Asymptotic analysis of an approximate model for time harmonic waves in media with thin slots

Abstract

Abstract.In this article, we derive a complete mathematical analysis of a coupled 1D-2D model for 2D wave propagation in media including thin slots. Our error estimates are illustrated by numerical results.Mathematics Subject Classification.

A mathematical analysis of the resonance of the finite thin slots

“…This is what we proved in [1] in the case of the semi-infinite slot (i. e. L = +∞), where we proved moreover that…”

“…Note that estimates (11) can be shown to be generically optimal as in [1]. We conjecture that the constant C blows up as 1/d(ω) where d(ω) denotes the distance to the set of resonant frequencies.…”

“…In [1], we studied the case of the 2D time harmonic wave equation with Neumann boundary conditions (which is physically relevant in acoustics and electromagnetism) and investigated the question of building an approximate model, in view of its numerical discretization, consisting of coupling a 1D model for the slot to a 2D model for the rest of the domain of propagation. We proposed a coupling strategy that we analyzed in detail in the case of a semi-infinite straight slot.…”

“…One observes that the two curves depicting respectively the L 2 norm of the diffracted field (normalized by the one of u 0 ) and the L 2 norm of the slot field present very sharp peaks corresponding to the resonant frequencies (ω/2π is an integer multiple of 1 2 ). One also observes the the height of these peaks decays with ω/2π for sufficiently large ω/2π.…”

Matching of asymptotic expansions for waves propagation in media with thin slots II: The error estimates

“…However, the complete understanding and evaluation of such models suffer, to our opinion, from a lack of mathematical analysis. In [9] we considered the case of the scalar Helmholtz Figure 1. Geometry of the domain of propagation.…”