6

S.B. Karavashkin and O.N. Karavashkina

Thus, we have proved in general case the

Theorem: In dynamical fields, the curl of vector is proportional to the vector product of an unit vector of the flux propagation direction by the derivative of the vector of flux with respect to time

4. Application of the curl theorem to dynamical fields

When proving the theorem, we agreed to omit the item of the studied fields potentiality. Both in the introduction and in proof we saw that if using the conventional concept of the vector potentiality (4), there arises an important question about the transformation of potential to solenoidal vector when transiting from stationary to dynamical fields. In the introduction, by examples of acoustical and electrodynamical models, we showed that two strongly potential dynamical sources can form the transversal dynamical field vector - i.e., by conventional concept, the solenoidal field.

To investigate this point in details, continue computations (6), decomposing the velocities ₁ and ₂  into the axes and (see Fig. 1). Then we yield

It follows from (27) that in an acoustical transversal field the vector of resulting velocity of molecules displacement remains potential too, though it is directed perpendicularly to the wave propagation direction, and the potential (r, t)   depends on r . A field described by this expression basically cannot be closed.

Such unexpected result showing a discrepancy with the conventional concept can be easy explained. When considering such points conventionally, one forms first the resulting field by potential fields superposition, and then, taking into account the far distance of the studied point from radiator, makes all necessary simplifications. With it by way of approximations the integral wave with the transversal vector is formed and analysed to determine the field potentiality. But writing down (6), we simplified it only partly, keeping the conditions that caused the transversal vector formation. It did not limit the solution a least but offered to consider more completely the question of vector potentiality.

The curl of given gradient of potential does not vanish; i.e., the lines of force of    do not close. As we showed in the proof, the production of non-zero right part in (25) is conditioned not by spatial distribution of the force lines, but by the time delay of process. Just this is why, when transiting to stationary fields, the right part of (25) vanishes in the region free of stationary vortexes.

As the existence of transversal dynamical potential vector has been proved to be possible, the question arises about its dynamical sources and sinks - the item which served as the grounds to R.V. Pohl and other investigators to close the lines of force of dynamical electrical field. To answer this question, refer to [10]. We have proved there that the divergence of longitudinal vector of dynamical field is non-zero. But according to the conventional concept, it can be only if within the studied region there are some non-compensated charges. As the problem was stated, they are absent there. The field itself takes their part due to its spatial delay. In this way the field ''retains'' the memory of the source at the moment of its generation. The same in case of potential transversal vector. In the near field this vector is undoubtedly potential, because for each point of field we always can determine its momentary source and sink which have formed the resulting direction of the field vector. At larger distances, when the wave leaves the source, the field also ''retains'' the memory of source at the moment of its generation. This memory might be absent only in case, if the wave propagation velocity was infinite. But in this case the right part of (25) automatically would vanish, which corroborates the said. Apropos, the progressive spatial wave process per se would be impossible at these conditions.