Electromagnetic scattering by a plane angular sector: I. Diffraction coefficients of the spherical wave from the vertex

Electromagnetic problem by a plane angular sector is studied. The perfectly conducting sector is illuminated by a plane electromagnetic wave. The scattered wave field consists of several components at large distances from the vertex of the sector. In the framework of the consequent and mathematically justified procedure, based on the Watson–Bessel and Sommerfeld integral representations of the electromagnetic Debye potentials, we develop expressions for the electromagnetic diffraction coefficients of the spherical wave from the vertex. To that end, we carefully study analytic properties of the Sommerfeld transformants and give a new procedure of analytic continuation for them. Like in the acoustic problem, the singularities of the Sommerfeld transformants on the real axis are responsible for the different components in the far field asymptotics.The expressions for the diffraction coefficients are specified in the form of integrals depending on the so called spectral functions, i.e. solutions of the boundary value problem for the Laplace–Beltrami operator on the unit sphere with the cut along a segment of big circle. The spectral functions are found in terms of series of “spherical harmonics”, whereas the coefficients of the series solve a Fredholm system of linear summation equations of the second kind. The properties of the spectral functions are also discussed and their interconnections with the “nonstationary” Green’s functions for the problems for some hyperbolic operator on the unit sphere with the cut are also established. The latter Green’s functions are exploited in the formulas for the analytic continuation of the Sommerfeld transformants. The results of the work form a theoretical basis for the numerical evaluation of the diffraction coefficients and for the description of the other components in the far field asymptotics to be given in the second part of the work.