ITD formulation for corner diffraction

A uniform diffraction coefficient is presented for the scattering in the near zone by a corner at the interconnection between two straight edges when it is illuminated by a plane wave. The solution is obtained by applying the incremental theory of diffraction (ITD). The high-frequency expressions provide the desired continuity of the total scattered field at those aspects where the leading ray-field contributions experience a jump discontinuity.<<ETX>>


INTRODUCTION
A high-frequency description of the scatterink, from complex structures that exhibits surface discontinuities su( h as edges and vertices, is of importance iu a wide variety of practical applications. To this end , the Geometrical Theory of Diffraction (GTD) [l] and its uniform extension (UTD) [2] provided very effectite tools for most engineering purposes. The practical applicability of this ray method depends on the solution of canonical problems that are used to locally approximate the actual configuration. Within this framework, an important canonical problem is that of a corner at tl-le interconnection of two straight edges, joined by a plane angular sector.
It should be noted that for most practical purpose's, the need for a corner diffraction coefficient in a UTD scheme mainly arises when the leading, edge diffracted field experiences a discontinuity, as it occurs when the diffraction point changes abruptly its location from one edge to the next adjacent edge. An even more seriow impairment is encountered in applying GTD to RCS calculations, di:e to the fact that the leading, edge contributions are restricted to lie on the pertinent Keller cones. A ray field description of the contribution from a corner may provide an effective tool to remove this major limitation as well as other inaccuracies, that may occur even when standard GTD edge contributions are applicable.
In this paper, a first order, high-frequency solution for the scattering by a corner is presented. Our formulation is based on the Incremental Theory of Diffraction (ITD), which has been recently developed [3]. The procedure consists of three basic steps. a) Fir+ a generalized Geometrical Optics (GGO) contribution is obtained; to this end, the incremental surface GO contributions (SGO) are integrated over the surface of the infinite plane angular sector, and the contribution from the linear integrations along the two semi-infinite edges of the incremental end-point GO fields (EGO) are extracted from the previous surface integral. b) Next, a diffraction contribution is obtained by a linear integration of the incremental diffraction coefficients (IDC along represented as the sum of the GGO field plus standard UTD edge diffracted fields and a diffracted field contribution from the tip. For the sake of simplicity, the scalar case is treated herein after. The same basic procedure can be extended, without any significant difficulty, to treat the more general electromagnetic vector problem. the two semi-infinite edges. c) Finally the total scattered 1 ield is

GENERALIZED GEOMETRICAL OPTICS C( )NTFUBUTION
The geometry at a corner interconnecting two edges is shown in Fig. 1. Let us denote by C the surface of the plane angular sector and by R the angle between the two edges. At each edge (n=1,2) it is useful tu define a local coordinate system (x,, y,, 2 , ) wit11 its origin at the vertex; the z,-axis is chosen along the edge, and the y,-axis is perpendicular to C. Accordingly a spherical coordinate system (r, 8,. &) is also defined; a plane wave illumination is assumed, with a direction of incidence (On', &'). Moreover, let us introduce a coordinate system (x,, yo, zo) with the zo axis perpendicular to the surface C ( Fig.1) and x, = zl; correspondigly, the spherical coordinates (r, 0, Q) and the coordinates (e', 4') are used to denote the observation point and the incident field, respectively. The GGO field is represented by In (4) $go is the standard GO contribution for the plane angular sector, which arises from the double stationary phasc, contribution, $: ? (n=1,2) are the stationary phase contributions from the boundary line of the integration domain, and &go is the asymptotic end-point contribution associated with the tip. Similarly, in (5) $',"," is the stationary phase contribution from the integration along the edge n, which asymptotically coincides with @$, and y5zg is the end-point contribution from the same integration. In particular and where either t= -1 or e = sgn(zo) for either soft or hard boundary conditions, respectively, T is the transition function defined in [4], which involves the generalized Fresnel integral, and 4 is the UTD transition function; their arguments are 6, = a s i n ( "-', 7) ; 6 , = , /~c o s (~) (8) in which cl= sgn(z,).

DIFFRACTION CONTRIBUTIONS
Next, in order to introduce appropriate diffracted field contributions that lead to a continuous and well-behaved description of the scattering phenomenon, incremental edge diffracted contributions F$(z,') are defined according to the ITD procedure. They are distributed and integrated along the two semiinfinite edges of the corner. This process yields two diffracted fields id, (n=1,2) that are given by 0 Its asymptotic evaluation provides a representation of the field as the sum of two contributions $1 $fa+ d t o (10) where +fs, which arises from stationary phase tontribution, is the standard UTD edge diffracted field when it exists, and is the end-point contribution, in which P(dn,Q3 = Finally, the diffracted field contribution from the tip of the corner is represented by i 0 --+;go -+;go -+ : g o + +: o+ $! o '

(13)
When the above field is summed to the UTD edge diffracted fields &, and to GO reflected field +go, a uniform, high frequency solution is obtained for the total scattered field.

CONCLUDING REMARKS.
A uniform diffraction coefficient has been presentei! for the scattering in the near zone by a corner at the interconnec1,ion between two straight edges, when it is illuminated by a plane wave. The solution is obtained by applying the Incremental Theory of Diffraction. The highfrequency expressions provide the desired continuity of the total scattered field at those aspects where the leading ray. field contributions experience a jump discontinuity.