Waves generated by Impact. 107 



must have v and 6 connected by the relation 



v 2 = ad + b0\ 



where a and b are two constants depending on the elasticity 

 of the spring and its initial strained condition. In actual 

 practice the mirror is initially in contact with the pointer 

 with a sensible pressure, and as the deflexion is usually very 

 small, the second term in the above expression is nearly 

 negligible in comparison with the first, v 2 is, therefore, 

 practically proportional to a6 ; in other words, the angular 

 deflexion of the mirror is approximately proportional to the 

 intensity of the sound incident on it. 



On account of mathematical difficulties, it seems to be 

 a hopeless task to attempt a numerical calculation of the 

 distribution of intensities in different directions round the 

 colliding spheres. But the analogous problem of two 

 vibrating spheres whose distance apart varies periodically 

 presents features similar to this problem, when the wave- 

 length of the disturbance produced is sufficiently smalL 

 For this case, however, we can approximately calculate the 

 distribution of intensities in different directions by the 

 following method. 



If we take as our origin the point symmetrically situated 

 between the two spheres and the line joining the centres as 

 our initial line, it is easy to see that the velocity potential 

 of the wave-motion will be given by 



^= [a„/„<» 



+ A: 



±{(ka)%(ka)} 

 (krYMkr) 



P 2 (cos#) 



+ Aj WW P 4 ( C0S ff) + &c. 1 «?* 

 Ta {(ka)%(ka )} 



where 



/m-^fl5 fl-f n ( n + 1 ) , (n-l)n(n + l)(n + 2 ) 



.2.3...2n 1 



: .6 ...2n(iQ n J ' 



+ ... + 



and -r is the wave-length. 



The unknown constants A , A 2 , &c. have to be determined 



