Sound Waves in tlie Atmosphere . 107 



0i = ©i w iU actually reach the ground *. It is easily seen 

 that under these conditions the ray 1 = ® 1 is a bounding 

 ray separating rays that reach the ground from those that 

 are totally reflected ; rays for which i7r>#i >®i reach the 

 ground, whilst rays for which ® 1 >^ 1 ^0 are totally reflected. 

 Beyond the point of contact of the ray 1 = ® 1 with the ground 

 the sound will be inaudible. Similarly, if at all heights 



a -W >a-W (32) 



the ray 1 = © 1 ', where ^7r<®i'<7r, will be a boundary ray 

 for rays in the opposite direction. If R denotes the range 

 of audibility measured along 0#, we find 



R = f Z (cot 6 + — cosec 6\d6, 



and on substituting for from (30) and assuming that the 

 variations in W and a are small compared with a, we have 

 approximately 



B-j X-« + Vw y &; • • • (33 > 



similarly for the range in the opposite direction 



R'=j X-Jwo + w )^- • • • ^ 



These are infinite integrals, the denominators becoming 

 zero for 2 = 0. They either may or may not converge;* 

 convergence indicates a limited range of audibility, 

 divergence an unlimited one ; in the latter case the 

 boundary ray has the #-axis for an asymptote and the 

 source is theoretically audible everywhere. 



Consider, as an example, the- common case in which the 

 "refracting wind" W + a is a linear function of z. It is 

 found that in this case 



R,R' = Z( % =,? . . . (35) 



approximately, the bars denoting mean values with respect 

 to the height. As an illustration, if 



Z~ 10,000 feet, a -d = lS ft./sec, W o -W = 50 ft, sec, 



then R = 47,600 feet and R' is infinite. 



* If condition (31) is not satisfied, the ray 0, = O X will be totally 

 reflected before reaching the ground, but there will be no ray for which 

 0o = O. See below, §7. 



