-13- 



259 



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The velocity potential $ describing the flow around 

 a moving, pulsating sphere in an Infinite body of water is 

 easily obtained from classical hydrodynamics. It is 



(2.1) 



^ = 



A^A' , A^B' cos e 



2 -75- 



where R,9 are coordinates as indicated in the diagram above, 

 and the "prlne" denotes a time derivative. The kinetic energy 

 "T* of the water is then 



(2.2) 



T' = 2n^A^[A'^ + |B'^] 



where p is the density of the water. For these classical 

 results, see, for example, [3] or Appendix II. 



The potential energy Xi of the system of gas bubble 

 and water is 



4_.3, 



U = §TTA^P^ + C}(A) 



(2.3) 



where the first term is the gravitational potential energy 



