# Electromagnetic Waves in Annular Regions
## 2 The Case of the Cylinder  
Our analysis starts from the classical set of [PERSON]'s equations in vacuum:  
\[\frac{\partial\mathbf{E}}{\partial t}\ =\ c^{2}\text{curl}\mathbf{B} \tag{1}\]  
\[\text{div}\mathbf{E}\ =\ 0 \tag{2}\]  
\[\frac{\partial\mathbf{B}}{\partial t}\ =\ -\text{curl}\mathbf{E} \tag{3}\]  
\[\text{div}\mathbf{B}\ =\ 0 \tag{4}\]  
where \(c\) is the speed of light and the two fields \(\mathbf{E}\) and \(c\mathbf{B}\) have the same dimensions. As usual, Equations (3) and (4) are automatically satisfied, when the electromagnetic fields are expressed through the potentials \(\mathbf{A}\) and \(\Phi\) such that:  
\[\mathbf{B}\ =\ \frac{1}{c}\ \text{curl}\mathbf{A}\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \mathbf{E}\ =\ -\ \frac{1}{c}\frac{\partial\mathbf{A}}{\partial t}\ -\ \
abla\Phi \tag{5}\]  
We also require that the vector and the scalar potentials are related by the Lorenz gauge condition:  
\[\text{div}\mathbf{A}\ +\ \frac{1}{c}\frac{\partial\Phi}{\partial t}\ =\ 0 \tag{6}\]  
Furthermore, one can easily deduce the wave equations:  
\[\frac{\partial^{2}\mathbf{A}}{\partial t^{2}}\ -\ c^{2}\Delta\mathbf{A}\ =\ 0 \tag{7}\]  
\[\frac{\partial^{2}\Phi}{\partial t^{2}}\ -\ c^{2}\Delta\Phi\ =\ 0 \tag{8}\]  
As we are examining periodic solutions, the cylindrical environment can be transformed in a 2-D problem in the system of coordinates \((r,z,\phi)\), where there is no dependence on \(z\) (see Figure 1). The electric field belongs the plane \((r,\phi)\). The magnetic field is orthogonal and lined up with the \(z\)-axis. We quickly recall some results concerning electromagnetic waves rotating around the \(z\)-axis. Before proceeding, we need to introduce the set of [PERSON]'s functions of the first kind \(J_{k}\), \(k\geq 0\), that are implicitly defined through the differential relation:  
\[J_{k}^{\prime\prime}(x)\ +\ \frac{J_{k}^{\prime}(x)}{x}\ -\ \frac{k^{2}J_{k}(x)}{x^{2}}\ + \ J_{k}(x)\ =\ 0 \tag{9}\]  
Other useful properties of [PERSON]'s functions to be recalled are:  
\[J_{k}^{\prime}(x)\ +\ \frac{kJ_{k}(x)}{x}\ =\ J_{k-1}(x) \tag{10}\]\[\text{div}\mathbf{E}\ =\ 0 \tag{2}\]0  
We remind that \(J_{k}(x)\), \(k\geq 1\), tends to zero for \(x\) tending to zero. [PERSON]'s functions of the second kind are denoted by \(Y_{k}\), \(k\geq 0\). They solve the same set of equations, but are unbounded in the neighborhood of \(x=0\). A classical reference for this set of special functions is for instance [21]. By denoting by \(\omega>0\) a parameter associated with the frequency of rotation of the wave \(\
u=c\omega/\pi\), it turns out that, for certain functions \(f\) of the variable \(r\), solutions of the [PERSON]'s equations in vacuum are obtained as:  
\[\text{div}\mathbf{E}\ =\ 0 \tag{2}\]1  
\[\text{div}\mathbf{E}\ =\ 0 \tag{2}\]2  
for any \(0\leq\phi<2\pi\) and \(r\geq 0\). Here, the prime denotes the derivative with respect to \(r\). As remarked above, there is no dependency on the variable \(z\). These solutions have been proposed in [22], however we do not exclude that they were present in the literature in earlier papers. As a first example, we set \(f=J_{k}\), \(k\geq 1\). The corresponding solutions can be checked by direct substitution into [PERSON]'s equations (it is necessary to use (9)), through a tedious but straightforward calculus exercise. They derive from potentials satisfying the [PERSON]'s condition (6), i.e.,  
\[\text{div}\mathbf{E}\ =\ 0 \tag{2}\]3  
\[\text{div}\mathbf{E}\ =\ 0 \tag{2}\]4  
where (10) and (11) need to be used. We observe that \(A_{1}\) and \(A_{3}\) have a phase difference of \(\pi/2k\). We can now give a bound to \(r\), obtaining in this way field distributions defined on a disk. This cut can be either done by imposing that \(f\) is zero at the boundary (Dirichlet type conditions) or by asking that \(f^{\prime}\) is zero (Neumann type conditions). The first option states that the boundary is a perfect magnetic conductor (i.e., \(\mathbf{B}\) has not tangential component on the boundary surface).