  This paper is devoted to the derivation and mathematical analysis of a
wave-structure interaction problem which can be reduced to a transmission
problem for a Boussinesq system. Initial boundary value problems and
transmission problems in dimension d= 1 for $2\times2$ hyperbolic systems are
well understood. However, for many applications, and especially for the
description of surface water waves, dispersive perturbations of hyperbolic
systems must be considered. We consider here a configuration where the motion
of the waves is governed by a Boussinesq system (a dispersive perturbation of
the hyperbolic nonlinear shallow water equations), and in the presence of a
fixed partially immersed obstacle. We shall insist on the differences and
similarities with respect to the standard hyperbolic case, and focus our
attention on a new phenomenon, namely, the apparition of a dispersive boundary
layer. In order to obtain existence and uniform bounds on the solutions over
the relevant time scale, a control of this dispersive boundary layer and of the
oscillations in time it generates is necessary. This analysis leads to a new
notion of compatibility condition that is shown to coincide with the standard
hyperbolic compatibility conditions when the dispersive parameter is set to
zero. To the authors' knowledge, this is the first time that these phenomena
(likely to play a central role in the analysis of initial boundary value
problems for dispersive perturbations of hyperbolic systems) are exhibited.
