SELF

28

S.B. Karavashkin and O.N. Karavashkina

To yield the sought solution for a distributed line, we have to determine the ultimate value of parameter _0r characterising the wave delay in a line as a whole. Substituting the values A and B from (30) to (18), we yield

As we see from (37), the same as in previous cases, the line resistance affection on _0r is limited by the low-frequencies band and depends on / too. In this connection, we can think this ratio determining the resistance affection on the pattern of vibration process in an elastic distributed line.

The typical plot for _0r ()  is shown in Fig. 5.

Fig. 5. Common delay phase _0r of the signal in a distributed line against the frequency f of external force

It fully corroborates the above analysis of _0r. We can add only one more feature. To determine the resulting phase delay of vibration process as a whole, we have to consider _0r in (15) together with the complex unity (- j), which shifts the solution by (- /2). Due to it, for a resistant line at  0 the total phase delay tends to (- /4), while in an ideal line it is (- /2). With growing frequency, at small resistance, the total phase delay quickly equalises with the value typical for an ideal line. But at large resistances this process naturally retards.

On the basis of determined parameters R, _0r and k_r we can write the general solution describing the vibration process in a semi-infinite distributed elastic line in presence of resistance . Substituting (32), (34) and (37) to (15) and using (29), we yield

The expression (38) has retained the main features of (15). Just as in (15), the resistance effects both on the vibration in a line as a whole and on the along-line excitation transfer, and this effect is especially strong at low and ultralow frequencies. This connection of solutions is quite natural, since the solutions for a distributed line correspond to an initial frequency band of solutions for a lumped line. As is shown in [13], to model a lumped line by means of that distributed is permissible at a condition

If disregarding this condition, the solution (38) will lose its accuracy of the line process description. In particular, (38) does not describe the processes at the bands of boundary and overcritical frequencies.