# Why EM wave equation does not conform to relativity?

Why EM wave equation does not conform to relativity?
1 August 2012

This question is not about real electromagnetic wave. Physical wave does not violate relativity principle and always travel at constant speed c. It is the electromagnetic wave equation that does not conform to relativity. By computing the speed of the solution of the wave equation, we find that its velocity is faster than that of light and varies with distance and frequency. This result proves that the wave equation does not faithfully describe the physical phenomenon of electromagnetic wave.

A rigorous mathematical proof is given below. The annex is provided just in case where someone wants to check the validity of the used equations and is not necessary to the proof.

We study the spherical wave sent by an element of antenna of length dl which carries a current I (see the Figure 1). As the magnetic field of this element is only in the phi direction, in spherical coordinates, the general wave equation simplifies to the polar wave equation for the phi component. The polar wave equation admits the an analytical solution (ref. Clayton R. Paul, Keith W. Whites and Syed A. Nasar, Introduction to Electromagnetic Fields, Mcgraw-Hill College; 3 Sub edition (December 9, 1997), p.590). We call this solution “antenna wave”.