ABSTRACT
This chapter introduces some fundamental concepts and methods required for the analysis of wave propagation in multi-scale solids. It begins by introducing the concept of dispersion, initially for the classical example of linear water waves, then followed by one-dimensional mass-spring chains. The chapter summarizes elements of Bloch-Floquet theory for infinite periodic media. It examines discussion of asymptotic approximations of high-contrast continua by discrete lattices. The chapter proceeds to study a one-dimensional transmission problem where two dissimilar lattices are joined by a "structured interface." It introduces the notion of the transmission matrix. An important connection is established between formally distinct Bloch-Floquet problem and the transmission problem. The chapter moves on to problems for defects in otherwise periodic media and analyses the associated localised waves. It also summarises the approach of Craste for the asymptotic homogenisation of periodic media at finite frequencies, which has proven extremely useful in the analysis of dynamic problems in multi-scale media.
