We introduce a new methodology, based on new quasi-periodic Green functions which converge rapidly even at and around Wood-anomaly configurations, for the numerical solution of problems of scattering by periodic rough surfaces in two-dimensional space. As is well known the classical quasi-periodic Green function ceases to exist at Wood anomalies. The approach introduced in this text produces fast Green function convergence throughout the spectrum on the basis of a certain "finite-differencing" approach and smooth windowing of the classical Green function lattice sum. The resulting Green-function convergence is super-algebraically fast away from Wood anomalies, and it reduces to an arbitrarily-high (user-prescribed) algebraic order of convergence at Wood anomalies.1 periodic Green function is not even defined (see Remark 2.2 for details on nomenclature concerning Wood anomalies). To address this difficulty we propose new quasi-periodic Green functions and associated series representation which converge rapidly even at and around Wood anomalies. More precisely, we present a set of rapidly convergent quasi-periodic Green functions G q j whose N -term truncated series converge, at Wood anomalies, at least as fast as (1/N ) (j−1)/2 for j even (resp. (1/N ) j/2 for j odd) as N → ∞; in view of the fact that this approach also incorporates the smooth windowing function methodology [36], the new Green functions also enjoy super-algebraically fast convergence (faster than any power of N ) away from Wood anomalies.The approach introduced in this text produces fast Green-function convergence at and around Wood anomalies on the basis of a certain order-j "finite-differencing" method (for positive integer values of j). To our knowledge, this is the first approach ever presented that is applicable to problems of scattering by diffraction gratings at Wood anomalies on the basis of quasi-periodic Green functions.It should be noted that a "method-of-images Green-function", which is related to our j = 1 Green function approach, was used in [15,48] to treat problems of scattering by nonlocal, non-periodic perturbations of a line in two dimensions. The j = 1 method, which suffices to yield (slow) convergence in the two dimensional case, does not give rise to convergence in three dimensions: for three dimensional configurations convergence only results for j ≥ 2 [12]. In this context we also mention the recent work [3,4] which, for two-dimensional problems, introduces an alternative integral equation which does not utilize a quasi-periodic Green function, and which is also applicable at Wood anomalies: in that approach quasi-periodicity is enforced through use of auxiliary layer potentials on the periodic cell boundaries. The practical feasibility of an extension of this methodology to three dimensional problems has not as yet been established.In order to demonstrate the character of the new approach we present efficient numerical methods, based on the new Green functions, for the solution of quasi-periodic scattering problems through...

This work deals with the scattering of acoustic waves by a thin ring that contains regularly spaced inhomogeneities. We first explicit and study the asymptotic of the solution with respect to the period and thickness of the inhomogeneities using so-called matched asymptotic expansions. We then build simplified models replacing the thin ring with Approximate Transmission Conditions that are accurate up to third order with respect to the layer width. We pay particular attention to the study of these approximate models and the quantification of their accuracy. RésuméCet article traite de la diffraction d'une onde acoustique par un structure constituée d'un anneau mince de diélectrique contenant un grand nombre d'hétérogénéités disposées périodiquement. On commence par écrire et justifier un développement asymptotique de la solution en fonction de l'épaisseur de l'anneau et de la distance entre les hétérogénéités. Puis, on construit des modèles approchés stables et bien posés d'ordres 2 et 3 dans lesquels l'anneau périodique est remplacé par une condition de transmission.