Wave motion in continuous mono-coupled locally periodic structures

This paper deals with wave propagation in locally (i.e., bounded) periodic continuous systems where the wavefield cannot entirely be described in terms of Bloch waves. To gain better insights into this problem, use is made of the analogy, from wave propagations perspective, between homogeneous systems and their locally periodic counterparts, by viewing the former as equally locally periodic but with arbitrary small periods (Saeed and Vestroni (1998)). The effect of truncation of periodicity is illustrated by the response of locally periodic rod and beam models. It is found that, the wave solution is made of both Bloch waves, carrying the contribution of the repeated discontinuities, and conventional wave modes defining the transformation from the physical domain to the wave domain. This transformation matrix is dependent on whether the number of interior degrees of freedom is equal to or greater than that of the coupling coordinates between cells. In the former case (the monomode rod) it simply corresponds to the conventional wave basis describing the properties of the medium at the (left) edge of the unit cell regardless of its specific form, while in the latter case (the beam model) it changes with change of the unit cell configuration because information carried by Bloch modes about the scattering process is incomplete. •The response in physical coordinates of locally periodic rod and beam models is analogous in form to the homogeneous mono-mode case where the Bloch transmission and generation modes replace the corresponding conventional wave modes provided that the outer ends have simple conditions.•Wave solution, obtained by converting the overall mechanical transfer matrix into its wave version via an appropriate conventional wave transformation matrix for each model, is expressed in terms of single cell scattering properties, Bloch wavenumber of the infinite problem and the number of cells.