Ship Form of Minimum Wave Resistance 



In order to make the wave resistance minimum under the condition of Eqs. (19) 

 and (20), the coefficients a^^ and k should satisfy the following equations. 



^=0, il. 0, ... ^= (22) 



da 2 oa^ dk ^ ' 



together with Eq. (19). These are equivalent to the simultaneous equation 



L 



M,„ ,» = kf-D" , n -- 1,2 (23) 



2m 2n, 2m 



If the infinite series of Eq. (16) is truncated at Nth term, Eq. (23) together with 

 Eqs. (19) and (20) presents N+l equations for n+ 1 unknowns. The coefficients 

 can be determined provided the characteristic determinant is non-zero. Assume 

 that the coefficients of the Fourier series, Eq. (16), satisfy the Eq. (23) and sub- 

 stitute in the integral 



i 



f(x') K(x - x') dx' . 



■I 



Making use of the Eq. (18), it is easily found that 



.1 



f(x') K(x- x') dx' = hi 



zz b-tk Y + 7] (-)" cos 2n 



L n=l 



f(x') K(x- x') dx' = b-t I o-(e') K [^(cos 6' - cos d)] 66' 



(24) 



where 



f(x) 



ha(0) 

 sin 



When one tends N toward infinity, Eq. (24) will give an integral equation which 

 the minimal solution f(x) or cr(e) should satisfy. However there is a relation 



1 ^ n ^ . (-f COS (2N+1)6? 

 ^+ 2^ (-) COS 2n^ = ^^^^ 



n= 1 



and the right hand side of Eq. (24) does not converge to a continuous function. 

 Bessho has proved that the solution diverges in the case of infinite strut. For 

 the infinite strut, the solution is expanded into a series of eigen-functions as 

 mentioned before. The coefficients can be determined analytically. By virtue 

 of the asymptotic behavior of Mathieu functions, a few terms at the beginning of 

 the series becomes dominant when the speed parameter ^^ increases. Though 

 the minimal solution gives a diverging series, the latter may be regarded as an 

 asymptotic expression for small Froude number. According to the numerical 

 results, the asymptotic value obtained by taking first three terms gives a 



221-249 O - 66 - 66 



1025 



