Sound Waves in the Atmosphere. 109 



azimuthal direction <f>. Then ignoring for a moment the 

 component of wind at right angles to </>, we see that 



R=z ( A_ — y 



\a —d + W — Wcos (/>/ 



(B-W cos <£)*' 



(36) 



say. A sketch of these curves of audibility for various 

 values of W is shown in fig. 3. When W<B, the curves 

 are closed ovals, and audibility is limited in all directions; 

 when W > B, the curve has a pair of asymptotes, 



</> = -Hare cos (B/W), 



which, divide the plane into regions in which the audibility 

 is and is not limited ; when W — B ? the audibility is limited, 

 save in the direction </> = 0, and there are two asymptotes 

 given by R sin<£= + A^B) - *. ^ Q effect of the cross 

 wind is to displace each point (R, <j>) inwards towards (f> = 

 through a small angle of which the value is found to be 



. W^ 



sm 



sin 4>[—\ 



The consequent displacement of the curves can easily be 

 pictured. 



§ 7. Further cases of limited audibility. 



In practice, when ® L or 0/ is real, the corresponding 

 condition (31) or (32) is usually found to be satisfied, but 

 interesting theoretical cases occur when this is not so. 

 For simplicity consider only rays in the positive direction 

 Ox, (0<^<i7r). Putting 



Q, = a l -a+W 1 -W, .... (37) 



we see that the refraction formula may be written 



a(sec6'-l)-a l (sec^ 1 -l)=n. . . . (38) 



Since sec#i>l, 6 cannot take the value zero along any ray 

 unless fl takes negative values. Let Q m (< 0) be the least 

 value of 12 between the ground and the source, and let 

 ® wl be the angle given by 



ai (sec ©„-!)=•- a» (39) 



