  We investigate here the interactions of waves governed by a Boussinesq system
with a partially immersed body allowed to move freely in the vertical
direction. We show that the whole system of equations can be reduced to a
transmission problem for the Boussinesq equations with transmission conditions
given in terms of the vertical displacement of the object and of the average
horizontal discharge beneath it; these two quantities are in turn determined by
two nonlinear ODEs with forcing terms coming from the exterior wave-field.
Understanding the dispersive contribution to the added mass phenomenon allows
us to solve these equations, and a new dispersive hidden regularity effect is
used to derive uniform estimates with respect to the dispersive parameter. We
then derive an abstract general Cummins equation describing the motion of the
solid in the return to equilibrium problem and show that it takes an explicit
simple form in two cases, namely, the nonlinear non dispersive and the linear
dispersive cases; we show in particular that the decay rate towards equilibrium
is much smaller in the presence of dispersion. The latter situation also
involves an initial boundary value problem for a nonlocal scalar equation that
has an interest of its own and for which we consequently provide a general
analysis.
