Propagation in a stratified medium with an extension of the circle diagram

Abstract

In previously described work, the circle diagram technique was extended to propagation whose characteristic impedance was approximated by a step-wise variation or was imaginary. This paper describes an extension in which the point impedance is found by solving a generalized Riccati equation. A normalized impedance is used so that one diagram can be applied to a class of variations. The solution is given by two orthogonal families of curves: curves of constant phase and curves of constant standing-wave ratio. The field can be obtained by a simple integration of an associated differential equation. The magnitude of the field can be obtained from a simple algebraic formula for the intensity. A simple solution is obtained for the case of a layer in which the variation of the characteristic impedance is linear; by a simple transformation, the impedance diagram is a bipolar plot. A brief connection between the graphical and analytic methods is made.