230 Prof. D. N. Mallik on the 



10. Thus the expression for the disturbance at a point O 

 can be written, for an element subtending d£l at 0, viz. f(t), 



where U^ is the resultant velocity at time t' at the sphere 

 centre and of radius c(t — t f ); and the above construction 

 shows that, if we put 



t-s = r ~, 



c 



then irx dS 



dil =—5 cos nr, 

 ir 



— £/K) 



=_^ if( t - r -) 



cos nr on V C J 

 [cos nr = angle between r and the normal, drawn outwards]. 



.-. ^).[|,|j/(.-*;)}~- r 1 |,/(-^. 



which is KirchhofPs solution. 



11. In particular, if r x is the distance of the centre of a 

 diverging spherical wave from d$, and 



/0)(at^)=^cos(^-^) 



in view of the fact that the disturbance (s) from a point- 

 source has to satisfy the differential equation 



d 2 (rs) 

 dt 2 



( \ v d\ . 



= [Y)'d? {rs) > 



then it can be shown that at P (whose distance from d$ is r) 

 the disturbance is 



A f 1 . ft r + r{\ . . , a 



sr I sm27r r -^ — Hcosnr — cos nr*) do. 



2\J rr x VT A. / v l 



12. Remembering that the direction of displacement must 

 be always perpendicular to the direction of propagation, the 



