80

S.B. Karavashkin, O.N. Karavashkina

This system has the following solution:

for the periodical regime ( < 1)

for the aperiodical regime ( > 1)

and for the critical regime ( = 1)

where

Comparing those with the corresponding solutions (13), (16) and (17) presented in [2], we see that in (50)-(52) the additional multipliers appear and the solutions themselves bifurcate into two ones describing the pattern of elements before and after the kth one, which reflects the formation of two standing waves in one line. With it the vibration diagram will essentially vary, depending on the number k and the frequency. Thus, in the periodical regime with

the vibration amplitude vanishes in some line sections, while in the other sections the vibrations remain. This is shown in Fig. 2 presenting the pattern of the process. In the left diagrams of Figures 2a and 2b, the vibrations in the left and right parts of the line are absent, while in the left diagram of Fig. 2c we see the standing waves of the complex structure. Depending on the vibration frequency, the pattern of standing waves can essentially vary, too. For example, with decreasing frequency there can form the standing waves having the oblique inflections divergent to both sides of the kth element.

In the aperiodical regime, the vibration structure also depends on the number k, though, in distinction from the periodical regime, vibrations themselves are non-annihilated in any line section. This is well seen in the right diagrams of Figures 2 a, b, c.