V.5 No 1

21

On longitudinal excitation of elastic medium having a moving boundary

Given the said, we can build the model of excitation propagation in a moving frame, doing not taking into account the transformation of space and time measures. But before we write down the equations of direct and reflected waves, let us determine the time necessary, excitation to pass some given interval from the viewpoint of moving frame. For it, let us consider the following scheme. At some moment of time t₁ the source radiated a wavefront. Select in the axis x of stationary frame some point A, so that it was located within the studied interval and shifted with this interval with the same velocity v.

Fig. 5. Diagram helping to calculate the time necessary the wavefront to pass the distance from the source S to an arbitrary point A within the studied interval

Then, according to the diagram in Fig. 5, the wavefront passes the distance l from the source S to the point A, and

where from

Similar calculation for the wavefront reflected from the boundary R gives

Basically, this calculation is far from being original, many authors did so. We intentionally repeated it with the aim to show that in the view of moving frame, (16) and (17) merely formally can be represented as a confirmation that the velocities of source and wave were added. None the less, as we can see from construction and from the derivation, there is no summation of velocities. We see here an effect of joint motion of the source and observer. While the wave is propagated with the constant velocity as to the elastic line and does not depend on the motion of source. But the shape of oscillations is transformed, and below we will follow this transformation.

Having determined the delay time of wave process, we, by analogy with the previous item, can write down the expressions for the direct and reflected waves immediately in the moving frame, given in transition between the frames, the time intervals do not transform.

By an analogy with (3) and noting (16), the shift of line elements in the view of moving frame will be the following:

and for the reverse wave

To provide further convenience of transformation, simplify the expressions for delay phases in (18) and (19). For it, multiply and divide the fractions of both delay phases by the multipliers complementing the denominators to the difference of squares; we will yield