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We consider the linear water-wave problem in a periodic channel $\Pi^h\subset \mathbb{R}^3$, which is shallow except for a periodic array of deeppotholes in it. Motivated by applications to surface wave propagationphenomena, we study the band-gap structure of the essential spectrum in thelinear water-wave system, which includes the spectral Steklov boundarycondition posed on the free water surface. We apply methods of asymptoticanalysis, where the most involved step is the construction and analysis of anappropriate boundary layer in a neighborhood of the joint of the potholes withthe thin part of the channel. Consequently, the existence of a spectral gap forsmall enough $h$ is proven.

Band-gap structure of the spectrum of the water-wave problem in a shallow canal with a periodic family of deep pools