A Contribution to Doubly Asymptotic Approximations: An Operator Top-Down Derivation

Abstract

As a contribution to the theory of Doubly Asymptotic Approximations (DAAs), a formal operator top-down derivation specialized to steady-state motions is proposed for the Neumann exterior problem associated to the Helmholtz equation. It generalizes previously published ones which relied either on a modal approach [1] or on a scalar approach for a model problem [2]. The proposed derivation is based on an integral representation of the solution of the Helmholtz equation in an unbounded domain: first, two asymptotic expansions of the representation with respect to the nondimensional wave number k are obtained in the low-k and the high-k ranges; then these expansions are matched. This procedure allows to point out that some geometrical physical and mathematical assumptions underlie the validity of high order continuous forms of the DAAs. Special attention is then devoted to their discretized counterparts which are compared to previously published ones. In both cases it suggests further investigation of some interesting and open geometry and numerical analysis problems.