**1. Introduction**

Diffraction of electromagnetic waves by canonical shapes and structures of more general and arbitrary shape is of enduring interest. The choice of an appropriate canonical structure to model the dominant features of a scattering scenario can be very illuminating. The study in this paper was originally motivated by the influence that the corners of buildings and their surface cladding have on electromagnetic wave propagation. A recent publication by Rawlins

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[1] considered an approximate model relevant to the understanding of signal strength for phones in this environment. It studied the diffraction of an E-polarised wave by an absorbing rectangular cylinder, based upon Keller's method of GTD and its extensions to deal with multiple diffraction. It utilized the diffraction coefficient derived for the canonical problem of diffraction by an impedance corner to obtain relatively simple high frequency approximate expressions for the scattered far-field resulting from a plane wave obliquely incident on an imperfectly conducting rectangle.

In order to validate the results of [1], Smith and Rawlins [2] undertook a numerical study of the scattering of an E-polarised plane wave by an infinite cylindrical structure in which an impedance boundary condition is enforced at all points on the cross-sectional boundary of the cylinder. It employed the integral equation formulation of Colton and Kress [3] for the unknown surface distribution comprising a single-layer potential and the adjoint of the double-layer potential. A Nyström method similar to that expounded by Colton and Kress [4] for the soft boundary condition was developed to obtain numerical solutions of this integral equation. The computed scattered far-fields were compared with the results of Rawlins [1] in order to validate his solutions over the range of impedances and wavenumbers examined. The study concluded that the approximations developed in [1] provide reasonably accurate patterns for rectangular structures for the range of wavenumbers and dimensions examined, but some divergences appear at smaller wavenumbers. There was a limitation to the study [2]: the method was applicable only to cylindrical cross-sections that are smooth (having a continuously varying normal vector at each point), and so the exactly rectangular structures investigated in [1] were treated by a replacing them by an appropriate "super-ellipse" that approximates the rectangle with rounded corners.

In order to clarify the effect of corner rounding this paper examines the diffraction from cylindrical scatterers which possess corners, that is, points at which the normal changes discontinuously. Specifically we develop a numerical method for the scattering of an E-polarised plane wave by such cylindrical structures. The work in [5] is significantly extended. We examine three different boundary conditions: soft, hard and an impedance loaded boundary condition. In each case the boundary condition is enforced at all points on the cross-sectional boundary of the cylinder. We implement the Nyström method expounded by Colton and Kress [4] for the soft boundary condition to obtain numerical solutions of this integral equation. We then develop other Nyström methods similar to [4] for the hard and impedance boundary conditions to obtain numerical solutions of the respective integral equations.

We use these numerical methods to examine the difference between a test structure with a corner and a rounded corner to assess the impact on near and far field scattering, as a function of the radius of curvature in the vicinity of the rounded corner point. We then extend the numerical methods developed thus far to examine a test structure with two corners. We conclude by examining the effect on the scattered field of rounding these corners as a function of the radius of curvature in the vicinity of the rounded corner points.
