Ship Form of Minimum Wave Resistance 



f(x',z')K (z+z',x-x')dx'dz' --^ k,z+k„-^+'^p,(x). (39) 



J- I' Jo 2 ^3z 1 



Integrating again with respect to z , one may obtain a linear integral equation as 



f(x',z') K^^\z+z',x-x')dx'dz' =-2l<jz2 + k2f(x,z) + z'p ^(x) + 't> ^(x) (40) 



where 



K (z + z , x-x ) = e ^ cos L7^(x-x )J i^i^ 



^y^Ji 7/2 - 1 



and cpjCx) is another arbitrary function of x only. Since K^^Vz+z'. x-x') is 

 absolutely integrable in the domain -'Cixi-t, <z <T, one can write 



I ( 2 ) ; I / / 



|K (z + z , x-x ) |dx dz < M. 

 I Jo 



M is the maximum of the integral in this domain. Now assume that 



— kjZ^ + zcpj(x) + cp2(x) 



is bounded. Then the Neumann series for the integral Eq. (40) converges uni- 

 formly if Ik^l > M. Therefore the linearized integral Eq. (40) has a solution, 

 and the latter is unique except for the arbitrary constants k^ and k^ and the ar- 

 bitrary functions cpj(x) and cp2(x) . These unknowns are determined by side con- 

 ditions. There are already two of them, the given volume and the given effective 

 area. However the solution is still indeterminate owing to the functions fPjCx) 

 and cpjCx) . Two other conditions are necessary in order to determine the solu- 

 tion. It is understood easily that the vanishing ordinate at the keel line, i.e., 



f(x,T) = (42) ' 



can become one of the required conditions. The other can be a condition im- 

 posed on the shape at the water line z = o . As the integral on the left hand side 

 of Eq. (39) is bounded in the domain -^t^x^-t, 0<z<T, one may have 



fCx,z')K (z',x-x')dx'dz'=k2 ( "^ ) + ^i(^) ■ 



(43) 



Therefore cpj(x) can be determined by giving the slope of the surface ?f/3z at 

 z = 0. The vertical sides for instance corresponds to df/dz = o at z = o. This 

 condition is equivalent to the implicit assumption employed by Lin, Webster and 

 Wehausen in their calculation. As the non-linear factor has always a non-zero 

 denominator ,/i + (3f/3z) ^ , the integral equation at any stage of the iteration has 



1029 



