A Luneburg-Kline representation for wave propagation in a continuously inhomogeneous medium

We derive a Luneburg-Kline (GK) asymptotic series representation for wave propagation in a one-dimensional (1-D) continuously inhomogeneous medium. We set the solution up so that the classical phase function common to the Wentzel, Kramers, Brillouin, and Jeffreys (WKBJ) approximation multiplies all terms of the L-K series. We develop an error criterion for the WKBJ approximation based on the magnitude of an ignored term relative to retained terms in the governing differential equation. Finally, we note that while the validity of the GK series solution is dependent upon a large free-space wavenumber for small to moderate spatial gradients in the index of refraction, large spatial gradients can be accommodated by increasing the free-space wavenumber. Hence, there is a strong similarity of this situation to boundary diffraction problems where rounded edges approach, but do not achieve, absolute sharpness. Loosely speaking, in both instances a scale size in terms of the wavelength is maintained constant