GENEKAL THEORY OF SOUND WAVES 



217 



equal wave- velocity being supposed to be horizontal, each ray 

 will travel in a vertical plane. The cur- 

 vature of a ray may be calculated directly 

 by a method due to Prof. James Thomson*. 

 If R be the radius of curvature, the two 

 wave-fronts passing through the extremities 

 of an element 8s of the path will be 

 inclined at an angle 8s/R, and if 8s be the 

 length intercepted on an adjacent ray in 

 the same vertical plane, we have 



are 



(2) 



.(3) 



where 8n denotes the distance between 

 the two rays, the standard case being 

 that shewn in the figure. Since the elements 

 described in the same time we have 



8s _ 8s 

 iT+8c"~"c' 

 whence, by comparison with (1), 



1 1 dc 



IR ~ ~cdn' 



When the temperature diminishes upwards, 9c/9n is negative 

 and the curvature l/R is positive, as in the figure, and the rays 

 are curved upwards. But if the temperature increase upwards, 

 the curvature is downwards, so that an observer at the level of 

 the source may hear sounds which would otherwise have been 

 intercepted by obstacles. 



The formula (3) leads to the ordinary law of refraction. If 

 >Jr be the inclination of the ray to the horizontal we may write 



dc dc dc dc . 



5 = -j- cos y, = -j-sm-\lr, (4) 



on dy ds dy 



if y be the vertical coordinate. Hence, along the course of 

 a ray, 



i?*lr 1 1 tJ.r. 



.(5) 



ds 



1 

 ^D 



R 



Idc 

 -y- 



cds 



* James Thomson (1822 92), professor of engineering at Belfast 1857 72, 

 and at Glasgow 1872 89. 



