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Spectral gaps for the linear surface wave model in periodic channels

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