Some of the components of the fields produced by an incident plane monochromatic wave scattered by a wedge diverge near the edge of the wedge. Rigorous solutions for the fields scattered by a perfectly conducting infinite wedge have been obtained, but this is not the case for a dielectric wedge. Numerical experiments were carried out to determine the behavior of the fields near the edge using singular integral equations (SIEs) or hypersingular integral equations (HIEs) for a finite wedge in the TE and TM modes. For the SIE, the unknown boundary functions also diverges near the edge, which creates a problem when solving more general two-dimensional problems with sharp edges, such as scattering by a strip on a substrate. If the behavior of the unknown function were known, it could be built into the numerical approximations for patches near the sharp edge to avoid problems that arise from getting too close or not close enough to the edge. Rounding off sharp edges can be useful in some cases, but it can also cause worse problems due to sharp spikes that are created. For the HIE, the unknown boundary functions remains constant near the edge and the numerical difficulties come from the highly singular behavior of the integrand, especially in the computation of the contributions of the self-patch and neighboring patches. For a perfectly conducting wedge, the behavior of the computed fields near the edge is that determined by Meixner. Nevertheless, the form of the rigorous solution of the scattering problem differs for wedge angles that are or are not rational multiples of pi, a distinction that is not relevant to computers. For a dielectric wedge, power series solutions indicate that the behavior remained essentially that of static fields. Numerical experiments showed disagreements between the computed and the expected behavior which depended on the direction of approach to the edge. Furthermore, the zero-frequency limit of the fields does not necessarily correspond to static fields, a dichotomy that is especially evident for oblique incidence. Numerical experiments for TE and TM modes generate fields near the edge of a finite wedge that agree only at times with the expected behavior. The results are even less clear for oblique incidence, as detailed in a following paper. The difficulty of matching two waves propagating at different speeds in different media on the two sides of a dielectric wedge and on the free space in the other half of the planes suggests that there may be no general rigorous solution for this scattering problem. Other numerical and theoretical approaches have been used to address the problem of electromagnetic scattering by a dielectric wedge, sometimes approximating the original problem by, for instance, using impedance boundary conditions.
January 1, 2003
PIERS 2003: Progress in Electromagnetics Research Symposium Proceedings