Usual textbook on electromagnetism derives and shows us the two solutions from the original wave equation with source electronic density distribution function and source electric current density distribution function. Each solution is called "retarded wave" and "advanced wave." The former solution propagates temporaly normally, that is, it propagates forward in time. For example, if we generate and send the retarded wave which involves a message "ABC...XYZ", from on the surface of the Moon toward the Earth, then one on the Earth surface receives the retarded radio signal about one second later as "ABC...XYZ." That is, the retarded wave departs fom present time and arrives future time.
Reversely, the advanced wave behaves completely inversely. If we could generate and send the advanced wave which involves a message "ABC...XYZ" from the Moon toward the Earth, then one on the Earth surface receives the advanced radio signal about one second before the one on the Moon sends it, and the contents of the message becomes reversed order like "ZYX...CBA." That is, the advanced wave departs now and arrives past.
In our everyday life we do not have ever heard such an advanced wave signal on radio. Is there or can there be the advanced wave really? or is the level of our radio technology not yet high enough to detect the advanced wave? Or is there some mistakes in deriving the solution for the original wave equation?