106 



Mr. E. A. Milne 



layers, and total reflexion occurs. If W and a change 

 continuously with the height (as is usually the case) the 

 ray has a horizontal tangent at the point where it is reflected, 



Fi- 2. 



and the path of the reflected portion is symmetrical with 

 that of the incident portion. 



It is well known that the range of audibility of an aerial 

 source of sound (with regard to points on the ground) 

 may be limited owing to total reflexion. Let T be the source 

 and let 6 be the angle which the normal to the wave-front at 

 any point makes with the. horizontal. Denote by 1? O the 

 initial and final values of 6 along any ray. Then (with the 

 axes as in fig. 2) 



«i sec 6 X — a sec = W — W 1? 



(29) 



where W, is the wind and a x the velocity of sound at the 

 height of the source. In this equation set # = 0, Oq — it in 

 turn, and let © x , ©/ be the corresponding values of 6 lm We 

 have 



sec©! 



Wo-W^flo 



sec©i' = 



Wo-W^ao 



Along this ray 



The angle ©! is real provided sec O x — 1 >0, or 

 W -W! + « -«i>0. 

 a sec 6 = a + W — W 1? 

 and hence provided in addition 



a + W >a + W (31) 



for all heights between the source and the ground, the ray 



(30) 



