On some periodicity effects

The talk is concerned with the modelling of wave propagation in and vibration of periodic elastic structures. Although analysis of wave-guide properties of infinite periodic structures is a well establish research subject, some issues have not yet been fully addressed in the literature. The aim of the talk is to illustrate these issues in simple examples and to discuss possible applications and generalisations. First, the eigenfrequency spectra of finite periodic structures are compared with the location of stop-bands for their infinite counterparts for a hierarchy of four mathematical models. In each case, special attention is paid to eigenfrequencies and eigenmodes of a single periodicity cell with appropriate boundary conditions. The influence of the amount of periodicity cells in a finite compound structure on its eigenfrequency spectrum is analyzed. Several features common for the considered models are found and discussed in the context of the existing knowledge on the subject. Another issue addressed in the lecture is an influence of the corrugation shape and amplitude on the broadness and location of the stop- and pass-bands in an infinite periodic structure. Explicit asymptotic formulae for the stop band borders are given and the topology of the stop band pattern is explained.