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We consider the spectral Dirichlet problem for the Laplace operator in theplane $\Omega^{\circ}$ with double-periodic perforation but also in the domain$\Omega^{\bullet}$ with a semi-infinite foreign inclusion so that theFloquet-Bloch technique and the Gelfand transform do not apply directly. Wedescribe waves which are localized near the inclusion and propagate along it.We give a formulation of the problem with radiation conditions that provides aFredholm operator of index zero. The main conclusion concerns the spectra$\sigma^{\circ}$ and $\sigma^{\bullet}$ of the problems in $\Omega^{\circ}$ and$\Omega^{\bullet},$ namely we present a concrete geometry which supports therelation $\sigma^{\circ}\varsubsetneqq\sigma^{\bullet}$ due to a new non-emptyspectral band caused by the semi-infinite inclusion called an open waveguide inthe double-periodic medium.

The spectrum, radiation conditions and the Fredholm property for the Dirichlet Laplacian in a perforated plane with semi-infinite inclusions