High-Frequency Asymptotics, Homogenisation and Localisation for Lattices

A two-scale approach, for discrete lattice structures, is developed that uses microscale information to find asymptotic homogenised continuum equations valid on the macroscale. The development recognises the importance of standing waves across an elementary cell of the lattice, on the microscale, and perturbs around the, potentially high frequency, standing wave solutions. For examples of infinite perfect periodic and doubly periodic lattices, the resulting asymptotic equations accurately reproduce the behaviour of all branches of the Bloch spectrum near each of the edges of the Brillouin zone. Lattices in which properties vary slowly upon the macroscale are also considered and the asymptotic technique identifies localised modes that are then compared with numerical simulations.