# Electromagnetic Waves in Annular Regions
## 2 The Case of the Cylinder
Clearly the above expression solves the scalar wave equation in vacuum. In fact, the second derivative in time produces the multiplicative factor \(-c^{2}\lambda\), while the application of the Laplace operator in cylindrical coordinates is equivalent to a multiplication by \(-\lambda\). Extensions of the previous results may be obtained when the rotation around the axis is combined with a radial oscillation. Here, in order to find exact solutions, we must assume that the section has a central hole. The study in [24] is connected to the search of eigenfunctions for the Laplace operator in such a way that the dimension of the corresponding eigenspace is equal to four. In such a circumstance, \(f\) is a linear combination of \(J_{k}\) and \(Y_{k}\). Note that we have to stay away from the point \(r=0\), since Bessel's functions of the second kind are singular at the origin. Again, the magnetic field is parallel to the cylinder's axis and the electric field belongs to the plane \((r,\phi)\). One may consider sections where \(R_{m}=1\leq r\leq R_{M}\) with the boundary conditions imposed at \(r=R_{m}\) and These last can be either of Dirichlet or Neumann type. Geometries compatible with the existence of solutions can be obtained by adjusting \(r=R_{m}\) and \(r=R_{M}\), as suggested by the location of the zeros of suitable combinations of Bessel's functions. Several options are possible and some of them are documented in [24]. Once the frequency of rotation (depending on \(\omega\)) is fixed and the type of boundary conditions decided, the size of the domain is automatically determined. The magnitude scales inversely with the constant \(\omega\) (high frequencies, small objects, and vice versa). This is done by preserving the geometrical proportions, thanks to the linearity of [PERSON]'s equations. There is however a reduced number of domains allowing for the creation of these periodic waves. If the proper shape is not respected, the wave does not follow a time-periodic motion, i.e., it will not complete an entire cycle preserving the phase. As said above, a sufficient condition allowing for such a guided behavior is related to the multiplicity of the eigenvalues of the Laplace operator on the specific domain. Concerning the solutions proposed so far, there are some peculiarities to be pointed out. First of all, the information travels along concentric circumferences at constant angular velocity and there is not a proper wave-front that envelops the lines of force of the electric field. The speed of rotation varies from zero at the center with no upper limit when moving far away. Assuming zero boundary conditions for the magnetic field, the smallest domain containing the wave has a radius where the peripheral velocity exceeds that of light. Contrary to what is commonly believed, [PERSON]'s equations are perfectly compatible with this type of behavior. Such a paradox can be solved in the context of general relativity, where we may suppose that a suitable alteration of the space-time ensures the development at constant velocity \(c\) along curved geodesics. The problem is however not trivial and requires further analysis (not available at the moment). The second comment concerns with the fact that \(\mathbf{E}\) is orthogonal to \(\mathbf{B}\), but not to the propagation fronts. Indeed, during the evolution, \(\mathbf{E}\) has both transverse and longitudinal oscillating components. This is also in contrast with popular belief. Locally, this behavior is similar to the one displayed by a wave traveling in a medium with dielectric constant different from that of vacuum, which is not however the case we are dealing with in this section. On the other hand, curly waves are the primary ingredients that inspired [PERSON] to develop his theory (see [25]). The aim was to investigate the electromagnetic structure of matter in terms of everlasting circulant \"molecular vortices\".