V.2 No 1

25

Homogeneous 1d resistant line

Considering an ideal elastic line as some limiting case of a resistant line, we can state that in this model, in the aperiodical regime also, the progressive wave will propagate, and its length will stabilise at  = 2a. We can judge of it, particularly, by a limiting curve to which the plots v_f () in Fig. 3 are tighten at r 0. With it we have to note the peculiarities that appear in the solutions as a result of idealisation. Particularly, in Fig. 3 we see that when the frequency tends to zero, the phase velocity also tends to zero. It will occur otherwise in ideal lines, and at 0 the phase velocity will tend to the conventional value (T_i/)^1/2, where T_i is the line stiffness. It is easy to check it, substituting the value from (11) to (27) instead _r and finding the limit at 0. None the less, as we will see below, real systems have some features of solutions, which one cannot yield on the idealised models, but it is desirable to take them into consideration, when analysing real systems.

The same for the group velocity of wave propagation. As we know (see e.g. [6, 9, 14]), at present, the energy transferred by a wave is associated with this velocity, considering that ”when the neighbouring atoms move in antiphase, which is realisable for

i.e. for the wavelength = 2a, the group velocity appears zero'' [14, p. 109]. Due to it, at the overcritical band the along-line energy transfer is absent, since ''the bound of Brillouin zone

will correspond not to a progressive but to a standing wave'' [14, p. 110].

To plot the group velocity with respect to frequency v_g () in case of elastic lumped line, we have to account that not the phase delay _r but is in (17) an independent variable over which we differentiate. So

Finding the derivative _r / from (17) and substituting its value to (28), we will determine the group velocity of wave propagation with respect to frequency. A typical plot of this regularity is shown in Fig. 4.

Fig. 4. The group velocity v_g of wave propagation in the line against the frequency f of external force affecting the line

First of all we see the alike patterns of regularity v_f () in Fig. 3 and v_g () in Fig. 4 at the subcritical band and small r. In both plots we see an abruptly increasing velocity at the ultralow frequency band followed by the anomalous dispersion up to the boundary frequency ₀. At  =₀ the phase velocity takes some finite value, while the group velocity in case of an ideal line really vanishes. But only for an ideal, not resistant line. In presence of resistance it does not take place, and at the boundary frequency, the more resistance is the more group velocity differs from zero.

At the overcritical band, the more group velocity increases the less resistance r is. But despite the large growth, its value stays finite. Hence, for all resistant elastic lines (even at an infinitesimal resistance), at the overcritical band the along-line energy propagation takes place, though with a large amplitude damping.

In a limiting case of an ideal elastic line, the group velocity at the overcritical band turns to infinity, and we have to take this peculiarity into account when analysing the results obtained with a definite idealisation of modelled process.

At the large resistance, v_g () becomes a monotonously growing function of frequency without peculiarities at the point of critical regime. The pattern of its dependence on frequency is also similar to v_f ()at large r.