not quite, exist in an exact sense. These false solutions must be eliminated from considera- 

 tion, and this may be done in the case of the cavitation problem by adding the following "junc- 

 ture condition." 



The flow at the body-cavity juncture must be "smooth," or, more specifically, the 

 slope of the cavity must be continuous with the slope of the body. In the analysis that fol- 

 lows, this condition will be enforced by eliminating those linearized solutions for the cavity 

 source distribution which "blow-up" at the juncture. The paradoxical result that had at first 

 appeared will thus be resolved, and, as in reality, a nonarbitrary correspondence between cav- 

 ity length and cavitation number will be obtained. 



The admissible linearized flows that remain still exist in contradiction to the Brillouin 

 Paradox, for they are flows about closed bodies with constant pressure afterparts. Their ex- 

 istence is explained by pointing out that the linearized theory is invalid near the end of the 

 cavity where the cavity has roughly an elliptical shaped trailing end, and the real flows cor- 

 responding to the linearized flows do not, of course, have constant pressure there. 



It is to be emphasized that the use of linearized theory makes unnecessary the selec- 

 tion of a specific finite cavity model. 



THE GENERAL SOLUTION FOR THE SOURCE DISTRIBUTION AND CAVITY SHAPE 



The solution of the integral Equation [9a] for the cavity source distribution is (see 

 Reference 5, Table of Singular Integral Equations and their Solutions) 



m(x) = 



ir'Vx{l-x) 



'' Vx'il-x ') 

 (x - x') 



naU^ 



2U.^dt 





dx' + a 



[11] 



To satisfy the juncture condition it is necessary that the term in brackets in Equation 

 [11] vanish at a; = 0, i.e.. 



a + 

 Jo 



\x'(l-x') ,, , , ['V xil-x) , , 

 , 7T aUx dx — I : a x 



Jo 



2U,^ dt 

 dt 



^(x'-t) 



= 



[12] 



so that 



m(x) = 



Vx 



' }fr^' 



naU^ 



TT^il — x \]{)Vx^{x—x') 

 or, using Equations [32] and [33] of Appendix 1, Part A 



2U^'^^df 

 dt 



ix'-t) 



dx' 



[13] 



