SELF

18

S.B. Karavashkin and O.N. Karavashkina

The resulting shift of line elements is

As we see, the sought wave in the studied region of line is a progressive wave with the time-variable amplitude that propagates from the source to the boundary (since the variable parameter x is negative). The frequency of this wave is proportional to the velocity of boundary motion, which is equal to

not to the frequency of exciting force.

The wave amplitude varies harmonically and has a complex form. Its value is

the frequency of amplitude variation is

and the phase of amplitude for all elements of considered region of the elastic line depends on the initial position of the boundary

We see that (6) describes the modulated wave with carrying frequency _p and modulation frequency  _A. The modulation parameters considerably depend on the initial position of boundary and its shift velocity. At

the modulation disappears and solution (6) transforms into

As we see, in this case there will take place a progressive wave propagating from the point of external force application to the boundary, doing not changing its amplitude.

Formally, (11) is the boundary of amplitude inversion of studied wave, in transition through which the phase of modulating oscillations changes by 180^?. Factually, the meaning of (11) is other. If we substitute (7) into (11), we will yield

or

where c is the propagation velocity of wave process along the line. Thus, (11) determines the condition of inversion at the speed of boundary motion equal to the wave propagation velocity. And as far as the inverse wave reflected from the boundary is not independent but determined by the fact that the straight wave catches up with the boundary, from the physical view, if (11) was true, and if _p exceeds , the solution (6) becomes impossible because of absence of conditions of inverse wave formation. So (11) determines the limits in which the studied solution is realisable.