AN OPERATOR TOP-DOWN DERIVATION OF "DOUBLY ASYMPTOTIC APPROXIMATIONS"

Abstract
Some problems involved in fluid-structure interaction in unbounded domains require the computation of the response of an homogeneous acoustic medium to prescribed harmonic motions. The complexity of the submerged structure studied sometimes suggests the use of approximate methods for the numerical analysis of this acoustics problem [1]. The present paper intends to propose a derivation of the so-called "Doubly Asymptotic Approximations" (DAA's). This formal top-down derivation, specialized to steady-state motions, relies on an integral representation of the solution of the Helmholtz equation in an unbounded domain. Two asymptotic expansions of this representation are obtained in the low- and the high-frequency ranges, then these expansions are matched. This procedure allows to point out that some geometrical assumptions underlie the validity of high order continuous forms of the DAA's. It suggests further investigations of some interesting open geometry and numerical analysis problems.