rod25

V.2 No 2	25
Compression waves in a rod

To find the regularity (, t) we have first to determine the regularity of a cross-section variation caused by the rod longitudinal deformation. Noting the results of item 2, we can write
	(30)
where and ' are the longitudinal and corresponding transversal deformations of the rod. Further, introducing ' and in the form
	(31)
we yield finally the dependence between the longitudinal and transversal deformation:
	(32)
or in tending , and correspondingly to zero,
	(33)
Finding the value of derivative / on the basis of solution (29), we can find the regularity of the rod radius variation under longitudinal vibrations in the form
	(34)
As we have predicted in the item 2, the solution (34) jointly with (29) shows that the propagation velocity of longitudinal and transversal waves is the same. In this way we get over the above contradiction connected with the possibility of a rod local thinning when shortening. To obtain the full pattern of a transversal wave propagation, we need noting that the value in (34) is determined at the points , not , the same as in case with the infinite point of a line. Thus, the regularity (, t) must be presented parametrically:
	(35)
The system (35) describes the transversally deformed wave propagating along the rod lateral surface. Its general form is shown in Fig. 3. We see from the construction that the wave deformation increases visually with the growing affecting force amplitude, and these waves have a form of swell propagating along the rod. Thus, the same as in case of 1D elastic line, we can state that a number of processes which up to now were considered as non-linear are quite describable in the frames of linear model.
Fig. 3. General form of transverse wave propagating in semi-infinite rod having finite section, under different amplitudes of external force F₀