A body immersed in an ocean of large depth is assumed to vibrate and to radiate a
time-harmonic acoustic field of small amplitude in the presence of gravity waves of
small amplitude. Assuming both waves to have lengths of the same order (which in
practice corresponds to very low acoustic frequencies) it is shown that the diffraction
of acoustic waves by the corrugated free surface generates a second-order acoustic
pressure field p2. The computation of p2 involves a difficulty: a non-homogeneous
Dirichlet condition to be satisfied on the mean free surface up to infinity which implies
the absence of any clear indication about the condition that should be imposed at
infinity to have a well-posed problem. In order to get an insight into this difficult
problem the simple case of a point source is studied. We first judiciously choose one
solution and then show it is the physical solution using a limiting-amplitude procedure.
Coming back to the general case of a vibrating body the calculation of p2 is split
into two successive steps: the first one consists in computing an explicit convolution
product via numerical methods of integration, the second one is a standard radiation
problem that is solved using a method coupling a Green integral representation and
finite elements. A peak of the second-order pressure appears just above the vibrating
body.
The same concepts also apply to other second-order scattering problems, such as
the sea-keeping of weakly immersed submarines.