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

Andrea Cancedda, Valeria Chiadó Piat, Sergei A. Nazarov and Jari Taskinen

Mathematische Nachrichten, 2022, vol. 295, issue 4, 657-682

Abstract: We consider the linear water‐wave problem in a periodic channel Πh⊂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)$O({h})$.

Date: 2022
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