The floating structure problem describes the interaction between surface water waves and a floating body, generally a boat or a wave energy converter. As shown by Lannes, the equations for the fluid motion can be reduced to a set of two evolution equations on the surface elevation and the horizontal discharge. The presence of the object is accounted for by a constraint on the discharge under the object. We address the shallow water approximation under the assumption that the flow is axisymmetric without swirl and we consider a solid which moves only vertically. In the second part of the talk, we deal with the decay test. It consists in releasing the solid body in a fluid initially at rest and letting it evolve towards its equilibrium position. It turns out that our previous theory does not work in this particular configuration. For this reason, we use a linear-nonlinear hydrodynamic model and we show that the solid equation can be written as a nonlinear second order integro-differential equation.