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.