This paper is concerned with waves in locally periodic media, in the high-frequency
limit where the wavelength is commensurate with the period. 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 suggest an asymptotic approach for overcoming this difficulty, which
we develop in detail in the case of time-harmonic waves in one dimension. The method entails match-
ing adiabatically propagating Bloch waves, captured by a two-variable Wentzel–Kramers–Brillouin
(WKB) approximation, with complementary multiple-scale solutions spatially localised about disper-
sion singularities. The latter solutions, obtained following the method of high-frequency homogeni-
sation (HFH), hold over dynamic length scales intermediate between the periodicity (wavelength)
and the macro-scale. In particular, close to a spatial band-gap edge the solution is an Airy function
modulated on the short scale by a standing-wave Bloch eigenfunction. Asymptotically matching the
WKB and HFH solutions in this scenario yields a detailed description of Bloch-wave reflection from
a band gap, which is shown to be in excellent agreement with numerical computations for a layered
medium