The features of inclined force acting

V.6 No 4, pp 13 - 19	15
The features of inclined force acting

From the carried out brief investigation we can see that far from always inclined vibrations arise as a consequence of non-linear processes in an elastic system, as it was supposed before. Inclined waves can arise quite naturally under action of a force inclined to the wave propagation direction. And this conclusion can be quite simply extended to a most wide spectrum of the vibration processes. 3. The line elements motion trajectory Drawing our attention to a considered line elements motion trajectory, we see that this trajectory has a form of an ellipse circumscribed around the point of rest of an element. And the wave inclination forms due to the shift phase of the element-to-element motion along these elliptic trajectories. The presented vibration structure is well-known in physics, particularly in wave processes in unbounded volumes of liquid. “In a wave, the motion of liquid is non-stationary. So the separate particles trajectories are far from being time-coinciding with the lines of current. They have absolutely other form. With the small amplitudes they are circumferences in a great approximation. We find these circular trajectories both on a surface and in depth of liquid. Only in the most upper layers the diameters of circular ways are the most large” [4, pp. 300-301]. Indeed, the spatial vibration processes have their peculiarities. None the less, it is typical that the basic regularities can be run down already in a 1D model. It also follows from the obtained solutions that the elements vibration ellipsoidal pattern retains as in the critical as in aperiodical regimes. Due to it, in this last case there in the line forms a complex wave fast-damping along the line, and this is one more feature that the exact analytical solutions demonstrate. 4. The new class of functions being the solution of the wave equation [5] We can extend the above generalisation also to the solution of the wave equation on the whole. It is known that the hyperbolic-type differential equation
	(12)
has general solution [6, p. 300]
	(13)
where c = /k is the wave propagation velocity - i.e., in the form of two explicit functions with respect to (x - ct) and (x + ct) correspondingly. Up to now this solution was thought the only and complete, due to the theorem of uniqueness of solution of differential equation. None the less, there exists one more class of functions being the solution of (12) but not taken into account by the solution (13). We can present the general form of this class of functions as
	(14)
here ₁(y) and ₂(y) are some double-differentiable functions. In other words, (14) belongs to the class of implicit functions whose regularities of behaviour and technique of differentiation and integration essentially differ from such for explicit functions. It is important that, while for explicit functions definite systematisation of differential equations has been created and for definite class of these equations the regularities and schemes to obtain solutions have been defined, for implicit functions all these developments are absent. Naturally, for today the correspondence of (14) to (12) can be checked only by the most simple way – by direct substitution (14) into (12). For it, basing on the known laws of implicit functions differentiation, find first and second particular derivatives of (14) with respect to x and t. To simplify the calculation, consider a half of the right part of (14)
	(15)
where

The first derivatives have the form
	(16)
The second derivatives after transformation and substitution of expressions (16) take the form
	(17)
Substituting (17) into (12), we obtain the required. Similarly we can prove the second part of expression (14) to be corresponding to the equation (12).