16

S.B. Karavashkin

Fig. 4. Inclined vibrations described by the implicit function being the solution of a wave equation at ₁ = 45^o   and ₂ = 60^o

Thus, the solution (14) 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 nonlinear 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 fully corresponds to the inclined vibration in an 1D line considered above.

5. The change of divergence theorem 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.

Fig. 5. The time-dependent diagram to study the vector of flux through the selected volume

The basic 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. In its turn, the Ostrogradsky-Gauss theorem is known 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

Select within this region four surfaces a₀, a₁, a₂, a₃ perpendicular to the wave propagation direction, and form with their help three selected volumes V₀₁, V₀₂, V₀₃ bounded by the related surfaces and lateral surface connecting them. Noting that the wave is 1D and (x,t) is parallel to the lateral surface of the selected regions, in the further consideration we will disregard the lateral surfaces.

Basing on 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 selected regions V₀₁, V₀₂, V₀₃ were finite, study first the expression

where i =1, 2, 3 . For it, plot the variation of (x,t) in space and time, noting the progressive pattern of wave process, and determine _0i = _i - ₀  for all selected regions. This will be easy, given 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 selected moment of time at the studied point into the area of related surface of the selected region. The yielded results are shown in Fig. 5 (bottom and right). As we see, _0i varies in time for all selected regions and is different for all surfaces a₀, a₁, a₂, a₃ , though we calculate in relation to the surface a₀ common for all regions. Furthermore, if on the basis of yielded values _0i we plot the regularity _0i (t), we will see that both amplitudes and phases of the fluxes difference variation _0i are different for all selected regions, the same as these parameters are different for G_i. Moreover, with the decrease of selected region, G_i  grows, tending to some finite limit value. This corroborates that the flux of vector through the selected region is time-inconstant in dynamical fields. And the fact of _0i variation is caused not by the spatial parameters but namely by the progressive pattern of the wave process, and we can easy prove it mathematically.