  The coupled motion is investigated for a mechanical system consisting of
water and a body freely floating in it. Water occupies either a half-space or a
layer of constant depth into which an infinitely long surface-piercing cylinder
is immersed, thus allowing us to study two-dimensional modes. Under the
assumption that the motion is of small amplitude near equilibrium, a linear
setting is applicable and for the time-harmonic oscillations it reduces to a
spectral problem with the frequency of oscillations as the spectral parameter.
It is essential that one of the problem's relations is linear with respect to
the parameter, whereas two others are quadratic with respect to it.
  Within this framework, it is shown that the total energy of the water motion
is finite and the equipartition of energy holds for the whole system. On this
basis, it is proved that no wave modes can be trapped provided their
frequencies exceed a bound depending on cylinder's properties, whereas its
geometry is subject to some restrictions and, in some cases, certain
restrictions are imposed on the type of mode.
