Maruo 



a solution. It can be shown by the Eq. (40) that f(x, z) is finite (or zero) at both 

 ends X - tl. 



ASYMPTOTIC FORM OF THE OPTIMUM ELEMENTARY 

 SHIP FOR VANISHING DRAFT 



It has been shown that the elementary ship of given vertical distribution has 

 a minimal solution for modified Michell's integral under the single condition of 

 constant displacement. The wave resistance is given by 



R = 2PU274 X(x) Z(z) XCx') Z(z') K(z + z', x-x')dxdx' dzdz'. (44) 



J -I J-lJo Jo 



Letting 



T ^T 



7(7.) Z(z') K(z + z',x-x')dzdz' = K(x-x') (45) 



-^0 



and writing f^x) in place of X(x), one obtains i 



R=2pU2 74 f(x) f(x') K(x-x') dxdx' . (46) 



J. I J-l 



Therefore the optimum form is given by a solution of an integral equation such as 



f(x') K(x-x')dx' = k . (47) 



I 



As mentioned before, the above integral equation have a solution which can be 

 determined only by a numerical way. Though there have been some examples, 

 the solving procedure requires very tedious and extensive calculation any way. 



As the basic assumption of Michell's theory is that the beam of the ship is 

 very small in comparison with the length, it applies to the thin ship. However 

 actual ships have draft which is smaller than the beam. The slender ship stands 

 on the idea that the draft length ratio is of the same order of amount as the 

 order of the beam length ratio which is much smaller than unity. The lineari- 

 zation is achieved by means of these parameters. Attempts have been made to 

 find out a slender ship form of minimum wave resistance [13]. They seem not 

 to be successful from the practical point of view. The reason is that the solu- 

 tion involves only ship forms of a very restricted class and is by no means the 

 best among whole admissible ship forms. 



Results which will be reported here deviate from the original slender ship 

 assumption. The basic idea is to return to Michell's integral and to look for an 

 asymptotic form of the minimal solution when the draft becomes infinitesimal. 



1030 



