  We consider a family $\{\Omega^\varepsilon\}_{\varepsilon>0}$ of periodic
domains in $\mathbb{R}^2$ with waveguide geometry and analyse spectral
properties of the Neumann Laplacian $-\Delta_{\Omega^\varepsilon}$ on
$\Omega^\varepsilon$. The waveguide $\Omega^\varepsilon$ is a union of a thin
straight strip of the width $\varepsilon$ and a family of small protuberances
with the so-called "room-and-passage" geometry. The protuberances are attached
periodically, with a period $\varepsilon$, along the strip upper boundary. For
$\varepsilon\to 0$ we prove a (kind of) resolvent convergence of
$-\Delta_{\Omega^\varepsilon}$ to a certain ordinary differential operator.
Also we demonstrate Hausdorff convergence of the spectrum. In particular, we
conclude that if the sizes of "passages" are appropriately scaled the first
spectral gap of $-\Delta_{\Omega^\varepsilon}$ is determined exclusively by
geometric properties of the protuberances. The proofs are carried out using
methods of homogenization theory.
