18

S.B. Karavashkin and O.N. Karavashkina

The equations (2) and (3) are usual wave equations, a₁ and a₂  are the velocities of excitation propagation. We see that these propagation velocities of the transfer component w₁ and components w₂ and w₃ are different. Hence, a plane elastic wave consists of two waves propagating independently. In one of them the shift (w₁) coincides with the direction of propagation of the wave itself. Such wave is called longitudinal and propagates with the velocity a₁. In the other, the shift (w = w₂ j + w₃ k) belongs to the plane, orthogonal to the direction of its propagation. Such wave is called transversal and propagates with the velocity a₂. Thus, in an elastic medium two velocities of sound exist" [3, p. 413-414].

At the same time, in registering seismic vibrations, "the records reveal three wave groups. First come the waves in which vibrations are mainly longitudinal. These are extension waves, their propagation velocity is large. Later the distortion waves arrive, the motion in them is mostly transverse. And finally, the surface waves whose amplitude is high in comparison with the amplitudes of two other types of wave. If this last group consists only of Rayleigh waves, there must be both vertical and horizontal components with these first predominating. It was practically found that it is not so, sometimes the vertical component is totally absent. For Rayleigh waves the direction of horizontal component vibrations must be parallel to the propagation direction, while we often reveal horizontal components parallel to the wave front" [2, p. 29]. "The difference between the constructions and plots of earthquakes is striking, where both the preliminary shocks and the main shock consisting of a large quantity of vibrations going here and there one tries to explain by the wave dispersion (dependence of the wave propagation velocity on the wavelength) caused by the insufficient homogeneity of medium" [4, p. 24]. "Love [5] has expressed the thought that these waves can be explained, if supposing that the elasticity and density of the outer layers of the Earth differ from their values inside. He showed that transverse waves can propagate along such outer layer without propagation in depth. Stoneley [6] has considered a more general problem of the wave propagation at the separation surface of two solid media. He showed that in media there must propagate the waves similar to Rayleigh waves, and their amplitudes must reach the maximum at the separation surface. Stoneley has studied also the generalised type of Love wave which propagates along the interior stratum limited from both sides by the thick layers of material distinguishing by its elastic properties" [2, p. 30]. In the most general form the problem of surface waves propagation was considered by Sobolev [7].

None the less, the endeavours to correlate the theoretical and experimental results by taking into account the nonlinear features of medium does not lift the principal problem of the approach which we considered in the beginning of this section. If the difference of propagation velocities was caused by the inhomogeneity, stratified elastic structure, but not by Poisson deformation, then obviously the conception of the solution needs to be reconsidered, beginning with the level of linear modelling of processes.

These peculiarities reveal also in case of a wave propagating within the cylindrical rods. Consider a small element of a rod having the length x   and cross-section area A, then, according to [2, p. 47],

"

"

This equation describes the longitudinal waves propagation along the rod with the velosity . In the course of derivation of (6) one did not suppose that the rod is cylindrical, so this equation is equally applicable to the thin rods and beams having any shape of cross-section and not varying in length.

The approximation of such approach resides in the supposition that plane cross sections of the rod remain plane in the strength waves transmission, and the strength is distributed equally in each cross section. However the longitudinal lengthening and shortening of the rod sections are certainly accompanied by the transversal thinning and thickening, and the relation of transversal and longitudinal deformation is equal to Poisson's ratio v. This transversal motion results the inhomogeneous distribution of strength in the rod cross-section, so that the plane cross-sections distort" [2, p. 47- 48].