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On longitudinal electromagnetic waves

Substituting the first equation by the second, we yield

We can see that the electric and magnetic fields and of a plane wave are directed perpendicularly to the direction of wave propagation" [2, p. 147].

Mathematically it seems to be proved. However if we assume a plane (or quasi-plane) longitudinal wave propagating in space (see Fig. 1) and vector of this wave directing along and  = (t - (x/c))  (i. e., the formulation of the problem coincides with that considered by Landau), Landau's proof will

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give us = 0, whereas, according to the Maxwell equation, the magnetic field must exist in this flux and must satisfy the equality

This contradiction can be resolved in two ways: either the longitudinal wave has no wave properties or the way of Landau's reasoning is incorrect. If we agree with the first (as all physicists have done unanimously), the whole number of other contradictions connected with the displacement current /t  will impose on this one, because any attempt of applying these mathematical operations, even in quasi-static cases (e.g., between the armatures of a discharging capacitor) will lead us to the similar result doubting the validity of Maxwell's regularity itself. But if we return to the adduced Landau's computations and take the divergence of the picked out region from the right and left parts of

i.e

and use the Lorenz calibration

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then, noting that in space free of charges div = 0 , we yield

whence only one solution can coincide with the boundary condition

Substituting now the value (, t) obtained from (12), we will automatically yield

where is the radius-vector. This means, (9) does not define the wave process, since it follows from (14) that

i.e., the same result that we know for the longitudinal waves.