Sound Waves in the Atmosphere. 99 



and let I, m, n be considered as constant in differentiating with 

 regard to &, y, z. Then (5) may be written 



I \dt B# ) m \ dt + ~dy J n\dt + 3* ) 

 explicitly, 



( l h +m &+ n h) Y > w 



* / , ,x9 y ,9V , BV 



* = -( m, + " i ^ + fo, % + jB V • • (7) 



It is clear that equations (1) express the direct propagation 

 of the sound, as influenced by the bodily displacement of 

 the medium, whilst equations (6) express the refraction 

 of the sound caused by the variation of the normal speed V 

 over the surface of the wave-front. 



§ 3. The partial differential equation satisfied by a family 

 of wave-fronts. 



Let the family of surfaces successively occupied by a 

 wave-front be given by an equation 



F(*,.y,jr;9«0 (8) 



Take a particular point (#, y, z) on a particular surface t, 

 and let (V, y', ?') be the point on the corresponding* ray 

 where it is met by the surface t + dt. By equations (1) 



x' = a: + (u + al)dt, (9) 



etc., where 



7= ±F x /(F/+F/ +!?)', • • • (10) 



and the sign is selected in accordance with the direction of 

 propagation. The new wave-front, obtained by eliminating 

 x, y, z between (8) and three equations of type (9), namely, 



F{x'-{u' + al')dt, , ;t)=0, 



must be identical with 



¥{x',y\z'; t + dt) = 0. 



Comparing these two we have 



^ +(« + „!)!? + («, + «>») |? + (w'+an) g =0, 



ot O'V oy o- 



or 



BF BF BF £F^ C/BF\ a , /BF\ 2 /3F\V-0 



H 2 



