21

Compression waves in a rod

3. The limiting process from the lumped elastic line to that distributed

Realising the succession to find the solution presented in the item 2 and describing the vibrations in a finite-cross-section rod, we will first use the results obtained in [14] in studying an ideal elastic lumped line, and on this basis we will determine the solutions for an infinitely thin distributed elastic line (rod).

In [14] we presented two blocks of solutions for a semi-finite elastic lumped line for forced and free vibrations correspondingly. For the present problem we are interesting in forced vibrations, because just this regime relates to the process of propagation of longitudinal compression waves. While free vibrations relate to the standing waves taking place in case of non-zero energy density in a line, as it was shown in [14].

In their turn, for forced vibrations in [14] three solutions were presented, dependently on relation between the parameter and the unity (where is the frequency of external force, m  is the line elements mass, s is the stiffness coefficient of a line). Three vibration regimes correspond to these three solutions: periodical ( < 1), aperiodical ( > 1) and critical ( = 1).

To process the limit passing to a distributed line, according to [15] we will be interesting in the periodical regime, because the critical and aperiodical regimes are impossible in a distributed line.

Thus, for the case < 1 the exact analytical solution for a semi-finite ideal elastic lumped line will be the following [14]:

where F₀  is the amplitude of the external force, _n is the absolute shift of the nth body from the state of rest, = arcsin , and n = 1, 2, 3, ... .

To make the limit passing to the solution for a distributed line, by analogy with [15] we have to introduce the correspondence between the parameters m, s,  _n, n  determining the lumped line and the following parameters: the line density , stiffness T , location of the studied point in an excited state and in the state of rest , which determine the processes in a distributed line. Conveniently do it by the following simple way, introducing

where a is the distance between the line elements in the state of rest.

With taking into account (13) and (14), the expression (12) takes the form