16

S.B. Karavashkin

Thus, the solution (17) determines a whole class of implicit functions satisfying the linear wave equation. And the presence of a new class of functions being the solution of differential equation (12) does not violate a least the theorem of uniqueness of solution of differential equation, because under definite conditions

the expression (14) degenerates into (13). Hereby it is proved that the solution known before is a particular case of more general class of functions.

The found class of implicit functions determines a non-linear wave; its degree of deformation depends on the type of functions ₁(y) and ₂(y). For example, in the following particular case of expression (14) (see Fig. 4)

the solution of a wave equation (12) describes a progressive wave propagating along the axis x and inclined by the angle , which completely corresponds to the inclined vibration in an 1D line considered above.

4. The divergence theorem changing in dynamical fields

Studying the changes appearing in the wave physics solutions, we must touch briefly the changes in the vector algebra equations appearing when taking into account the dynamical processes in power fields.

The main conservation laws related to the vector flux are known to be formulated on the basis of Ostrogradsky-Gauss theorem about the flux of vector through a picked out space region. The Ostrogradsky-Gauss theorem is known in its turn to be formulated for stationary fields, i.e., for the case when the vector of flux does not depend on time directly. In dynamical fields the pattern essentially changes.

Actually, let in some bounded, connective, free of sources spatial region propagate a plane-parallel wave whose power vector (x,t)   has a standard form

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 volumes V₀₁, V₀₂, V₀₃ bounded by the corresponding surfaces and lateral surface connecting them. Noting that the wave is 1D and (x,t)   is parallel to the lateral surface of the picked out regions, in the further consideration we will not take into account the lateral surfaces.

On the basis of this model, consider a conventional definition of the divergence of vector (see, e.g., [5, p.166])

where V is the studied region containing the point (); S is the closed-loop surface limiting the region V;  here denotes the most distance from the point ()  to the points of the surface S.

Since the picked out regions V₀₁, V₀₂, V₀₃ were finite, study first the expression

where i=1, 2, 3. For it, plot the time-spatial variation of (x,t), noting the wave process progressive pattern, and determine _0i = _i - ₀ for all picked out regions. This will be easy, noting the wave front plane pattern and that the flux of vector is 1D. Therefore the integration is reduced to a simple multiplying of the vector parameters at the fixed moment at the studied point into the square of corresponding surface of the picked out region. The obtained results are shown in Fig. 5 (bottom and right). As we see, _0i varies in time for all picked out regions and is different for all surfaces a₀, a₁, a₂, a₃ , though we calculate relatively to the surface a₀ common for all regions. Furthermore, if we plot the regularity _0i(t) on the basis of obtained values _0i, we will see that both amplitudes and phases of the fluxes difference variation _0i are different for all picked out regions, the same as these parameters are different for G_i. Moreover, with the diminishing size of the picked out region, G_i grows, tending to some finite limit value. This corroborates that the flux of vector through the picked out region is time-inconstant in dynamical fields. And the fact itself of _0i   variation is caused not by the spatial parameters but just by the progressive pattern of the wave process, and we can easy prove it mathematically.