A High Order Method for Analyzing the Electromagnetic Response of an Array of Thin Wires
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The problem of evaluating the electromagnetic response of wire-antenna systems by means of integral equations is one of great importance and significant mathematical complexity. The literature on this subject is extensive, including contributions from the 19th century work of Pocklington, the classic works of King, to some of the most recent work by Davies et.al., Bruno and Haslam and others. In this thesis, we develop a new high order numerical method to treat the problem corresponding to a parallel array of thin straight wires. A number of significant difficulties arise in this problem as a result of certain singularities and near-singularities that are inherent in its integral equation formulation. In particular, no satisfactory quadrature methods exist for the high-order evaluation of the integrals which arise from the thin wire equations when two wires in an array are separated by a small distance but finite distance. Such a configuration requires the evaluation of integrals whose integrands are logarithmically singular not only at a point inside the domain of integration, but also at a point just outside of the domain integration. The quadrature formulas we derive as a main contribution of this thesis explicitly treat both of these cases. A full numerical implementation of our algorithm was developed and results corresponding to high, moderate and low excitation frequencies are presented.