69

On solution for an infinite heteroheneous line

In the first section there propagates the progressive wave whose phase shift is determined by the complex multiplier in the square brackets of (42):

In the second section we see a complex vibration process determined by the similar expression in brackets in (43). On one hand, the wave process in this section has the phase

depending on parameter x₀ nonlinearly. On the other hand, the wave amplitude depends on this parameter nonlinearly too. Its value is proportional to

The amplitude reaches its extreme values at the points

where p = 0, 1, 2, ... . In the first case the expression (47) is equal to v₂, and in the second case – to v₁. Depending on the relationship between the wave propagation velocities in related sections, at these points we observe the maximums and minimums of the wave process.

The results obtained for the first and second sections essentially differ from the conventional concept based on a simple superposition of the direct and reverse waves. According to this concept,

“the amplitude reflection coefficient

the amplitude transmission coefficient

(where A₁, A₂, B₁ are respectively the amplitude coefficients of direct, reflected and transmitted waves;Z₁=₀₁₁ =T/ ₁ is the mechanical impedance of the first section of an elastic line; Z₂=₀₂v₂ =T/ v₂ is the mechanical impedance of the second section of an elastic line,– Authors). These coefficients do not depend on and are the same for the waves of any frequency. They are real and do not introduce any phase shifts except the shift by changing the sign of the term” [11, p.123]. As we see from this analysis, when reflecting from the heterogeneity transition, there forms a complex wave process whose phase and amplitude depend directly on the external force frequency, the same as on elastic line parameters. These conclusions are true also for the transmitted wave.

In the third section of elastic line we observe the progressive wave having the along-the-line phase delay x₀/v₂ and common phase shift determined in (44) by the unit imaginary number and x_k/₁.

To see the typical vibration pattern, conveniently pass from the longitudinal to transverse vibrations. It will be sufficient for it to direct the external force perpendicularly to the line axis and to substitute in the solutions (42) – (44) the longitudinal shift x - x₀ by the transversal shift y . The typical diagram of transversal vibrations in a heterogeneous elastic distributed line is shown in Fig. 4. The pattern presented in it completely corroborates the above analysis of solutions (42) – (44).