Double diffraction coefficients for source and observation at finite distance for a pair of wedges

A closed form, high-frequency solution is presented for describing the double diffraction mechanism at a pair of parallel wedges, when they are illuminated by a spherical wave. The solution is obtained by using a spherical waves spectral representation of the first order diffracted field from each wedge. For the sake of simplicity, only the scalar case is considered when either hard or soft boundary conditions may be imposed on the faces of the two wedges. This provides a basic step for constructing the solution in the more general electromagnetic case. Although the procedure is applicable to any couple of parallel wedges, the case of two wedges sharing a common face is explicitly considered in the solution presented. The asymptotic evaluation of the double spectral integral leads to transition functions involving generalized Fresnel integrals. Numerical calculations show that the high frequency formulation fails so gracefully that it gradually blends into the solution of a single wedge when the distance between the two edges vanishes. This is a desirable property for analysing the shadowing effects of a thick screen.


INTRODUCTION
A closed form, high-frequency solution is presented for describing double diffraction mechanism at a pair of parallel wedges, when they are illuminated by a spherical wave.The solution is obtained by using a spherical waves spectral representation of the first order diffracted field from each wedge.For the sake of simplicity, in this paper only the scalar case is considered when either hard or soft boundary condition (b.c.) may be imposed on the faces of the two wedges; this provides basic step for constructing the solution in the more general electromagnetic case.Although the procedure is applicable to any couple of parallel wedges, the case of two wedges sharing a common face is explicitly considered in the presented solution.The asymptotic evaluation of the double spectral integral leads to transition functions involving generalized Fresnel integrals.Numerical calculations show that the present high frequency formulation fails so gracefully that it gradually blends into the solution of a single wedge when the distance between the two edges vanishes.This is a desirable property for analizing the shadowing effects of thick screen.

FORMULATION
Let us consider a pair of wedges with parallel edges, illuminated by a spherical source.It is useful to define a cylindrical (pi.&, zi) and a spherical (ri, pi, 4;) coordinate system at each edge (i=1,2).The position of the source is denoted by (pi', q$' , zi') or (ri', pi', 4;') (i=1,2).Our description of the double diffraction mechanism is constructed as the superposition of two analogous mechanisms; a field diffracted from edge 2 when it is illuminated by the field diffracted from edge 1 (12), and that from 1 when it is illuminated by 2 (21).For the sake of simplicity, only the contribution 12 is considered in herein after.The ray geometry for the field doubly diffracted at Q1 and Q2 is depicted in Fig. 1; the distance between the two points of diffraction Q1 and Q2 is denoted by l=dJsin PI' where d is the distance between the two edges.
The singly diffracted field from the first wedge illuminated by a spherical source is conveniently expressed by The integrand in (1) represents a spectral spherical source; this is used for illuminating the second wedge.Then, by analytic continuation of (1) the near field response of the second edge to a spherical spectral source is introduced and used to weight the spectrum of (1).Thus, a double spectral integral representation is obtained as This rappresentation which explicity satisfies reciprocity, exhibits a two dimensional saddle point at (Cl, C2) = (0, 0); its asymptotic approximation is presented next.

ASYMPTOTIC EVALUATION
The integral in ( 2) is now asymptotically evaluated by stationary phase approximation.To this end, it should be noted that in the soft case the spectrum GIG* vanishes at the saddle point.Thus, this latter case requires a higher order term in the asymptotic evaluation.This is indeed expected since in the soft case double diffraction is essentially a slope contribution.A rigorous asymptotic analysis leads to the following expression for the doubly diffracted field where are the incident field at Ql and the spreading factor for the doubly diffracted ray, respectively and Di$ is the diffraction coefficient for either soft (s) or hard (h) case In (6) and ( 7) Tm are the two dimension transition functions, that are defined herein after; they become unity for k large, away from the transition regions.
It worth noting that at these aspects the hard coefficient Dh w , while the soft coefficient Dh N The transition functions are defined as 1 as a slope contribution.

NUMERICAL RESULTS
The formulation presented in this paper has been applied to calculate double diffraction contributions from a strip illuminated close to grazing aspects by a TE field &om a dipole.Calculations of the total field observed at a fiNte distance are presented in Fig. 2. The numerical results are compared with first order UTD calculations, that provide a discontinuous