The wave processes in the infinite and finite periodic shells (cylinder shell, beam, and rod) are explored. Look-alike systems can model different elements of buildings, hydro-technical constructions, bridges, oil rigs, different pipes, etc. The statement of the problem is considered the rigorous statement. In the infinite systems, Floquet solution is founded. The comparison of the wave processes in the rod and beam is fulfilled. The energy fluxes in them are calculated. The main effects are explored with the attraction of the analysis of vibrations, corresponding to different pass- and stopbands. The dependence of character and «heterogeneity degree» of wave process in the finite systems via the position of corresponding wave number in relation to passbands is considered. The modes of free vibrations of a periodic cell in the case of its asymmetry are analyzed with special attention to edge effects via the parameters of the problem.