SELF

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S.B. Karavashkin and O.N. Karavashkina

4. Limit passing to a distributed line

The exact analytical solutions (19) yielded for a lumped line can be easily extended to that distributed. To do so, we have to substitute the parameters characterising the discrete mass distribution for a corresponding system of parameters characterising 1D system of distributed masses.

Introduce

where  is the elastic line density, x is the momentary coordinate of the studied point of a line in the state of rest, x₀ is the studied point coordinate in the state of rest, T_l  is the line stiffness, is the mechanical specific resistance of a line, a is the distance between the elements of a lumped line.

Noting (29), we can determine the transformation of parameters (16)–(18) at a 0. First of all, at small a , the parameters A,  B,  C, D  can be written down with regard to (29) as follows:

Substituting (30) into (16), we obtain accurate to a:

where the damping coefficien

contains only the values corresponding to a distributed system.

In further studying the parameter R, we have to note its power in (15). Then from (29) and (31) we can write

It means, when transiting to a distributed line, the along-line damping is described by an exponential regularity. In transiting to an ideal line, i.e. at 0, according to (31), also tends to zero, as it was expected. The dependence () especially reveals at low frequencies, when /  is comparable with .