Snell's law of refraction and sound rays for a moving medium

摘要

Fermat's principle of least time is used to derive an improved law of refraction for a stratified medium moving horizontally with different temperatures in each layer and includes velocity of sound, wind speed, and the angle (vertical directed) between the vectorial sum of sound velocity and the wind speed. The new equations are compared with the usual approximations for the different laws of refraction of a moving medium occasionally mentioned in literature as "Snell's law for a moving media". The propagation of sound rays is calculated with the improved law of refraction and compared with the classical results. For upward-orientated sound rays and a moderate thermical stratification with high wind speeds the difference in the angle of refraction between the improved refraction law and the "usual approximation" is roughly 3 degrees and neglectable for grazing incidence. The sound ray trajectories are expressed as analytical equations for a constant sound velocity combined with a wind speed profile proportional to height; the results are shown in a figure. Furthermore, the sound ray trajectories for a linear sound speed profile combined with a linear height-dependent wind speed profile are calculated and the results are shown graphically. An approximation of the sound ray trajectories in a moving medium is given for the three-dimensional case. [References: 26]