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If an active resistance appeared in the line, the aperiodical and critical vibration regimes become impossible. There appears the damping, but not exponential, as in case of distributed lines. This damping has a complex power depending on the line parameters and external force frequency. And the value of resistance r effects not only on the amplitude but also on the phase characteristics of the vibration process. In Fig. 6 we show the damping and phase delay as a function of frequency for a semi-finite elastic resistant line. One can see that at small resistance these plots fully repeat the processes as they occur in an ideal line, despite at the entire range the periodical regime is inherent in the line. At the band lower than critical frequency f₀ the damping is absent and phase varies with frequency as arc sine. At the overcritical band a power-type damping takes place in the line, and the phase stabilises at the value (-) causing the antiphase vibrations of the neighbouring elements. It is typical that these plots having kinks and variation of the functional regularities are described by the common analytical functions. In this way the exact analytical solutions show again their ability to transform when transiting from the complex models to those simpler.

The same for the phase delay. Conventionally, with incomplete solutions, it is thought that “when the neighbouring atoms move in antiphase, … the group velocity appears zero. So it is senseless to describe the wave propagation causing the phase delays exceeding 180⁰ ” [15, p.109] (i.e., exceeding the first Brillouin zone). This is a settled mind reflecting in the description of the physics of vibrations, in the modelling and in the calculations in crystal lattices. However one can see from Fig. 6b that at the overcritical frequency the delay phase never exceeds the Brillouin zone. This conclusion is basically important, since it enables us to improve essentially our concept of the group and phase velocities of the wave propagation in crystals.

In Fig. 7 we show the group and phase velocities of the wave propagation in a resistant line. One can see that in case of practically ideal line, at the subcritical band, the amplitudes of both velocities first abruptly grow in a very narrow initial domain and then fall with the growing frequency, which evidences the anomalous dispersion. At the critical frequency the group velocity vanishes, however at the overcritical band it does not remain zero but abruptly ascends up to infinity. With it the phase velocity linearly increases with the frequency, though on the above models we could observe only the antiphase damping vibrations of the elastic line neighbouring elements. But if the line resistance is not so small, which is more typical for real-world models, then the group velocity does not vanish at the critical frequency, and at the overcritical frequency its amplitude is finite, though large, and growing with frequency. It contradicts the conventional concept basing on the incomplete solutions. It is practically impossible to obtain these solutions on an ideal model, because when transiting to the aperiodical regime, the direct relation between the periodical, aperiodical and critical regimes losses and they are presented as the system of solutions. But using in this order the resistant line model, when the solutions are presented as a common functional regularity, and passing to an ideal (non-resistant) line as to the limit, one can see clear, are there the group and phase velocities at the overcritical band.

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