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S.B. Karavashkin, O.N. Karavashkina

The same as in previous example, the force acting on the line interior element has bifurcated the solution into two intervals. On the interval 1 i k, in the periodical regime, there has arisen a standing wave having some phase delay (2k - 1) depending on the number of the picked out element k, and on the interval k i the progressive wave has retained, and its amplitude at the line specified parameters also depends on the number k. Hence, at

the progressive wave amplitude vanishes, and the vibrations are located in a line quasi-finite section bounded by the line end and the external force application point. At the same time, unlike the finite line, vibrations in a quasi-finite section are non-suppressible.

These distinctions are reflected in the diagrams presented in Fig. 4.

Going on comparing finite and semi-finite lines, note that at k = 1 , i.e. in case when the force acts upon the first line element, the solutions (58)- (60) transform into the corresponding solutions of [2] losing the multipliers being specific namely for the given structure of the generalising model, which makes the reverse transition also impossible.

4. The feature of the line heterogeneity at the external force application point

On the presented examples we can run down a number of common features caused by the external force acting on the line interior elements. In both cases the point of application mattered as a heterogeneity on which the vibrations pattern transforms. Due to it, in a finite line there have formed two sections with different vibration patterns. In a semi-finite line there formed a quasi-bounded section with the standing wave, and in a infinite line in [2] there arose two progressive waves propagating in opposite directions, which enables us speaking about the feature of a line heterogeneity at the external force application point.

5. The progressive wave parameters in an infinite elastic line

Investigating the features of models for homogeneous elastic lines, we should note one more peculiarity connected with the parameters of progressive waves propagating along the infinite lumped lines. For these lines the wavelength has quite conventional pattern, because not always it is connected with the specific lumped masses; i.e., far from always we can select two bodies at a distance of wavelength vibrating in phase. However, the body-to-body phase delay equal to 2  indicates also the definite wavelength fully characterising the line processes.

To determine mathematically, we have to suppose that at the wavelength distance from a picked out element there exists some fictitious line element vibrating in phase with this picked out element.

Noting that, in case of non-damping process, the distance between the elements vibrating in phase does not depend on the amplitude, the wavelength is determined by the expression

On the grounds of (62), the propagation velocity v in the line is determined by the expression

It is typical that in distinction from distributed lines, in the investigated models the propagation velocity does not remain constant and depends on the vibration frequency. Therefore, when a complicated-spectral-composition mechanical pulse is fed to the line input, this pulse does not hold its structure when propagating along the line. And only at << 1 the propagation velocity stabilises, coming in the limit to the known value

With it the wavelength will be

which also corresponds to the known value.