The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on October 27, 2006
The Quarterly Journal of Mechanics and Applied Mathematics 2006 59(4):517-550; doi:10.1093/qjmam/hbl014
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A spectral approach for scattering by impedance polygons
( Département de Physique Théorique et Appliquée, CEA/DIF, BP 12, 91680 Bruyères le Châtel, France )
jean-michel.bernard{at}cea.fr
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Abstract |
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We study a new spectral approach for scattering by two-dimensional polygonal objects with arbitrary surface impedance conditions. In this delicate exterior problem, the Wiener–Hopf method cannot be applied, while asymptotic methods can only be used if corners are widely spaced compared to wavelength. A new method based on the Sommerfeld–Maliuzhinets integral representation is presented to reduce the problem to simple spectral equations in the complex plane. For this, we use an expression of the spectral function, where we can isolate the contribution of any element of an arbitrary surface. Considering polygons with impedance boundary conditions, it then becomes possible to derive functional equations on spectral functions of Maliuzhinets type for finite or infinite objects. We apply this approach to an important class of three-part impedance polygons composed of a finite segment attached to two semi-infinite planes, and reduce this problem to non-singular Fredholm integral equations, suitable for approximation or numerical inversion. In the particular cases of a three-part impedance plane or symmetric impedance polygon, we show that the system of integral equations in the spectral domain can be simply uncoupled.