Calculation of reflection and transmission coefficients for a class of one-dimensional wave propagation problems in inhomogeneous media

Making use of an extension of Langer's method, we have solved the one‐dimensional Helmholtz equation ${\mathrm{w″+k}}_0^2\text{g(z)w(z)=0}$ , k₀ = 2π/λ₀, in which λ₀ is the free space wavelength and in which g(z) represents any one of the three characteristic symmetric profiles. The main objective of this paper is the computation of the reflection and transmission coefficients for such a family of profiles making use of one and the same method throughout. First, we introduce our modification of Langer's method that reduces the problem to the solution of a comparison differential equation of the parabolic cylinder type, which we choose to solve in terms of confluent hypergeometric functions. Then, we set up a solution w₊(z) valid for 0 ≤ z which, as z → ∞, becomes the transmitted plane wave. At this juncture we apply the method of analytic continuation, as z is changed into − z, and we obtain a linear combination w₋(z) = w_r(z) + w_i(z) valid for z ≤ 0 which, as z → − ∞, yields the superposition of the incident and reflected waves. The reflection and transmission coefficients then ensue readily upon division by the coefficient of exp(i k₀z) in the incident wave. The body of the work is first carried out for symmetric profiles, which we then proceed to extend to nonsymmetric profiles. Because we are using Langer's method in lowest approximation, we examine in detail the conditions underlying the applicability of the method and compare our results with the exact results derived from symmetric Epstein profiles. Although this paper is concerned mainly with the solution of the reflection‐transmission problems, we have in fact set up all the necessary mathematical machinery for the computation of the actual wavefunctions in the vicinity of the origin. We have undertaken to do so, for specific parameter values, in a companion paper to appear shortly.