École Nationale Supérieure de
Techniques Avancées

Abstract

This thesis dealswith the propagation ofwaves inmedia that comprise thinwires the thickness of which ismuch smaller than thewavelength.We apply thematched asymptotic expansionmethod and derive an expansion of the solution to the two dimensional Helmholtz equation around a small obstacle with Dirichlet boundary condition. Then we present a simplified model for this problem involving an averaged boundary condition and analyze two non-standard numerical methods for computing accurately the corresponding solution : the first one is a variation of the singular function method, and the second one is a scalar version of the Holland method. We prove the consistency of both methods in this case. Then we provide comparable results for the 3D Helmholtz problemwith Dirichlet boundary condition on a wire-shaped obstacle with ellipsoïdal tips.We also derive a simplifiedmodel in the latter setting and this leads to a justification of a scalar version of the Pocklington equation.