110 



Mr. E. A. Milne on 



Then it is clear from (38) that the ray Q x — 6 m divides the 

 rays into two groups ; rays for which O<0, <^@ ni are totally 

 reflected at some level before reaching the ground, whilst 

 rays for which ® m <6\<\ir reach the ground without 

 reflexion. As before, the range of audibility is given 

 approximately by 



h:(ap- 



(40) 



and the boundiDg ray 0- 1 =® m arrives at the ground with a 

 value of 6 given by 



a (sec6> -l)-n o -n wi . 



If H takes its minimum value Vl m at the ground, is zero, 

 (31) is satisfied, and we have the case there considered. But 

 if this is not so, the bounding ray arrives at the ground with 

 a non-zero value of 6. The range of audibility is limited or 

 not, according to the convergence of (40) in the vicinity 



Fig. 4. 

 Total reflexion with limited range of audibility. 



of 12 = £2 m . Figs. 4 and 5 illustrate the different cases. 

 In fig. 4 the boundary ray arrives at the ground at a 

 point P with a non-zero value of # c , and beyond P the 

 source will be heard (if at all) only by diffraction. The 

 existence of a non-zero value of 6 at the point of furthest 

 audibility is curious, as it offers a marked contrast to the 

 cases of § 6, where the corresponding value of O was zero. 

 The distribution of wind and temperature thus cast a definite 

 sound shadow extending beyond P, the boundary ray being 



