322 MOTION OF THE GAS SPHERE 



is required, the first few of which are shown in Fig. 8.17.^° This neces- 

 sity for successive images indicates that surface and bottom effects are 

 not simply additions. 



The distances of the successive images from the original source are 

 easily seen to be 



(OP) n = n{d -\- b) if n is even 



= (n - 1) {d + 6) + 26 if n is odd 



{OQ)n = n{d -{- h) if n is even 



= (n - 1) {d -\-b) -\- 2d if n is odd 



ZERO PRESSURE 



BOTTOM 

 ^ ^ ^ V 



Fig. 8.16 Geonietry for a sphere pulsating between the surface and bottom. 



^° The infinite set of images required for this case were first given by Kennard 

 (55) ; the effect on bubble motion has been further developed by Shiffman and Fried- 

 man (102) and by Friedman (37). 



