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The convergence of spectra via two-scale convergence for double-porositymodels is well known. A crucial assumption in these works is that the stiffcomponent of the body forms a connected set. We show that under a relaxation ofthis assumption the (periodic) two-scale limit of the operator is insufficientto capture the full asymptotic spectral properties of high-contrast periodicmedia. Asymptotically, waves of all periods (or quasi-momenta) are shown topersist and an appropriate extension of the notion of two-scale convergence isintroduced. As a result, homogenised limit equations with none trivialquasimomentum dependence are found as resolvent limits of the original operatorfamily, resulting in limiting spectral behaviour with a rich dependence onquasimomenta.

Quasi-periodic two-scale homogenisation and effective spatial dispersion in high-contrast media