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In the preceding chapters, the Sommerfeld method has been extended from its original application—a plane wave incident on a perfectly conducting halfplane—to the case of a spherical wave incident on a variety of open surfaces, all perfectly conducting. For sufficiently great distances from the spherical wave source P , the spherical wave approximates a plane wave. Alternatively, in a Fresnel construction, one may imagine a plane wave as the envelope of spherical waves with centers on a plane. In all cases considered, finding a suitable coordinate system is an essential part of the problem’s solution: first, to obtain a branch curve of the desired kind, and second, to obtain branch points that vary correctly with the angle variable  . To confine oneself to well-known coordinate systems like the cylindrical, bipolar, and toroidal coordinate systems is to severely limit the kinds of branch curves to which this method can be applied, as shown in the foregoing chapters. Moreover, in such a case, the desired goal of providing a flexible tool of analysis for applications in optics is also very limited. In this chapter, it is proposed to show how the Sommerfeld method can be freed of some of the restrictions of coordinate system choice and to extend the discussion to create a new method of analysis that can be applied to closed surfaces of more arbitrary configurations.
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