Introduction

Abstract

In this chapter we will briefly present a survey of some of the well-known wave equations, describing waves traveling along inhomogeneous media and the attempts to arrive at solutions under different approximations. The electromagnetic fields and the wave propagation associated with them have been studied in great detail. These are vector wave equations for the electric $\mathop {\text{E}}\limits^ \to$ and magnetic $\mathop {\text{H}}\limits^ \to$ vector fields [1]. The scalar Schrödinger wave equation of non-relativistic quantum mechanics has been the subject of intense analysis [2]. The study of the passage of an electromagnetic wave through an inhomogeneous medium is vital to the great advances made in optics and hence we introduce below the ideas relating to the passage of a plane wave through a stratified medium, the electric and magnetic fields being perpendicular to each other in a plane perpendicular to the direction of the wave. They can be shown to satisfy linear equations provided the dielectric constant ε and magnetic permeability μ are independent of the applied fields. The case in which these quantities depend on the applied fields leads to nonlinear equations, which can be analyzed by the imbedding techniques presented in this monograph.