We address a spectral problem for the Dirichlet-Laplace operator in a waveguide II ^ε. II ^ε is obtained from an unbounded two-dimensional strip II 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, O(1) and O( ^ε) respectively, where 0 < ^ε ≪ 1. We look at the band-gap structure of the spectrum σ ^ε as ε → 0. We derive asymptotic formulas for the endpoints of the spectral bands and show that σ ^ε has a large number of short bands of length O( ^ε) which alternate with wide gaps of width O(1).