Abstract :
Recently the author has undertaken the classification of continuous solutions to
some common vector PDEs. These include the simplest of Beltrami solutions to
hydrodynamic flows and electromagnetic wave equations which are, in fact, closely
related.. In this paper, we consider linearly polarized transverse electric wave
solutions to the electromagnetic wave equation: ==EsivH; ==HsyivE.
Using Clebsch functions and differential geometric techniques the author is able to
give a discussion of the most general possible forms of propagation of linearly
polarized waves. A zero curvature condition which asserts that the Clebsch
functions necessary to represent the solutions are expressible in terms of Cartesian
variables. reduces the study of vector PDEs to nonlinear PDEs, which may be
solved in special cases. The benefit is that the Clebsch representation is the one
that is the most useful for analyzing the structure of the flow. This method of
analysis is now a feasible solution method due to the tremendous advances in
mathematical software allowing one to compute curvatures, given metric functions.
For the special case of linearly polarized transverse electric waves the analysis may
be applied to a great extent.
