About the radiation diagram of an underwater acoustic source in the presence of gravity waves

Abstract

A time-harmonic acoustic source of amplitude E being located at a point A of depth a, the corresponding acoustic field in the absence of gravity waves is well-known : it can be obtained by introducing a sink of amplitude -E at the point symmetrical of fl with respect to the free surface. If there are also gravity waves of small amplitude ,Ll and if the ratio f? of the acoustic to the gravity wave lengths is of order 1 , the reflexion of the acoustic waves by the free-surface generates an acoustic perturbation of order E A. In a previous report (Euvrard and Mechiche Alami 1992) an explicit expression of this second-order acoustic pressure has been found, and then justified via a limitingamplitude procedure: it is a Helmholtz double layer distribution on the mean free-surface. Here the asymptotic behaviour of the second-order acoustic pressure for large values of the horizontal distance is exhibited with help of matched asymptotic expansions, and then proved using Lebesgue's theorem together with stationary phase upper bounds. Finally this asymptotic behaviour can be written in a closed form as a function of f ; it does not satisfy the standard radiation condition but exhibits some interesting features.