359 



Sn is the reflection of S in the bottom 

 1 o 



S -, is the reflection of S in the surface 



S T is the reflection of S in the bottom 

 n+l -n 



S_ is the reflection of S -, in the surface. 



Note that a reflection in the bottom, to satisfy equation 

 (4.2), gives an image of the same strength, while a reflection 

 in tiie free surface to satisfy equation (4.o) gives an image 

 of the same strength with the opposite sign. (A source of 

 negative strength is to be considered a sink.) V.'e find that 



the strength of Soj, is (-)" at a distance 2nD from S 



the strength of S2J^_^ is (-)" at a distance 2nD-2B from S. 



the strength of 3 ,, is (-) at a distance 2nD from S^ 

 -^ -2n o 



the strength of S r, t is (-)^''' at a distance 2nD+2B from S^. 

 ^ -2n-l o 



$ (0), the potential at the center of the bubble due to 

 this collection of sources and sinlcs can be obtained by com- 

 bining the effects of S axid S „. 



° n -n 



The potential at S due to Sg^^ and S_2^ is 



nD 



while that due to S^ -, and S_o ^^ is 



(4 7) ( ^nr 1 L_l= (-)" (43-2D) 



^ ' ^ ' ^(2n+2)D-23 2nD+2B^ [ (2n+l)D+D-2B] [ (2n+l)D-(D-2B) ] 



The total potential due to all the image sources and sinks 

 is, therefore, 



00 , <n T .„ 00 I xn 



^ W-^D^B - ^ (2n4.1)2 - (1 -f)2 



