Confusion in the showing EM wave exist from Maxwell equation When deriving the mathematical description of a field, we set the current density and charge to zero in Maxwell's equations.
However, this condition is not absolutely true anywhere on earth.
Yet, we are able to apply EM waves for problems in communication, medicine etc. How is that possible that instead of ignoring the sources of the fields the fields are calculated obviously properly nevertheless?
 A: First of all, a general solution of a system of inhomogeneous linear equations (such the Maxwell equations with sources) can be always decomposed into a particular solution of the inhomogeneous equations and the general solution of the homogeneous ones (i.e., the Maxwell equations without sources, of which EM waves are solutions).
Secondly, one needs to include the sources (i.e., the currents and charges) to describe the generation/absorption of the EM waves, but not their propagation. This is widely studied, but it is more complicated (mathematically) than studying the free Em waves, which is why one usually starts with the latter.
A: Free space solutions of maxwells equations show that wave like solutions can theoretically exist. Plane wave solutions to the homogenous wave equations are not created by charges and currents, and thus these solutions don't prove that EM waves are generated by charges.
This is not the only way that we can show wave like solutions exist, for example, using the retarded potentials for the hertzian  dipole. This is the best way to show EM waves are generated by accelerating charge
I also think a point can be made, that any REAL EM wave generated by charges and currents, will ALSO satisfy the homogenous wave equation, in regions where charge density and current density are zero. However these  real waves generated by charges, will only satisfy the homogenous equation on a restricted domain.
This is similar to using $\nabla^2 V = 0$
In order to solve the the electric field on a restricted domain, in regions where there are no charges and currents.
An example of this, (I think) is solving the homogenous equations in spherical coordinates. I believe that you get solutions that are caused by a charge distribution, as there is a singularity in the center. However I think the plane wave solutions are the only solutions that are valid on all domains, and thus charge density IS infact zero everywhere.
