MOTION OF THE GAS SPHERE 



323 



The correction to the velocity potential function at any point is then 

 obtained by adding the potentials of the image sources, and for evalu- 

 ation of the kinetic energy it suffices merely to know the mean values 

 at the original source. Since the source is, in the present approxima- 

 tion, a point, only the values at this point are required. Summing the 

 values of 1/OPn, 1/OQn with proper regard to sign gives for the correc- 

 tion A^a, remembering that the original source has a strength a^, 



(8.43) 

 where 



A<^a = 



X = 



d-h 



d + h 



[2xF{x) - log, 2] 



F{x) = E 



n = 0(2^ + l)^ 



SURFACE BOTTOM 

 / \ 



^etc0 k 



SOURCE 



© 



: © 



© 



etc 



Fig. 8.17 System of images for a pulsating sphere between a rigid bottom 



and free surface. 



8.9. The Equations of Motion for a Sphere and Bounding 

 Surfaces 



In the last section, the distributions of sources which represent the 

 flow of a fluid between a moving sphere and infinite boundaries were ob- 

 tained. In order to determine the motion of the sphere, it is next 

 necessary to write the dynamical equations of the motion, and this is 

 most conveniently done by use of the energy principle. The kinetic 

 energy T of the fluid is, from section 8.4, given by 



(8.44) 



-t/j' 



dn 



dS 



where the surface integral is to be extended over all boundary surfaces. 

 The velocity potential (p is calculated from the source distributions for 

 pulsation and translation of the sphere: 



da 

 di 



U<p. 



