **What is unchanged:** 1. The integer $m$ is enforced by $\phi$-periodicity. This is exact for any $\epsilon$. 2. The integer $n$ is enforced by $\theta$-periodicity. The Hill equation has a discrete Floquet spectrum; the periodicity condition selects integer Fourier indices regardless of $\epsilon$. 3. The mode-counting rule — one integer pair $(m,n)$ per allowed standing wave — holds for all aspect ratios. **What changes:** 1. The exact mode frequencies receive corrections of order $\epsilon^2$ relative to the flat-torus values. For a fat torus ($\epsilon \sim 1$), these corrections are $O(1)$ and the flat formula is quantitatively wrong, even though it is qualitatively correct. 2. The curvature correction couples neighboring Fourier modes in $\theta$, meaning the exact eigenmodes of the full equation are not pure $e^{in\theta}$ but superpositions with small admixtures of $e^{i(n\pm 1)\theta}$, $e^{i(n\pm 2)\theta}$, etc. The leading admixture is $O(\epsilon)$. 3. For a fat torus, the mode shapes are distorted: energy density is higher on the outer equator ($\theta = 0$) than on the inner equator ($\theta = \pi$), because the outer equator has larger $\eta$ and therefore a longer path around the major cycle. ## A2.8 Implications for the Self-Consistent Closure Appendix A0 shows that the self-consistent closure has aspect ratio $$ \frac{r}{mR} = \frac{\sqrt{n^2-1}}{n}, $$ which for the minimum closure ($m = 1$, $n = 2$) gives $\epsilon = r/R = \sqrt{3}/2 \approx 0.87$. In this regime, the flat-torus frequency formula of Chapter 8 is a qualitative guide but not a quantitative prediction. Computing the exact spectrum of the minimum closure requires solving the full Hill equation of Section A2.6 at $\epsilon = \sqrt{3}/2$. This is a numerical exercise that does not change the topological structure of the spectrum (one mode per integer pair) but does change the quantitative spacing between levels. For higher $m$ at fixed $n$, or for closures confined by external loading to $\epsilon \ll 1$, the flat-torus formula is quantitatively accurate to $O(\epsilon^2)$. ## A2.9 Summary 1. The full Laplacian on a toroidal surface contains a position-dependent major-cycle term $1/\eta^2$ and a curvature correction $-(\sin\theta)/(r\eta)\,\partial_\theta$. 2. Chapter 8 drops both by replacing $\eta \to R$. The resulting flat-torus equation is exact in the limit $r/R \to 0$. 3. The integer mode counting $(m, n)$ is topological: it is enforced by periodicity in $\phi$ and $\theta$ and holds for all aspect ratios. 4. The first-order frequency correction vanishes. The leading correction to the mode frequencies is $O(\epsilon^2)$ where $\epsilon = r/R$. 5. For the self-consistent minimum closure ($\epsilon \approx 0.87$), the corrections are large and the exact spectrum requires numerical solution of the Hill equation. 6. The qualitative picture of Chapter 8 — discrete integer modes, Rydberg-type scaling from standing-wave partitions — is robust. The curvature terms change the exact frequencies, not the counting.