We address a spectral problem for the Dirichlet-Laplace operator in a waveguide \begin{document}$ \Pi^ \varepsilon $\end{document} . \begin{document}$ \Pi^ \varepsilon$\end{document} is obtained from repsilon an unbounded two-dimensional strip \begin{document}$ \Pi $\end{document} which is periodically perforated by a family of holes, which are also periodically distributed along a line, the so-called "perforation string". We assume that the two periods are different, namely, \begin{document}$ O(1) $\end{document} and \begin{document}$ O( \varepsilon) $\end{document} respectively, where \begin{document}$ 0 . We look at the band-gap structure of the spectrum \begin{document}$ \sigma^ \varepsilon $\end{document} as \begin{document}$ \varepsilon\to 0 $\end{document} . We derive asymptotic formulas for the endpoints of the spectral bands and show that \begin{document}$ \sigma^ \varepsilon $\end{document} has a large number of short bands of length \begin{document}$ O( \varepsilon) $\end{document} which alternate with wide gaps of width \begin{document}$ O(1) $\end{document} .