Thus far, the method of ensuring that the branch points of the distance function D1 vary as ub i , an essential feature of the Sommerfeld method, has not differed greatly from the procedure used by Sommerfeld. However, generalizing the method from one branch curve to two or more requires a more readily applied formulation than that used in Ref. . The original procedure made use of transcendental functions in an essential manner and provided little indication of how the method could be extended.
In this chapter, the algebraic formulation discussed briefly in previous chapters is generalized in order to deal with several branch curves. As an example of this general approach, a multiple-valued generalization of a point radiation source, defined on a space bounded by two coplanar and concentric branch circles, is constructed. Then, by a procedure described in the preceding chapters, such a solution of the wave propagation equation can be used to construct a solution of a boundary value problem for a perfectly conducting, flat annular ring corresponding to a spherical wave incident on the ring (Fig. 6.1). A minor alteration in the imaging procedure leads to a solution for Fresnel diffraction by a circular annular aperture in a perfectly conducting plane (Fig. 6.2). The boundary conditions can also be imposed on surfaces other than those illustrated in Figs. 6.1 and 6.2 and spanning the branch circles. The surfaces need only be specified by a constant value 0 , as is discussed in Sec. 5.5 and Eq. (6.3).
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