An asymptotic procedure based upon a two-scale approach is developed for wave propagation in a doubly periodic inhomogeneous medium with a characteristic length scale of microstructure far less than that of the macrostructure. In periodic media, there are frequencies for which standing waves, periodic with the period or double period of the cell, on the microscale emerge. These frequencies do not belong to the low-frequency range of validity covered by the classical homogenization theory, which motivates our use of the term ‘high-frequency homogenization’ when perturbing about these standing waves. The resulting long-wave equations are deduced only explicitly dependent upon the macroscale, with the microscale represented by integral quantities. These equations accurately reproduce the behaviour of the Bloch mode spectrum near the edges of the Brillouin zone, hence yielding an explicit way for homogenizing periodic media in the vicinity of ‘cell resonances’. The similarity of such model equations to high-frequency long wavelength asymptotics, for homogeneous acoustic and elastic waveguides, valid in the vicinities of thickness resonances is emphasized. Several illustrative examples are considered and show the efficacy of the developed techniques.

This study revisits Timoshenko beam theory (TBT). It discusses at depth a more consistent and simpler governing differential equation. The so-called second spectrum is also addressed. Then, we provide the asymptotic justification of the aforementioned differential equation along with detailed discussion of the boundary and initial conditions. The paper also presents remarks of historical character, in the context of other pertinent studies.

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.