V.5 No 1

19

On longitudinal excitation of elastic medium having a moving boundary

To show it visually, in Fig. 2 we present the animated diagram of variation of oscillation amplitude in the elastic line, at the allowed range of subcritical frequencies.

Fig. 2. Wave amplitude variation in the line with the stationary source and moving boundary, under the following line parameters: = 1 kg / m , T_f = 1 N ,  = 3 1 / s , x₀ = 12 m   and the boundary speed  v = 2c / 3

When we compared the visually shown pattern of amplitude variation with the above analysis of solution, it can outwardly rise a feeling of some contradiction, as the diagram shows some standing wave that is as if extracted from the range bounds, while when analysing, we said of progressive wave pattern in the line. But there really is not a contradiction. If we take any point within the studied interval (1), a progressive wave will go through it towards the boundary, propagating at the velocity v along the line. But just at the boundary, due to the absolute reflection from it, we will see the node - and this makes an impression of standing wave extracted from the range bound. Moreover, if the observation point moves with the boundary, we will see its oscillations typical for a standing wave, as in this case the propagation velocity and observer velocity will be equal.

Thus, in this first study we revealed that in case of stationary source and moving boundary, a progressive wave modulated in amplitude by the frequency of external excitation will propagate along the elastic line with the velocity of boundary motion.