  This paper is concerned with the asymptotic description of high-frequency
waves in locally periodic media. A key issue is that the Bloch-dispersion
curves vary with the local microstructure, giving rise to hidden singularities
associated with band-gap edges and branch crossings. We describe an asymptotic
approach for overcoming this difficulty, and take a first step by studying in
detail the simplest case of 1D Helmholtz waves. The method entails matching
adiabatically propagating Bloch waves, captured by a multiple-scale
Wentzel-Kramers-Brillouin (WKB) approximation, with complementary
multiple-scale solutions spatially localised about dispersion singularities.
Within the latter regions the Bloch wavenumber is nearly critical; this allows
their homogenisation, following the method of high-frequency homogenisation
(HFH), over a naturally arising scale intermediate between the periodicity
(wavelength) and the macro-scale. Analogously to a classical turning-point
analysis, we show that close to a spatial band-gap edge the solution is an Airy
function, only that it is modulated on the short scale by a standing-wave Bloch
eigenfunction. By carrying out the asymptotic matching between the WKB and HFH
solutions we provide a detailed description of Bloch-wave reflection from a
band gap. Finally, we implement the asymptotic theory for a layered medium and
demonstrate excellent agreement with numerical computations.
