# Non Maxwellian examples

### The following animations show the evolution of the electric field E (red) and the corresponding Poynting vectors (black). The magnetic field B is orthogonal to the screen.

#### Here below is a portion of a plane sinusoidal wave. Each propagating front is an entire plane;  the electromagnetic fields belong to it and are everywhere constant . #### One can easily check that plane fronts, carrying (almost) any type of vector information, are admissible by the new set of equations. There is no hope instead to get this result  by the standard Maxwell’s equations. Here below is a train of wave-packets with compact support, modulated by a sinus function.  The transversal electromagnetic fields belong to  each propagating front and are zero outside a bounded region. #### Perfect spherical fronts, carrying any type of vector information on the local tangent planes, also satisfy the new set of equations. Nevertheless, Maxwell’s equations are  not compatible with waves such that, in vacuum, the direction of the energy flow (associated to the Poynting vector) is the same as the one of the evolving fronts. For example, in spherical coordinates, a classical setting is the following one: #### Here  below is the vector field V (normalized Poynting vectors) in the case of the above spherical example. The situation is stationary. The information escapes radially at constant speed. The fronts evolve according to the rules of geometrical optics, hence they exactly satisfy the Huygens principle. #### A fragment of a perfect spherical wave, also satisfies the whole set of equations. For example, E can be chosen according to the following expression: #### for an arbitrary function f. The fronts now evolve along a narrower path. As in the previous case, the corresponding vector field V is constant, radial and stationary. Concerning Maxwell’s equations such a wave cannot be modelled. The situation is even worse than the case, previously considered, of a global spherical front. In fact, we cannot cut a piece of wave without altering the magnitude of div(E). #### The last two movies show the evolution of the electric field E and the velocity vector field V (normalized Poynting vectors) in the case of the Hertzian wave. The information spreads along concentric spheres, but the behavior of the Poyinting vectors is quite uncontrolled, especially in proximity of the vertical axis.  