Waves generated by Impact. 105 



(2) The case of two spheres. 



The solution for the case when both the spheres undergo 

 instantaneous change in velocity U cannot, however, be so 

 easily obtained. In a recent paper * published in the 

 ' Bulletin ' of the Calcutta Mathematical Society, I have 

 given a method by which a solution for this case can be 

 obtained. For our present purpose, however, we see from 

 symmetry that if we take as our origin the point of contact 

 of two equal impinging spheres, the velocity potential of the 

 wave-motion which satisfies the boundary conditions over 

 the surfaces of both the spheres can be written in the form 



^ = A„ U " 2 ^V) + A 3 U„'(^L) 2 . 

 r \rQrJ { 



2( X 2 (~) 



P 2 (cos6>) 



+ &C. + A 2n Va 2 ^ 



-f&c, . . 



\r-br) 



*/, 



"2n 



P:n(cOS 6) 



(8) 



due regard being paid to the dimensions of both the sides. 

 A , A 2 , &c. ; \ , \ 2 , &c, are certain constants not 

 depending on the radius of the spheres to be determined 

 by the boundary conditions. 



At a great distance from the source of sound we can 



neglect all powers of - in this expression beyond the first. 



So that at a great distance we have approximately 



Uo 2 

 r 



■\ir : 



[Ao^° 



A \ n 



) +A 2 X 2 2 /^^)p 2 (cos6>) + &c."j. (9) 



Also, since the sound-pulse is practically confined to the 

 wave- front, and also since in this region we have either r 

 equal to ct or less than ct by a few diameters, we see that 

 the expression within the bracket may be regarded as a 

 simple function of the time and the inclination 6, independent 

 of the radius of the spheres or the distance r. 



The intensity of the sound for large values of r therefore 



IPa 4 

 varies as — «-. 





* On " Sound-waves due to prescribed Vibrations on a Spherical 

 Surface in the presence of a rigid and fixed Spherical Obstacle," 

 Bulletin of the Calcutta Mathematical Society, vol. iv. 



