Advanced properties of spectral functions in frequency and time domains for diffraction by a wedge-shaped region

The scattering by objects with singular geometries is a particularly delicate problem, and the Sommerfeld-Maliuzhinets representation of field, when it is associated to Maliuzhinets inversion [3]-[4] and single face representation [7]-[8] methods, is a powerful tool to investigate the acoustic and electromagnetic waves diffraction by a complex wedge-shaped region, in particular based on remarkable properties of spectral functions in this representation. In first step of this chapter, we review some general expressions and properties in scalar case for these problems : general properties in complex plane, spectral representation of Green function, single face representation of spectral function and consequences, are analysed for scatterers with imperfectly reflecting surfaces that can extend to infinity. In a second step, we consider the general solution for a wedge with face impedances of arbitrary signs (passive or active case), then the solution for the diffraction of a skew incident plane wave by a passive anisotropic impedance wedge of any angle in vectorial case [9] (2D 1/2 electromagnetism problems), and detail efficient expansions of special functions used for them and their properties. In a third step, an explicitely causal representation of field in time domain is developed within a large domain of validity, including the case of a dispersive wedge with multimode boundary conditions.