We study waves governed by the planar Helmholtz equation, propagating in an
infinite lattice of subwavelength Dirichlet scatterers, the periodicity being
comparable to the wavelength. Applying the method of matched asymptotic
expansions, the scatterers are effectively replaced by asymptotic point
constraints. The resulting coarse-grained Bloch-wave dispersion problem is
solved by a generalised Fourier series, whose singular asymptotics in the
vicinities of scatterers yield the dispersion relation governing modes that are
strongly perturbed from plane-wave solutions existing in the absence of the
scatterers; there are also empty-lattice waves that are only weakly perturbed.
Characterising the latter is useful in interpreting and potentially designing
the dispersion diagrams of such lattices. The method presented, that simplifies
and expands on Krynkin & McIver [Waves Random Complex, 19 347 2009], could be
applied in the future to study more sophisticated designs entailing resonant
subwavelength elements distributed over a lattice with periodicity on the order
of the operating wavelength.