Abstract
AbstractWe consider the linear water‐wave problem in a periodic channel , 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 : the endpoints are determined within corrections of order . The width of the first spectral band is shown to be .
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