The mathematical problem of reflection solved by an extension of the WKB method

Abstract

Starting with two first order linear differential equations having slowly varying coefficients and mutually connected by the dependent variables, the reflection problem is solved approximately as in the cases of electromagnetic and acoustic waves. An extension of the WKB method is developed and applied to study this problem up to any higher order of accuracy one requires. The solution of the second order differential equation in normal form by the extended WKB method is used to find the characteristics of propagation at a point of discontinuity of higher order derivative of the parameter.