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On longitudinal electromagnetic waves

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We should notice that whereas S₂   is an equiphase surface,  ₂  coincides with the flux vector of the vector function .

Substituting (28) into (21), we yield the final expression

Thus we have proved

THEOREM: When the wave flux propagated in a source-free space, the divergence of vector function of the flux is proportional to the scalar product of this function derivative with respect to time into the direction of the vector of flux of this function.

Returning to the beginning of this item, we yield:

for the electric field strength

and for the magnetic field strength

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Independently and at the same time as me, Prof. V.A. Atsukovsky yielded almost similar result but in some other way. He stated it in his book "General electrodynamics" (Energoatomizdat, Moscow, 1990, Russian, pp. 166- 182). The difference is in application of divergence theorem to the vector flux of magnetic field which Prof. Atsukovsky did in the form

(formula (7.79) in his book). As I see, this is not quite legitimate by a simple reason: in case when the magnetic field vector was perpendicular to the wave propagation direction, it determines not so much the vector itself or its direction in space (though it is presented in such form in this problem) as its time variation in the vector flux. (I added this paragraph in 1994 when preparing this paper for publication).

Go on considering the formulas (30) and (31). Some sceptic can notice: well, suppose this is admissible for  , but one of indisputable postulates of electrodynamics is that the divergence of vector of magnetic field is zero, which is determined by its pure solenoidal shape.

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So it was though up to now. But look again at (24). The first integral of the sum determining    as zero in absence of charges within the region, as well as the solenoidal pattern of magnetic field, in such or other way falls out of consideration, and the right-hand part of (31) is formed not by path-tracing the surface of a selected region, but by a simple integration over the space that includes the vector function variable within this region. So in the right-hand part of (31)  /t   appears as if in evasion of conventional reasoning and cannot vanish with the time-variable  . Thus we have proved the second statement.

Pay a special attention that the scalar products in right-hand parts of (30) and (31) cannot vanish irrespectively of the direction of vector function, or the very idea of vector of flux through a region would lose its sense. Thus   /t  and  /t determine those components of vector functions with respect of    and t whose vector is directed along the flux. In absence of such component (in accordance with the statement of problem) the flux and consequently the wave process are absent.

Finishing this item, I would like to add that if we use the above technique for the integral form of Ostrogradsky-Gauss theorem, for the electrodynamic process it will take the following form: