Gradual Reflection of Short Waves

Short wave asymptotic methods (geometrical optics, WKB, averaging and two-timing, etc.) fail to give information on the wave reflection caused by gradual variation of propagation speed with position. This paper examines plane wave propagation according to the classical wave equation in a medium of optical index of refraction changing analytically smoothly in the direction of propagation. The reflection coefficient is shown to be exponentially small in the ratio of wavelength to modulation scale. It is calculated to the first approximation for a large class of analytical index functions.