MOTION OF THE GAS SPHERE 335 



/W\l/3 



L* = (5.36 X lOhQyi^ f — j , if L* and Zo are in feet, 

 C* = 0.216 L*2,-i/2, in seconds 



as shown in the derivation of Eqs. (8.54), (8.55). 



The period T in seconds using these values of L* and C* is then 



(8.61) 



0.561 



{rQW) 



1/3 



r ().Ml{rQWYi^F{x) l 

 L hoZo'" J 



1.0 



05 



3 



b -0.5 



Z 

 D 



UJ 



o -10 



2 



■1.5 



•2.0 



■0.5 



•0.25 



025 



05 



0.75 



1.0 



V d+b / 



d+b 



Fig. 8.1 The image function F{x) for effect of surface and bottom. 



The function F(x), which varies between — oo and +1 over the range 

 — 1 < a: < + 1, is plotted in Fig. 8.18. It is interesting to note that, 

 contrary to what one might expect, the effects of free and rigid surfaces 

 do not cancel i^F^x) = 0) for the value x = ^2, corresponding to a charge 

 midway between the surface and bottom, but rather for the charge % 

 of the depth below the surface (x = 3^). Above this depth the period 

 is decreased {F{x) negative), and the period is greater closer to the bot- 

 tom, in agreement with the qualitative discussion of section 8.9. 



