Refraction of Sound by Wind. 163 



characterizing the intermediate zones. It. should be noted, 

 however, that a cosecant cannot have a value between -f 1 

 and — 1, so that i£ any of the zones required this, the series 

 must cease there, equations (5) and (6) not holding for 

 higher zones. This brings us to the next topic. 



Total Reflexion. — Although the cosecant law for the wave- 

 front obtaining here differs from the ordinary optical law of 

 refraction, we still have, as in optics, the phenomenon of 

 total reflexion possible. And it is in this connexion that 

 the distinction between wave-front and direction of pro- 

 pagation is most striking. Thus, for the wave-front, if the 

 angle of refraction 6 n is put 7T/2, we have from equation (5) 

 the critical case expressed by 



coseo^ - ? ^^ =l (7) 



Hence any pair of values of O and (u n — u ) which violates 

 (7) affords an example of total reflexion, the last zone not 

 being entered by the beam. 



Suppose now that we have a series of wind-zones in which 

 the wind-speed increases as we ascend : then, provided the 

 initial inclination of the wave-front were finite, it is clear that 

 at some point we must have total reflexion. But if, on the 

 other hand, the wave-front were initially horizontal, we 

 should then have # = 0, and, by equation (5), all the } s 

 would be zero also. That is, we should have no refraction 

 of the wave-front, and consequently no total reflexion any- 

 where, whereas the ray, as we ascend, deviates without limit 

 from the vertical, for equation (6) now reduces to 



tan (/>„ = u n /v (8) 



So that in this case we have zero refraction of the wave-front 

 associated with unlimited refraction of the rays, total reflexion 

 being impossible. 



On consideration of the case by Huyghens' principle, it 

 is seen that where a zone cannot be penetrated, and total 

 reflexion occurs, the reflexion follows the ordinary optical 

 law, angle of reflexion equals angle of incidence. 



Path of Rays ivhere Wind increases continuously with 

 Height. — Let the wind be everywhere horizontal, and in the 

 same vertical plane, but let its speed vary from one level to 

 another according to the equation 



u/v = c + ay, (9) 



where u is the speed of the w T ind, v that of sound, c and a are 

 constants, and y is measured vertically upwards. The x co- 

 ordinate will be taken horizontally to leew T ard. It is now 

 required to determine the inclinations of the wave-front and 

 of the rays at any point, i. e., we require 6, </>, and x for any 

 given y. The equations for 6 and (f> are derived immediately 



