Modelling microstructured media: periodic systems and effective media

Abstract

My aim in these lectures is to give a broad overview of the Mathematics and Physics of perfectly periodic systems, drawing heavily upon the literature of solid-state physics: it is essential to understand how structure on a micro-scale affects longer scale macro-scale behaviour and periodic systems are a naturally place to begin. Periodic systems are, on one hand, quite special and the constructive interference created by periodicity leads to strong effects that we shall see later, but on the other hand many natural and man-made structures exhibit, at least some, general periodic structure. After developing the language of periodic systems we will turn our attention to the development of asymptotic “effective” media that are posed entirely upon the macro-scale. Importantly we will develop asymptotic theories valid at high frequencies. A general approach valid for continua, semi-discrete (frame) and fully discrete (mass-spring) systems will be developed. If time allows we will then look further into some of the remarkable physics that can be seen when waves move through structured media: defect states, all-angle negative refraction and ultra-refraction.