  In this paper, we investigate the boundary layer arising from the fast
internal waves in the Boussinesq equations with the Brunt-Vais\"al\"a frequency
of order $ \mathcal O(1/\varepsilon) $. For the homogeneous-in-height
stratification, previous work by \emph{Desjardins, Lannes, Saut, 3(1):153--192,
Water Waves, 2021} establishes uniform-in-$\epsilon$ estimates locally in time,
with additional constraints on the boundary data initially, which essentially
restricts the dynamics in the spatially periodic domain. Removing such
constraints, our goal is to investigate the general near-boundary behavior. We
observe that the fast internal waves will give rise to large growth of the
spatial derivatives in the normal direction of the solutions in the vicinity of
the boundary. To capture this phenomenon, we introduce an inviscid boundary
layer using a natural scaling. In addition, we investigate the well-posedness
of such a boundary layer system in the space of analytic functions.
