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”.
Please read the following document
This comment is not intended to cast doubt on the accuracy of your solution to the wave equation, which I haven't checked. I just want to point out that the fact that the phase velocity of a wave exceeds the velocity of light is not a contradiction to relativity theory. Even though Fourier theory implies that a physical wave can be thought of as a superposition of sinusoidal waves, that does not imply that the velocity of the signal is greater than c, the velocity of light.
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