Maruo 



2 r"' r^ c^ 



R ^^ , f(x) f(x') cos [7\(x- x')] dxdx'. ('^^ 



From this equation, the condition of the minimum wave resistance for a fixed 

 water plane area gives the integral equation 



J 1 J}J^ J- 



f(x') COS [7>v(x -x')]dx' +k = 0. 



(8) 



Because of the integral representation of the Bessel function of the second kind, 

 the kernel is expressed by a known function. 



5 1, 



f(x') Y^ [7(x- x')] dx' = k . (9) 



This equation was dealt numerically by Pavlenko [8] without regard of the exist- 

 ence of the solution. Wehausen pointed out that the solution of the integral equa- 

 tion has a type of 



U(x) 



where U(x) is bounded. Karp, Kotik and Lurye calculated the function U(x) nu- 

 merically for several Froude numbers. It was found that U(x) did not vanish at 

 X = ±-t, so that the solution becomes singular at both ends. If f(x) gives the 

 ordinate of the surface, infinite horns appear and the condition f(±^) = is 

 violated. A similar situation appears in the case of finite draft because of the 

 logarithmic singularity still existing in the kernel. As far as original Michell's 

 assumption is employed, there is no admissible solution of the present problem. 



However the formula of the wave resistance may have a different interpre- 

 tation from the definition of original Michell's integral. It can be shown that the 

 Eq. (7) gives the wave resistance of a distribution of x -directed dipoles over 

 the vertical plane y = 0. Then f(x) does not mean the shape of the strut but 

 gives the density of the dipoles. Karp and others calculated the boundary 

 streamline when such a dipole distribution was placed in a uniform stream. 



The integral Eq. (9) belongs to the family of equations of the type 



f(x') Y„ [y{x- x')] dx' = g(x) (10) 



I 



which was solved by Dorr [9]. By the change of variables 



X = -'{■ COS 1^ , x' = --{ COS 6' 



Eq. (10) is converted into 



1022 



