The Classification of Linearly Polarized Transverse Electric Waves

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:∇×E=iωH;∇×H=−iωE. 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.