SIMPLE-HAKMONIC WAVES. DIFFRACTION 229 



at the mean position of the centre of the sphere, and the axis of 

 x along the line of its vibration ; and we denote its velocity 

 by U. The velocity of the fluid in contact with the sphere at 

 any point P, resolved in the direction of the normal, must be 

 equal to the normal component of the velocity of the point P of 

 the sphere itself, i.e. to UcosO, where is the angle POx. 

 This gives 



- d ^=Ucos0 [r=a], ................ (1) 



if a be the radius. The velocity due to a double source at in 

 an unlimited mass of incompressible fluid is of the form 



sfl; ..................... (2) 



and in order that this may be consistent with (1) we must have 



C=27ra*U. ........................ (3) 



With this determination of C the effect of the sphere on the 

 fluid is exactly that of the double source, and the solution of 

 our problem is 



*=g- 3 cos0. ..................... (4) 



This depends only on the instantaneous value of U, as we should 

 expect, since under the present hypothesis disturbances are 

 propagated with infinite velocity. It should also be noted that 

 there is so far no assumption that U is small. 



The directions of motion at various points of the field may 

 be shewn by tracing the " lines of motion," which are lines 

 drawn from point to point, always in the direction of the 

 instantaneous velocity. In the case of small vibratory motion, 

 which we have especially in view, each particle oscillates 

 backwards and forwards through a short distance along the line 

 on which it is situate. If Sr, rS0 be the radial and transverse 

 projections of an element of such a line, these quantities must 

 be proportional to the radial and transverse components of 

 velocity, viz. d<f)/dr and -d<l>/rd6, respectively. Hence 



Sr rS0 

 cose Jsinfl' ' 

 the integral of which is 



........................ (6) 



