## 5 Free-Waves  
An interesting subset of solutions is obtained when \(D\mathbf{V}/Dt\) and \(p\) are vanishing, though \(\rho\) will be allowed to remain different from zero. These will be called free-waves, and in general they have the property that the triplet \((\mathbf{V},\mathbf{E},\mathbf{B})\) is orthogonal with the additional conditions \(|\mathbf{E}|=|c\mathbf{B}|\) and \(|\mathbf{V}|=c\). In this case, Equation (10) reduces to:  
\[\rho\big{(}\mathbf{E}\,+\,\mathbf{V}\,\times\,\mathbf{B}\big{)}\,=\,0 \tag{20}\]  
from which one could easily deduce the orthogonality of the couple \(\mathbf{E}\) and \(\mathbf{B}\), and that of the couple \(\mathbf{E}\) and \(\mathbf{V}\). As it is shown in Chapter 2 in [14], the reduced system of equations is invariant under [PERSON]'s transformations and contains several interesting solutions not available in the Maxwellian context. We are going to introduce those that are more relevant in this discussion. In the three dimensional cylindrical coordinate system \((r,\varphi,z)\), we set:  
\[\mathbf{E}=(E_{r},\,0,\,\,E_{z})=\Big{(}\,g(r)f(ct-z),\,0\,,\,0\Big{)}\]  
\[\mathbf{B}=(0,\,B_{\varphi},0)=\Big{(}\,0\,,\,g(r)f(ct-z)\,,\,0\,\Big{)}\qquad \qquad\mathbf{V}=(V_{r},\,0,\,V_{z})=(0,\,0,\,c) \tag{21}\]  
Here \(g\) and \(f\) are arbitrary functions (\(g\) is only required to tend to zero as \(r\) goes to zero). Waves of this kind solve the whole set of Equations (6)-(10). The solution travels in the direction of the \(z\)-axis at speed \(c\) (see Figure 1, left). By choosing both \(f\) and \(g\) with compact support, we are able to describe solitonic solutions similar in their behavior to \"photons\" of any size. Some plane waves of infinite extension trivially belong to this typology. The expressions in (21) are at the foundation of the research carried out in this paper. As a matter of fact, the device proposed in Section 9 is finalized to generate  
where \(f\) and \(g\) are arbitrary smooth functions. These are genuine free waves, according to our definition. Note that the electric and magnetic fields have no component along the radial direction. Moreover: \(\mathbf{E}\cdot\mathbf{B}=0\). Since \(g\) does not depend on \(\varphi\), the behavior is similar to that generated by an infinitesimal oscillating dipole lined up with the \(z\)-axis. This family of solutions moves along the radial direction with velocity \(c\), and they are modulated by the function \(f\). The wave-fronts are the surfaces of spheres centered at the origin. By explicitly computing the divergence of the electric field, we get:  
\[\
abla\cdot\mathbf{E}=\frac{c}{R^{2}\sin\vartheta}\frac{\partial\big{(}g\sin \vartheta\big{)}}{\partial\vartheta}\ f(ct-R) \tag{25}\]  
It turns out that there are no possibilities to impose the free divergence condition required from Equation (1) in a vacuum, unless one chooses a \(g\) value proportional to \(1/\sin\vartheta\) (which implies the existence of singularities at the poles). Therefore, waves with spherical fronts cannot be of Maxwellian type. Note that the well-known spherical Hertz's solution for an infinitesimal dipole solves the entire set of [PERSON]'s equations, including \(\
abla\cdot\mathbf{E}=0\). Nevertheless, its wave-fronts (i.e., the surfaces enveloping the electromagnetic field) are not spherical and \(\mathbf{E}\) is not orthogonal to the propagation direction. More comments on this delicate subject are provided in Appendix D in [15]. Let us also observe that the field \(\mathbf{V}\) in (24) has a nonvanishing divergence; i.e., \(\
abla\cdot\mathbf{V}=2 cR^{-1}\). Note that this is also true for the field in (22). This aspect reveals that the model equation are not directly related to classical fluid dynamics, where the condition \(\
abla\cdot\mathbf{V}=0\) implies conservation of mass. No masses are involved in our case--instead, purely electromagnetic propagation in a vacuum. We are assuming however that there may be regions where \(\
abla\cdot\mathbf{E}\) is different from zero.