  The convergence of spectra via two-scale convergence for double-porosity
models is well known. A crucial assumption in these works is that the stiff
component of the body forms a connected set. We show that under a relaxation of
this assumption the (periodic) two-scale limit of the operator is insufficient
to capture the full asymptotic spectral properties of high-contrast periodic
media. Asymptotically, waves of all periods (or quasi-momenta) are shown to
persist and an appropriate extension of the notion of two-scale convergence is
introduced. As a result, homogenised limit equations with none trivial
quasimomentum dependence are found as resolvent limits of the original operator
family, resulting in limiting spectral behaviour with a rich dependence on
quasimomenta.
