We consider the linear water-wave problem in a periodic channel which consists of infinitely many identical containers connected with apertures of width ɛ. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the continuous spectrum. We show that for small apertures there exists a large number of gaps and also find asymptotic formulas for the position of the gaps as ɛ → 0: the endpoints are determined within corrections of order ɛ^{3/2}. The width of the first bands is shown to be O(ɛ). Finally, we give a sufficient condition which guarantees that the spectral bands do not degenerate into eigenvalues of infinite multiplicity.