Panel Discussion 



been assumed to provide a sufficiently 

 realistic shape for the present purpose. 



If we use an approximate relation 



for a short interval of x, we can find the 

 volume inside the cylindrical column and 

 outside the original ship hull between x - a 

 and X ^ b by an integration of A(x) 



A(x) dx 



-2v - 



'-^r 



,2v2 _ _ „2v3 



3a. 



/( 



a2x2)3 



Fig. 2 - 

 of a ship 

 cal bulb 



Cross section 

 with cylindri- 



where a^ and b^ are values corresponding to a and b due to the approximation 

 y = ax, or aj - y(a)/a and bj = y(b)/a. Since the volume of a sphere with the 

 radius v^ is ^-nr^ /3, using the radius- strength relation of doublet bulb we can 

 add a proper doublet to fill in v^ at a proper position, say, z= -H, y=0, x = d 

 Then this would produce another negative sine wave which can be reduced by 

 adding a sine ship which has the bow stem at the position of the doublet. Thus, 

 for this sine ship the original Froude number F will be increased to 



Fi 



if the stern position is kept the same for each elementary ship. Thus, r^ and 

 are known. Therefore one can obtain the optimum a^ from the optimum rela- 

 tion mentioned before for r, , f , and a^, . 



For a given radius of cylindrical bulb r, we took 



d =— nr^ d(n), n=0,l,2,3,..., N 

 3 



2 4 , 



+ rr n I r = a(n) , 



b = 



2 4 



— + — (n+1) 



3 3 ' 



a(n+l) 



n = 0, 1, 2 ... N , 



where N is such that v^ is always positive. Thus, N + i is equal to the number 

 of point doublets distributed. Although the idea behind the choice of d, a, and b 

 is that a cylindrical column whose radius is r and whose volume is the same as 



1563 



