Archivum mathematicum

235 - 236

S.B. Karavashkin

This difference leads to such a fact that, for example, in the basic equations of the method used for studying the EM radiation attenuating above the Earth surface, the vertical dipole radiation is presented as [6, p. 130]:

(where w(r)   is the ‘attenuation factor’ and k₀  is the wave coefficient - S.K.), in which the time dependence is deleted from a conventional expression for a travelling wave, and the radiation field is considered as a stationary one. Such a simplification allows the authors to express “the field at the arbitrary point of a plane with the help of Green function” [6, p. 130].

At the same time, when studying the vector potential (, t)  produced in the far of dipole having the dipole momentum d(₀) (where ₀ is a time parameter), as a result of calculation, Levitch [7, p. 107] obtains

(where c   is the light velocity; r  is the distance from dipole; is the direction from the dipole to the studied point). The expression (5) basically differs from (1) by having a time-dependent right-hand part. This is due to the difference between dynamical and stationary fields shown above.

In some cases, as for example, when studying the attenuation of EM radiation from a stationary source, the difference is negligible. But in a number of problems, if the source amplitude varied in time or if the problem requires knowing the momentary wave parameters value at some point or region, such an approximation becomes incorrect, and we need to take into account the dynamical pattern of process. The more that in the beginning of the 20^th century Eichenwald noted that for Pointing vector also [8, p. 123], “if we apply the equation

(where f_n   is the projection of vector f  to the external normal to the surface element ds; W  is the density of EM energy; is a picked out region - S.K.) to a finite region, then, generally speaking, its right part will be non-zero, and the electromagnetic energy within this region will vary in time”.

This problem is solvable if introducing the correspondence between the conventional definition of divergence (2) and the formulation of theorems proved on the basis of this definition. This will generalise the divergence definition per se for the dynamical case and will refine the methods to investigate the EM fields in space.

The investigation that we will carry out in this paper will be targeted to this problem, and the most important its results will be described below.

2.  Preliminary analysis for the case of 1D flux

The divergence of vector in dynamical fields is convenient to be studied, beginning with a simplified model of 1D flux. Proceeding from the fact that the divergence definition “relates to any vector function, not only to the electrical field; to denote this function, we will use the symbol F(x, y, z). In other words, in the meanwhile we will give a preference for mathematics before physics and will name F simply – a vector function in general form, meaning, of course, 3D space” [9, p. 69].

p. 236

Let in some bounded connective, source-free space region   propagate a plane-parallel wave, whose force vector (x, t) has a conventional form

where   is the frequency of the given vector variation and k   is the wave coefficient. Pick out of this region four surfaces a₀, a₁, a₂, a₃  perpendicular to the wave propagation direction , and form with their help three picked out regions V₀₁, V₀₂, V₀₃  bounded by the corresponding surfaces and the lateral surface connecting them, as shown in Fig.1, above. Considering the 1D pattern of wave and that (x, t) is parallel to the lateral surface of the picked out regions, further we will not take into account the lateral surfaces.

Fig. 1. The time-dependent diagram for the investigation of vector flux