# Electromagnetic Waves in Annular Regions
## 3 The Case of the Ring
Other conditions, such as perfect conductivity (\(\mathbf{E}\) orthogonal to the boundary), can be easily implemented. In this case, we need to set \(E_{2}=0\) in (23), so that condition (22) becomes:  
\[\frac{\partial(ra)}{\partial z}\ =\ \frac{\partial(rb)}{\partial r} \tag{24}\]  
The domain \(\Omega\) (which is not necessarily a circle) is supposed to be \"centered\" at the point \((\eta,0)\), where \(\eta>0\) is large enough to avoid the intersection of \(\Omega\) with the \(z\)-axis. Thus, the quantity \(2\eta\) is related to the major diameter of the toroid. As done in [22] (Section 5.4), we set \(\ y=r-\eta\). By differentiating (20) with respect to \(z\) and (21) with respect to \(y\), we arrive at:  
\[ \begin{cases}\frac{1}{c^{2}}\frac{\partial^{2}u}{\partial t^{2}}\ =\ \frac{\partial^{2}u}{\partial z^{2}}\ +\ \frac{\partial}{\partial y}\left(\frac{\partial u}{ \partial y}+\frac{u}{y+\eta}\right)\\ \\ \frac{1}{c^{2}}\frac{\partial^{2}v}{\partial t^{2}}\ =\ \frac{\partial^{2}v}{ \partial z^{2}}\ +\ \frac{\partial}{\partial y}\left(\frac{\partial v}{ \partial y}+\frac{v}{y+\eta}\right)\end{cases}  \tag{25}\]  
where \(u=\partial a/\partial z\) and \(\ v=\partial b/\partial r=\partial b/\partial y\), with \(u-v=0\) on \(\partial\Omega\). Through the substitutions \(\tilde{u}=u\sqrt{y+\eta}\), \(\tilde{v}=v\sqrt{y+\eta}\), we then obtain:\[\left\{ \begin{aligned} \frac{1}{c^{2}}\frac{\partial^{2}\tilde{u}}{ \partial t^{2}}&=\ \frac{\partial^{2}\tilde{u}}{\partial z^{2}}\ +\ \frac{\partial^{2}\tilde{u}}{\partial y^{2}}\ -\ \frac{3}{4}\frac{\tilde{u}}{(y+\eta)^{2}}\\ \frac{1}{c^{2}}\frac{\partial^{2}\tilde{v}}{\partial t^{2}}& =\ \frac{\partial^{2}\tilde{v}}{\partial z^{2}}\ +\ \frac{\partial^{2}\tilde{v}}{\partial y^{2}}\ -\ \frac{3}{4}\frac{\tilde{v}}{(y+\eta)^{2}} \end{aligned} \right. \tag{26}\]  
to be solved in \(\Omega\), with the boundary condition: \(\tilde{u}-\tilde{v}=0\). Furthermore, we introduce a new unknown \(w=\tilde{u}-\tilde{v}\) and take the difference of the two equations in (26). Finally, we pass to the stationary eigenvalue problem:  
\[\left\{ \begin{aligned} & Lw\ =\ -\ \frac{\partial^{2}w}{\partial z^{2}}\ -\ \frac{\partial^{2}w}{\partial y^{2}}\ +\ \frac{3}{4}\frac{w}{(y+\eta)^{2}}\ =\ \lambda w \quad\text{in}\ \Omega\\ & w\ =\ 0\quad\text{on}\ \partial\Omega\end{aligned} \right. \tag{27}\]  
obtained by looking for time-periodic solutions of the system in (26). Similarly to the previous section, we would like \(\lambda>0\) to be an eigenvalue (with multiplicity two, at least) of the positive-definite differential operator: \(L=-\partial^{2}/\partial z^{2}-\partial^{2}/\partial y^{2}+\frac{3}{4}(y+ \eta)^{-2}\), with homogeneous Dirichlet boundary conditions. This request implicitly defines the set \(\Omega\). By introducing:  
\[\Big{[}u(t,y,z)-v(t,y,z)\Big{]}\sqrt{y+\eta}\ =\ \tilde{u}(t,y,z)-\tilde{v}(t,y,z)\]  
\[=\ w_{1}(y,z)\sin(ct\sqrt{\lambda})\ +\ w_{2}(y,z)\cos(ct\sqrt{\lambda}) \tag{28}\]  
where \(w_{1}\) and \(w_{2}\) are independent (suitably normalized) eigenfunctions corresponding to \(\lambda\), one gets viable solutions of the system (26). The electromagnetic fields are then recovered by the expressions (see (23) and (19)):  
\[\mathbf{E}=\frac{1}{c\sqrt{\lambda}}\left(-\frac{\partial}{\partial z}\Big{(} \frac{-w_{1}\cos\zeta+w_{2}\sin\zeta}{\sqrt{y+\eta}}\Big{)},\ \frac{1}{y+\eta}\frac{\partial}{\partial y}\Big{(}(-w_{1}\cos\zeta+w_{2}\sin \zeta)\sqrt{y+\eta}\Big{)},\ 0\right) \tag{29}\]  
\[\mathbf{B}\ =\ \frac{1}{c^{2}}\left(0,\ 0,\ \ \frac{w_{1}\sin\zeta+w_{2}\cos \zeta}{\sqrt{y+\eta}}\right) \tag{30}\]  
Figure 3: Electromagnetic fields circulating in a toroid cavity \(\Sigma\). The electric field lies on each vertical section, whereas the magnetic one is distributed parallel to the large circumference. The time evolution recalls that of a fluid dynamics vortex ring. with \(\zeta=ct\sqrt{\lambda}\). The above displacements rotate in the \((r,z)\) plane by varying the time parameter \(t\).