Growth of oscillations of a ray about the irregularly wavy axis of a lens light guide

If a ray is launched in a direction coincident with the axis of a lens light guide whose axis is somewhat wavy, the ray will soon begin to oscillate about the axis. The amplitude of these oscillations will grow in proportion to the square root of the product of the distance from the origin and the natural warn number of the oscillations, on the average. The growth rate is proportional to the amplitude of the components of the spectrum of the waves in the guide axis in the immediate vicinity of the natural wave number of the ray oscillations. The rest of the spectrum of waves in the guide does not contribute appreciably to the growth of ray oscillations after the first few oscillations. A tractable analytic expression with four adjustable parameters is used to approximate the wave spectrum of the shape of the guide axis. The expression is used to illustrate the relations between various factors such as mechanical stiffness of the guide, spectrum of external forces, over-all amplitude of waves in the guide, etc., and the rate of growth of the ray oscillations. Estimates of the order of magnitude of the oscillation growth rate in some realistic models of light guides are made from these relations.