63

On solution for an infinite heteroheneous line

3.2

When ₂ exceeds the unit value, in the heavy part of line the aperiodical vibration regime settles. With it all the terms in the right-hand parts of the system (2) – (4) transform as according to (8), and this system takes the following form:

for  i k

for  k i n

and for  i n + 1

where

In the first section related to i k, there has retained the periodical process of progressive wave propagation with the phase delay [2(n - i) +1] ₁ and the amplitude depending on the parameters of both sections of line. However, the general phase delay has changed. If in (2) the vibration process has delayed by relatively to the external force phase, then in (11) the general delay is determined by the parameter (-₂). At ₂ +1 we have ₂ , and at ₂ we have , not zero.

The less difference between the element masses m₁ and m₂ is, the more vibration amplitude is. Basically, at m₁ m₂ the amplitude of _i tends to infinity too, this is determined by the multipliersand cos ₁. But the amplitude does not reach the infinite value, since at m₁ = m₂ in the light section the aperiodical vibration regime will also settle, and the solution (11) will transform accordingly.

When the expression in square brackets in (11) vanished, i.e. at