General Solutions to Maxwell's Equations for a Transverse Field.

Abstract : The general solution to the wave equation for a transverse field is obtained in terms of the geometry of the wavefront surfaces S. Every solution to Maxwell's equation is a solution to this wave equation, but the converse is not necessarily true. Indeed, by using results from differential geometry and topology, it is found that smooth, singularity-free transverse solutions to Maxwell's equation cannot exist if S is a spheroid, a noncircular cylinder, or a surface or revolution. It is conjectured that smooth, singularity-free, transverse solutions to Maxwell's equations can only exist if S is a circular cylinder or a (flat) plane. Keywords: Electromagnetic propagation; and Harmonic fields.