Spectral gaps for the linear water-wave problem in a channel with thin structures

We consider the linear water-wave problem in a periodic channel pi h subset of R2$\Pi <^>h \subset {\mathbb {R}}<^>2$, which consists of infinitely many identical containers and connecting thin structures. The connecting canals are assumed to be of constant, positive length, but their depth is proportional to a small parameter h. Motivated by applications to surface wave propagation phenomena, we study the band-gap structure of the essential spectrum in the linear water-wave system, which forms a spectral problem where the spectral parameter appears in the Steklov boundary condition posed on the free water surface. We show that for small h there exists a large number of spectral gaps and also find asymptotic formulas for the position of the gaps as h -> 0${h} \rightarrow 0$: the endpoints are determined within corrections of order h3/2${h}<^>{3/2}$. The width of the first spectral band is shown to be O(h)$O({h})$.