Ship Form of Minimum Wave Resistance 



The solution to minimize Michell's integral for elementary ships under a 

 single condition of constant volume exists as mentioned before. Assume an ele- 

 mentary ship, f(x,z) : X(x) Z(z), and write Michell's integral 



IpV^y'* I X(x) X(x') KCx-x') dxdx' 



(48) 



where 



K( x - X ' ) 



/.OO 



COS [7\(x-x')] 



f 



Vn 



Z(z)e-^'^ 'dz 



The integral equation to determine the minimal solution is 



/. d/. 



(49) 



X(x') K(x-x') dx' = k . 



(50) 



Since the kernel has a logarithmic singularity at 

 form 



the solution takes the 



X(x) 



U(x) 



V^ 



(51) 



If U(x) is finite at x = ±-1, the function X(x) becomes singular at both ends. Now 

 let us consider the asymptotic behavior of the wave resistance when the draft T 

 tends to zero. The simple slender ship theory expands the integral 



r 



•Jo 



7(2) e 



by an ascending power series of T and takes the first term that makes the kernel 

 K(x - x') have the order of T^, Though the kernel has a higher singularity at 

 X = x' , the integral with respect to x and x' is regarded as the finite part due to 

 Hadamard. Then Eq. (48) becomes finite only when dxcx)/dx = o, otherwise the 

 integral diverges. Since the finite part of the integral is taken, the singularity 

 of the kernel does not matter except at the end points x = i-t. Therefore the in- 

 finity appears from the behavior of the integrand at the ends. This phenomenon 

 may be called the end effect. It has been shown that the end effect gives a term 

 of the order of T^ -tnT when dX(x)/dx is finite there. The order of the end effect 

 can be evaluated if Z(z) is assumed as a simple function such as Z(z) = i and 

 the behavior of the resulting integral is examined at the limit of zero draft. It 

 can be proved that the end effect has the order of T^^^ ^hen the water line func- 

 tion takes the form of Eq. (51). Since the volume is proportional to T, the re- 

 sistance per unit volume increases infinitely when the draft decreases. It seems 

 to be natural that this case is excluded from the admissible solution. The case 

 that X(x) is finite at both ends is also excluded by the same reason because the 

 order of the end effect is T -Cn T. Therefore the only case that the width of the 



1031 



