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Chapter 3:
Two-Leaved Generalization of a Spherical Wave: One Branch Line
In this chapter, the Sommerfeld method is applied to the construction of a solution of the wave propagation equation defined on a two-leaved space bounded by a straight line. The solution is a two-leaved generalization of a spherical wave, analogous to Sommerfeld’s generalization of a plane wave in physical space. (For the sake of comparison with Sommerfeld’s original work, a very brief summary of his original paper is given in Appendix B.) It is then possible to solve a boundary-value problem for a spherical electromagnetic wave incident on a perfectly conducting semi-infinite plane, which will be done in Chapter 4. A spherical wave solution (Fresnel diffraction) lends itself to experimentation more readily than Sommerfeld’s solution (Fraunhofer diffraction) since the generation of a spherical wave in the laboratory does not require a collimating lens to produce the incident radiation, providing a considerable advantage. Another advantage of the spherical wave generalization lies in that it is generated by a point source, and like its analog in the theory of static electricity, it can be used to construct extended source distributions. Finally, if the source point of a spherical wave is imagined to be very far from the point of observation, a plane wave is approximated and the corresponding boundary-value problem represents Fraunhofer diffraction. These fundamental ideas are developed in this chapter and applied to subsequent applications of Sommerfeld’s method.
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