  We study the canonical problem of wave scattering by periodic arrays, either
of infinite or finite extent, of Neumann scatterers in the plane; the
characteristic lengthscale of the scatterers is considered small relative to
the lattice period. We utilise the method of matched asymptotic expansions,
together with Fourier series representations, to create an efficient and
accurate numerical approach for finding the dispersion curves associated with
Floquet-Bloch waves through an infinite array of scatterers. The approach lends
itself to direct scattering problems for finite arrays and we illustrate the
flexibility of these asymptotic representations on topical examples from
topological wave physics.
