The fundamental equations governing the linear, oscillatory dynamics of structures consisting of repeated similar sub-structures are examined. The systems considered are mono-coupled and undamped, and they are examined by using wave, modal, receptance and finite element (F.E.) analyses. These analyses are aimed at providing an improved insight into the vibratory behaviour of engineering structures which exhibit periodicity. In particular they are directed towards the plated structures with evenly spaced welded stiffeners that are commonly employed in modern engineering designs: e.g., ships, aircraft, etc. Results originally derived from studies of the microscopic behaviour of the constituent atoms of crystalline solids are used in this work, and some of the differences of emphasis highlighted. The effects of deviations from perfect periodicity are considered, particularly with reference to the dramatic changes in mode shapes that sometimes occur. Nonetheless, it is demonstrated that in the present context, all the methods of analysis used are capable of giving similar results (within anticipated accuracies), indicating the importance of correctly modelling a structure, as compared with the exact details of how the model is then analyzed.