18-19

S.B. Karavashkin

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

When decreasing the size of selected region V_0i , i.e., when x_i and correspondingly 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 we give in [7, p. 28- 33] and [8, p. 234- 243]; it leads us to the following regularity:

i.e., when the wave flux propagated 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, p. 172- 173] has yielded an alike result through the differential forms.

5. Conclusions

We studied a semi-finite model of an elastic line on whose free end a force inclined to the line axis affects. The consideration showed that in such line, the inclined waves propagate, and they are described by an implicit function. We can easily extend this conclusion to the finite lines, to the free vibrations in an elastic line and to the vibrations in distributed lines.

We also found that the solution in the form of implicit function is the generalised for the conventional solution of a wave equation. This much changes the understanding of the pattern of nonlinear vibrations in the form of inclined waves propagating in elastic systems.

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

Accurately noting that the processes are dynamic, we derived that in dynamic fields the divergence of vector is not zero 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. In passing 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.

1. Karavashkin, S.B. Exact analytic solution of infinite 1D elastic lumped line vibration. Materials, Technologies, Tools (National Academy of Sciences of Belarus), 4 (1999), 3, 15- 23 (Russian). Electronic version: SELF Transactions, Archive (English, html)

2. Karavashkin, S.B. Exact analytical solution of finite 1D elastic lumped line vibration. Materials, Technologies, Tools (National Academy of Sciences of Belarus), 4 (1999), 4, 5- 14 (Russian). Electronic version: SELF Transactions, Archive (English, html) and 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 modelling of forced vibrations for 1D homogeneous elastic lumped lines. Materials, Technologies, Tools (National Academy of Sciences of Belarus), 5 (2000), 3, 14- 19 (Russian). Electronic version: SELF Transactions, 2 (2002), 1, 71- 85 (English, html) and 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. GITTL, Moscow, 1957 (Russian, translation from German)

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

6. Korn, G.A. and Korn, T.M. Mathematical handbook for scientists and engineers. Nauka, Moscow, 1968 (Russian; original 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), 15- 47. Eney, Kharkov, Ukraine, 1994

8. Karavashkin, S.B. Transformation of divergence theorem in dynamic fields. Archivum Mathematicum, 37 (2001), 3, 234- 243. Electronic version: SELF Transactions, Archive

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

The manuscript first received by "Materials. Technologies. Tools" on 03.02.2000