Second Approach to the Green Functions: The Self-Consistent Way

Abstract

In Chap. 3 we have demonstrated that the scattering problems of our interest are solved approximately if knowing a finite series expansion of the primary incident field at the scatterer surface ∂Γ in terms of the radiating eigensolutions of the Helmholtz and vector-wave equation, respectively. From this we could obtain an approximation of the scattered field everywhere in the outer region Γ₊ which is also given by a series expansion in terms of the radiating eigensolutions. Comparing this approximation with the integral representations resulting from the application of Green’s theorems we could derive corresponding approximations for the scalar and dyadic surface Green functions G _∂Γ and G_∂Γ. In the next step, by employing Hugens’ principle formulated exclusively in terms of Green functions, we were able to derive the corresponding approximations of the actual Green functions G _Γ+ and G_Γ+. In the course of these derivations, we had just one restriction concerning the location of the source generating the primary incident field. This source must be placed outside the smallest spherical surface circumscribing the scatterer. In this chapter, we will now answer the question if it is possible to derive the approximations of the relevant Green functions without the fallback to the approximation of the scattered field. Conversely and more naturally, the approximation of the scattered field should be a consequence of its integral representation in terms of these Green functions. The justification of such a question is due to the fact that the Green functions themselves are solutions of certain boundary value problems related to the inhomogeneous Helmholtz and vector-wave equation, respectively, with Dirac’s delta distribution as inhomogeneity. The corresponding boundary conditions at the scatterer surface have been already formulated in Sects. 2.5.3 and 2.6.4. In the first part of this chapter, we will pursue this goal. The resulting formalism will be called “self-consistent Green function formalism” for obvious reasons.