## 1. Introduction

[2] The problem of calculating the diffraction of an electromagnetic or acoustic wave by a quarter plane (or more generally by a plane angular sector) is still a challenge. As a matter of fact, the 2004 URSI Commission B international electromagnetics prize dealt with a readily computable solution for the scattering by a quarter plane. Indeed, existing solutions consider the quarter plane as a degenerated elliptic cone and express the field as a spherical wave multipole series [*Kraus and Levine*, 1961; *Satterwhite*, 1974; *Sahalos and Thiele*, 1983; *Bowman et al.*, 1987; *Hansen*, 1990] (see [*Blume*, 1996] for a recent review). However, such a series exhibits a poor convergence rate when both the source and the observation point move far from the quarter-plane tip. In the plane wave incidence–far field observation regime, which is of great interest in high-frequency modeling of scattering, the series computation becomes critical [*Blume and Krebs*, 1998]. Furthermore, each term of the series requires the numerical solution of the transverse (on the sphere) eigenvalue problem with the calculation of eigenvalues and eigenfunctions (Lamè functions [*Whittaker and Watson*, 1990]). For this reason many works proposed alternative formulations based on heuristic models [*Sikta et al.*, 1983; *Pathak*, 1988; *Hill and Pathak*, 1991; *Maci et al.*, 1994; *Capolino and Maci*, 1996] or hybrid analytical-numerical approaches [*Hansen*, 1991; *Babich et al.*, 1996]. In 1961 Radlow published a paper [*Radlow*, 1961], followed by the improved version [*Radlow*, 1965], where the problem is attacked with a Wiener-Hopf [*Noble*, 1958] (convolution integral) technique, somehow inspired by Sommerfeld's half plane solution [*Sommerfeld*, 1896; *Born and Wolf*, 1964], but rephrased in two complex variables. The papers aroused a discussion because the order of the field singularity at the tip of Radlow's solution differs from the correct value [*Van Bladel*, 1991], revealing a suspicion of incorrectness. A researcher who gave him credit was Albertsen who extended the solution to the electromagnetic case [*Albertsen*, 1997], also providing a closed form vertex diffraction coefficient by means of an explicit calculation of the Wiener-Hopf factorization involved in Radlow's solution. From the same canonical problem, *Albertsen* [2000] also derived a uniform high-frequency description for the doubly diffracted rays, i.e., rays which experience two successive diffractions at the two edges. Indeed, the plane wave spectral representation which is obtained by Radlow is very appealing to derive high-frequency description of the scattering via asymptotic ray optics approximation, however, as we will demonstrate in this paper, it is not correct.

[3] In what follows we will adopt the notation of *Radlow* [1965] to which we address for all the details. For reader convenience we briefly abstract the statement of the problem and Radlow's solution in section 2. In section 3 we conduct a detailed analysis of the various functions introduced by *Radlow* [1961, 1965], providing a closed form expression for the Wiener-Hopf factorization and a description of the singularity topology of the factorized function. This discussion is fundamental for the understanding of the following sections. Next, in section 4 we briefly summarize why Radlow's demonstration that his solution satisfies boundary conditions is incorrect. Finally, in section 5 we shows that Radlow's solution does not respect required boundary conditions and therefore it is not the exact solution. By means of a direct check of the boundary conditions, an extra term, not accounted for by Radlow, is shown to be present, thus preventing the fulfillment of boundary condition on the quarter plane. Some conclusions about Radlow's solution end the paper.