  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 deep
potholes in it. 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 includes the spectral Steklov boundary
condition posed on the free water surface. We apply methods of asymptotic
analysis, where the most involved step is the construction and analysis of an
appropriate boundary layer in a neighborhood of the joint of the potholes with
the thin part of the channel. Consequently, the existence of a spectral gap for
small enough $h$ is proven.
