Ship Form of Minimum Wave Resistance 



^min = 2pU2ytk X(x)rix. (59) 



This is the case of a simple slender ship theory. The method of solution and 

 numerical results are given in literature [13]. In order to solve the integral 

 equation, let us employ the dimensionless coordinates 



X ~ -'i COS 6 , x' - -■t cos 6' . (60) 



The sectional area is non-dimensionalized and to facilitate the solution, one 

 may put 



dA(x) V diO) ■ 



dx 2l^ si" ^ ' 



Then Eq. (57) is equivalent to the following integral equation: 



-77 



cr{e') Y^ [y^Ccos e- cos d')]de' = k^ COS + kj cos 3(9 (62) 



Jo 



where y^ - g-t/u^. The displacement becomes 



r 



^ 



Jo 



1 dACx) ^ 

 V = - I ■ xdx 

 dx 



a(e) cos e de 



•'0 



SO that 



I 



cr(9) cos 9 d^ = 2 . (63) 



The integral Eq. (62) can be solved by means of Fourier expansion by Mathieu 

 functions. There is a relation for even Mathieu functions ce^(i9,q), that 



^n ce^id' ,q) Y^[2^{cos e - COS e')\Ae' = ce^(^,q). 



Jo 



Since ce^(c9,q) makes an orthogonal system such as 



:\ ce^{e,q) ce^(£^,q)d(9 =0 n :|; m 



•^ n 



'lo 



= 1 n = m 



1033 



