216 DYNAMICAL THEORY OF SOUND 



74. Refraction due to Variation of Temperature. 



Questions relating to wave-propagation in heterogeneous 

 media can only be discussed in a general way, and with the 

 help of conceptions borrowed from geometrical optics. If at 

 any surface there is an abrupt change of properties the law of 

 propagation is of course altered. If the dimensions of the 

 surface, and its radii of curvature, are large compared with 

 the wave-length, we have phenomena of regular reflection and 

 refraction, as in optics. Cases of absolute discontinuity are of 

 course not met with in the atmosphere, but the theory would 

 be practically unaffected if the change of properties were 

 effected within a space which is small compared with the 

 wave-length. 



When on the other hand we have a continuous variation 

 such that the change of properties within a wave-length is 

 negligible, the case is analogous to that of atmospheric 

 refraction of light, which is discussed in books on optics and 

 astronomy. In an atmosphere of the same gas, at rest, a 

 variation in the velocity of sound can only arise through a 

 variation of temperature ( 59). The refraction due to varia- 

 tion of temperature with altitude was first discussed by Osborne 

 Reynolds (1876). Suppose that, as usually happens, the 

 temperature diminishes upwards. Since the velocity of sound 

 varies as the square root of the absolute temperature, the lower 

 portions of a wave-front will be propagated faster than the upper 

 ones, so that a front which was originally vertical gets tilted 

 upwards more and more as it proceeds. The sound will there- 

 fore, for the most part, pass over the head of an observer at 

 a sufficient distance, such residual effects as he perceives being 

 referable to diffraction. On the other hand, whenever the 

 temperature increases upwards the waves will be tilted down- 

 wards, and the effect at a distance will be greater than if the 

 temperature had been uniform. This latter condition of the 

 atmosphere sometimes prevails on a clear night following a 

 warm day, when, owing to the cooling of the ground by 

 radiation, the lower strata of the atmosphere are reduced in 

 temperature relatively to the upper ones. 



The theory has been further developed by Lord Rayleigh, 

 by means of the conception of rays of sound. The surfaces of 



