New insights in integral representation theory for the solution of complex canonical diffraction problems

The problem of scattering by 2D and 3D canonical objects with imperfectly conducting surfaces requires some particular efforts on the representation of scattered and incident fields, and we present here some remarkable aspects of them in complex situations and their applications. We begin with the study of 2D problems. The Sommerfeld-Maliuzhinets integral and its inversion in the spectral domain of complex angles has opened a new way of investigation on diffraction by a wedge-shaped domain. In this frame, we present this spectral method in a new perspective, by giving some novel exact general expressions and properties of the associated spectral function attached to the total field : novel spectral expression of free space Green function, single face representation and its consequences, uniqueness, existence, reciprocity, spectral causal representation of field in time domain. We then analyze in a distinct section the diffraction by an impedance wedge with curved faces. The diffraction coefficient, when limited to its principal order in curvature, leads us to some miscalculation when the wedge approaches the discontinuity of curvature case for example. Thus, we give here a general asymptotic representation taking into account a development at several orders for arbitrary wedge angle, which is uniformly valid from the discontinuity of curvature to the curved half-plane. Concerning 3D problems, we begin with a section devoted to an efficient exact solution for the radiation of a point source above an impedance plane in electromag-netism. In classic approaches, we need to consider vertical and horizontal dipole cases separately, and we present here a method, consisting in filling a gap in representation of incident fields by potential theory, to finally permit an efficient global representation of the diffracted field for arbitrary primary sources in a direct manner. We conclude this chapter with the representation of fields for the diffraction by an imperfectly conducting cone. We present a general exact expression of fields with Debye potentials and Kontorovich-Lebedev (KL) integrals, and new general properties of spectral functions attached to them. We then insist on a novel compact representation of plane waves, which does not need any use of spherical harmonics series.