18-19

S.B. Karavashkin

It means that despite we study the wave process simultaneously on the entire surface of the picked out region at the only moment t₀ , due to the wave time-delay the value G_i does not vanish but depends on the wave delay time t_i in the picked out region.

When diminishing the size of picked out region V_0i , i.e., when x_i  and correspondingly at t_i tend to zero, on the basis of (20) we yield

As we see, in the considered model of a 1D wave process the divergence of vector does not vanish.

General proof of this theorem presented in [7, pp.28-33] and [8, pp.234-243] leads us to the following regularity:

i.e., when the wave flux propagates in a source-free space, the divergence of vector of flux is proportional to the scalar product of the particular derivative of this vector with respect to time into the unit vector of the studied flux direction.

V.A. Atsukovsky [9, pp.172-173] has obtained an alike result through the differential forms.

5. Conclusions

The carried out investigation showed that in a semi-finite model of an elastic line on whose free end a force inclined to the line axis acts the inclined waves propagate, and they are described by an implicit function. This conclusion can be easy extended to the finite lines, to the free vibrations in an elastic line and to the distributed lines vibrations.

The general solution in the form of implicit function is generalising for the conventional solution of a wave equation being the superposition of travelling waves; this essentially changes the concept of the non-linear vibration pattern in the form of inclined waves propagating in elastic systems.

We also have ascertained that under an inclined force action the line elements move by an elliptic trajectories.

If taking into account the dynamics of processes, we see that in dynamical fields the divergence of vector is not zero, as conventionally, but is proportional to the scalar product of the particular derivative of this vector with respect to time into the unit vector of the studied flux direction. When transiting to the stationary processes which do not depend on time directly, the divergence naturally vanishes, and for these stationary processes the Ostrogradsky-Gauss theorem is true.

References:

1. Karavashkin, S.B. Exact analytical solution on infinite one-dimensional elastic lumped line vibration. Materials, Technologies, Tools. The Journal of National Academy of Sciences of Belarus, v.4, 1999, #3, pp.15-23 (Russian)

2. Karavashkin, S.B. Exact analytical solution of the finite 1D elastic lumped line vibration. Printed version: Materials, Technologies, Tools. The Journal of National Academy of Sciences of Belarus, v.4, 1999, #4, pp.5-14 (Russian). Electronic version:  Mathematical Physical Preprint Archive http://www.ma.utexas.edu/mp_arc/c/02/02-79.ps.gz (English, Postscript)

3. Karavashkin, S.B. Some features of the forced vibrations modelling for 1D homogeneous elastic lumped lines. Printed version: Materials, Technologies, Tools. The Journal of National Academy of Sciences of Belarus, v.5, 2000, #3, pp.14-19 (Russian). Electronic version:  Mathematical Physical Preprint Archive http://www.ma.utexas.edu/mp_arc/c/02/02-89.ps.gz (English, Postscript)

4. Pohl, R.W.. Mechanics. Acoustics and heat theory. The State PH for Technical and Theoretical Literature, Moscow, 1957, 484 pp. (Russian, translation from German).

5. Karavashkin, S.B. On the new class of functions being the solution of the wave equation. SELF Transactions, 1 (1994), pp. 57-67. Eney, Kharkov, Ukraine, 1994, 118 pp. (English)

6. Korn, G.A. and Korn, T.M. Mathematical handbook for scientists and engineers. Nauka, Moscow, 1968, 720 pp. (Russian; from edition: MGRAW-Hill Book Company, Inc., New-York - Toronto - London, 1961).

7. Karavashkin, S.B. On longitudinal EM waves. Chapter 1. Lifting the bans. SELF Transactions, 1 (1994), pp.15-48. Eney, Kharkov, Ukraine, 1994, 118 pp. (English)

8. Karavashkin, S.B. Transformation of divergence theorem in dynamic fields. Archivum Mathematicum, 37 (2001), 3, pp.234-243

9. Atsukovsky, V.A. General etherodynamics. Energoatomizdat, Moscow, 1990, 278 pp. (Russian)