Excel Files for Teaching About Wave Fronts and Rays for a Wave Moving in Media with Changing Index of Refraction

Abstract

In teaching about waves the students are learning how to find the refracted rays by using wave fronts or finding the wave fronts by using the rays. The usual teaching is based on models where the speed of the wave is constant in one medium and changes abruptly as the wave passes from one medium to another. This paper deals with ways of calculation the wave fronts and the rays for the case of a continuous changing of the index of refraction. For this purpose, Fermat’s principle is applied for multiple layers of very small thickness. Two models are presented for the speed of the waves: A model on which the wave speed depends on the square root of the depth of penetration of the wave and the other model, where the speed depends on a linear dependence. In both cases it is found that as the wave progresses it is “totally reflected”. In the case of the “square root dependence” the solution is a kind of cycloid which shows this behavior. In the linear case it is found that there is a moment where the wave is reflected, which is found by the maximum of a quantity “Z”. By using this quantity, the coordinates x and y can be calculated. As an application the refraction of the light in the atmosphere is calculated, where the dependence of the distance from the center of the earth is calculated and again the 2 models are applied. In this case the “square root” model gives a stronger deviation from the linear model. This helps the student to understand the change on the perceived position of the celestial bodies.