38 Lord Rayleigh on the Incidence of Aerial 



x, 47T — L is infinitesimal. By (21), (24) we get in this case 



4ttT _ 8c 3 

 4tt-L~T ; 

 so that 



^=-^!L± e — (60) 



oat r r 



If the axis of the disk be inclined to that of x, yjr retains its 

 symmetry with respect to the former axis, and is reduced in 

 magnitude in the ratio of the cosine of the angle of inclination 

 to unity. 



In the case of the sphere the general solution is 



, vrTe- ikr Cm'— m Sx a f -a~) ,..* 



Waves in Two Dimensions. 



In the case of two dimensions (x, y) the waves diverging 

 from a cylindrical obstacle have the expression, analogous 

 to (41), " 



*=S D (*r)+S 1 D l (*r) + ,.., . . (62)f 



where S , Si . . . are the plane circular functions of the various 

 orders, and 



D »w=-(^) Va '{ 1 -i + ---} 





• • 5 



(63) 



8ikr J 



Pr 2 l , ikr\ C kr k 3 r 3 "1 



kr 3 AV 3 ,„.. 



+ i-i^i + < 64 > 



As in the case of three dimensions already considered, the 

 term of zero order in y\r depends upon the variation of com- 

 pressibility. If we again begin with the case of an unyielding 



* 'Theory of Sound/ §335. 



t See ' Theory of Sound,' § 341 ; Phil. Mag. April 1897, p. 266. 



