On the evaluation of quasi-periodic Green functions and wave-scattering at and around Rayleigh-Wood anomalies

Abstract

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Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

“…According to our time convention, it holds that for n ∈ U the modes u + n (x, y) := e iαnx+iβny and u − n (x, y) := e iαnx−iβny (7) in (5a) are upgoing and downgoing propagative planewaves, respectively. For n ∈ V, in turn, u + n (resp.…”

“…, corresponds to the so-called Rayleigh-Wood anomaly configurations [6,7,21]. For such n values it holds that u n (x, y)…”

“…In particular, it can be shown (cf. [19]) that upgoing and downgoing planewaves (7) for n ∈ U as well as horizontally propagating modes (8b) for n ∈ W satisfy…”

“…for the parametrized traces of the Rayleigh modes (7), and exploiting the linearity of the integral operators M p,q , we arrive at…”

Windowed Green function method for wave scattering by periodic arrays of 2D obstacles

“…BIE solvers that leverage this modified Green function [15][16][17] exhibit superalgebraic convergence away from RW anomalies and algebraic but arbitrarily high-order convergence at and around RW anomalies, at the cost of 𝑛-tupling the number of function evaluations where 𝑛 is the numbers of "shifts" utilized in the approximation. Recent developments in this direction 11,18 present a general methodology based on hybrid spatial/spectral Green function representations and the Woodbury-Sherman-Morrison formula that makes classical approaches such as Laplace-type integral and Ewald's methods, as well as the shifted Green function approach, applicable and robust at and around RW-anomaly configurations.…”