Perturbation of low-frequency underwater acoustics by gravity waves

A body immersed in an ocean of large depth is assumed to vibrate and to rad iate a time-harmonic acoustic held of small amplitude in the presence of gr avity 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 p(2). The computat ion of p(2) involves a difficulty: a non-homogeneous Dirichlet condition to be satisfied on the mean free surface up to infinity which implies the abs ence 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 j udiciously choose one solution and then show it is the physical solution us ing a limiting-amplitude procedure. Coming back to the general case of a vi brating body the calculation of p(2) is split into two successive steps: th e first one consists in computing an explicit convolution product via numer ical 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 th e vibrating body. The same concepts also apply to other second-order scattering problems, suc h as the sea-keeping of weakly immersed submarines.