## 8 Numerical Simulations
This is done by respecting the perfect conductivity relation. Again referring to Figure 6 (right), at each time-step, the (red) vector of components \((E_{r},E_{z})\) in proximity of the boundary is projected onto the direction normal to the surface, giving birth to the (green) vector \(\frac{1}{2}(E_{r}-E_{z},E_{z}-E_{r})\). This last vector is successively shifted up to the corresponding boundary grid point. The parameters relative to the numerical tests are provided in the figure captions. As far as Figure 7 is concerned, the simulations rely on the field \(\mathbf{V}\) given in the first plot of Figure 5. At the grid points where \(\mathbf{V}\) is not defined, we are actually solving the system (6) and (7) with \(\mathbf{V}=0\). At the end of the guide the solution turns out to be discontinuous, thereby producing sensible numerical instabilities. The general behavior of the wave is however reasonably well reproduced. Concerning Figure 8, this is obtained from the field \(\mathbf{V}\) as given in the second plot of Figure 5. Again we set \(\mathbf{V}=0\) at the remaining grid points. The solution is clearly focused in the horizontal direction. The approximation suffers from the discontinuity across the dielectric boundary. Improvements could be obtained through the adoption of a more accurate discretization scheme. Anyway, our purpose here was to show that numerical simulations are feasible and produce a consistent prediction. Figure 5: Displacement of the vector field \(\mathbf{V}\) within the computational domain for \(N=40\). The antenna boundaries are represented by the solid lines. The plot on the left-hand side shows \(\mathbf{V}\) when the dielectric is absent. In this case we have \(|\mathbf{V}|=1\). On the right-hand side, \(\mathbf{V}\) is radial with \(|\mathbf{V}|=1/\sqrt{\varepsilon_{r}}\) inside the dielectric, and horizontal with \(|\mathbf{V}|=1\) outside. The situation is referred to the choice \(e_{r}\approx 2\) (see the left-most curve in Figure 4), so that \(1/\sqrt{\varepsilon_{r}}\approx 0.7\). Finally, in Figure 9, we show the results when \(V\) is totally eliminated. We are not, however, enforcing the condition \(\rho=\
abla\cdot\mathbf{E}=0\). Since the initial guess itself does not satisfy \(\rho=0\), we cannot claim that we are actually solving the full set of [PERSON]'s equations in a vacuum. We are not even solving the classical wave equation, since we can just deduce that \(\partial^{2}\mathbf{E}/\partial t^{2}=-c^{2}\
abla\times(\
abla\times\mathbf{E})=c^{2} (\Delta\mathbf{E}-\
abla\rho)\). On the other hand, there are very few initial data recoverable from (24) such that \(\rho=0\). This is true, for instance, for \(g=1/\sin\theta\), which does not correspond with a consistent choice for a signal emerging from an antenna (see the discussion in [14], Chapter 1). The behavior of the solution shown in Figure 9, especially regarding the second plot (to be compared with the second plot of Figure 7), is very inefficient. As far as the successive plots are concerned, some of the energy remains concentrated at the endpoints of the guide. Noticeably, in most of the computational code available on the market, the checking of the zero divergence condition in vacuum is disregarded. This means that, if \(\rho\) at a certain time starts assuming nonvanishing values (because of imposing some boundary constraints deriving for example from a scattering object), the problem is not approached in the proper way. This is true, for instance, in the case of [PERSON]'s scheme, mentioned in Section 6, whose implementation strictly trusts the fact that \(\rho\) must remain zero for all times. The real trouble is that [PERSON]'s equations are too restrictive. In short, the solution of the vector wave equation, in a given domain with generic boundary conditions, has no chance to have its divergence be zero. That is why we adopted the extended model in our simulations. We end this section with a few remarks. We placed our initial guess at a distance from the real source (the gap between the two cones).