## 1. Introduction

[2] It has been widely recognized that high-frequency ray methods, such as the Geometrical Theory of Diffraction in its uniform version (UTD) [*Kouyoumjian and Pathak*, 1974], are useful in efficiently describing scattered or diffracted fields originating from complex objects. Complex structures may include two interacting edges in which case contributions involving field of rays doubly diffracted by the two interacting edges need to be introduced to improve the field estimate [*Tiberio and Kouyoumjian*, 1979; *Tiberio et al.*, 1989; *Ivrissimtzis and Marhefka*, 1991, 1992; *Capolino et al.*, 1997; *Albani*, 2005]. Furthermore, difficulties may occur in applying ray methods close to and at caustics, and one is required to introduce distributed incremental field representation to overcome these difficulties. One advantage of the incremental representation is that it may improve upon the field estimate whenever the stationary phase condition leading to the ray field regime, is not yet well established. In particular, it is desirable to use incremental descriptions of diffracted fields that naturally and smoothly blend into the ray field estimate. The Incremental Theory of Diffraction (ITD) [*Tiberio and Maci*, 1994; *Tiberio et al.*, 2004] provides a self-consistent, high-frequency description of a wide class of scattering phenomena within a unified framework. The ITD may be classified as a field-based method, since the incremental field contributions are directly deduced from the solution of locally tangent canonical problems that, generally, have a uniform cylindrical configuration. An important feature of the ITD is that it smoothly recovers the ray-field description and explicitly satisfies reciprocity, a feature that is typical of field-based methods. Satisfaction of reciprocity is a very desirable property, and is embedded in Maxwell equations. Let us recall the ITD procedure for defining incremental fields. This procedure, which consists of two steps, is based on the relationship between the incremental field description of the scattered or diffracted field and the solution of the canonical problem. First, the spectral integral representation for the exact solution of the relevant local canonical problem is cast in a convenient form as an inverse Fourier transform (FT) of the product of two spectral functions. By means of Fourier analysis, it is found that the same spectral integral may be represented as a spatial integral convolution along the longitudinal coordinate of the local cylindrical configuration. Second, the incremental field contribution is then naturally localized at the tangential point on the actual object. This is referred to as the ITD FT-convolution process. Explicit dyadic expressions of the incremental diffraction coefficients for wedge-shaped configurations are presented by *Tiberio et al.* [2004]. The asymptotic analysis performed here yields high-frequency, closed form expressions which explicitly satisfy reciprocity, and are well behaved at any incident and observation aspects, including those close to and at the longitudinal coordinate axis of the cylindrical configuration. An analytical verification for the case of a disk is provided by *Erricolo et al.* [2007]. It is also found that, for the scalar case, the same result was obtained by *Rubinowicz* [1965], by means of an entirely different method.

[3] This article was motivated by several actual scattering and diffraction problems in which the dominant field of a singly diffracted ray is shadowed by a second edge, or is near to the shadowing. The introduction of doubly diffracted rays becomes necessary in order to compensate for such a discontinuity. It is well known that the consecutive application of the ordinary UTD diffraction coefficients fails when the edge of the second wedge lies within the transition region of the field diffracted by the first wedge [*Tiberio et al.*, 1989]. This is due to the rapid spatial variation and to the non-ray-optical behavior of the field diffracted by the first wedge, which impinges on and illuminates the second wedge. The same limitation is expected to affect also the ITD representation, thus preventing a simple subsequent application of the coefficients for single diffraction. Therefore, it is necessary to develop a double diffraction coefficient which uniformly accounts for the different transitions that may occur. In some UTD formulations available in the literature [*Tiberio et al.*, 1989; *Albani*, 2005] these coefficients are derived by solving a proper canonical problem.

[4] In this paper, an ITD dyadic coefficient is presented for double diffraction between a pair of skewed separate wedges, as well as for double diffraction between the edges of joined wedges that have a common face. In both cases, the new formulation provides an accurate, first-order asymptotic description of the interaction between the edges by introducing an augmented incremental slope diffracted field. The ITD representation for the incremental doubly diffracted field (ITD-DD) requires a twofold numerical integration in the space domain, which arises from the ITD line integral representation for the diffracted field at each edge. It is also worth noting that the present ITD-DD description is valid in the general case where the edges are not necessarily coplanar.

[5] This paper is organized as follows. The ITD formulation for single edge diffraction is summarized in section 2. The extension to the case of double diffraction between a pair of skew or joined wedges is presented in section 3, where the question of how the even and odd components of the spectral Green's function contribute to the incremental field it is also discussed. There, it is shown that the double slope diffraction augmentation can be accounted for by the product of the odd parts of the ITD dyadic diffraction coefficients. Discussion about singularities and discontinuity compensation mechanisms of the proposed formulation is provided in section 4 for both skewed separate and joined wedges. In section 5, some numerical examples are presented and discussed.