Maruo 



show interest in the application of the above theory in recent years. It is not at 

 all simple to realize its direct application because the theory of wave resistance 

 is not necessarily an approximation accurate enough to describe the actual phe- 

 nomenon. The exclusion of viscosity should be the most serious defect. Never- 

 theless a great utility of the above theory can be anticipated. The present paper 

 is prepared in order to place emphasis on the feasibility of the theoretical result 

 on the ship form of minimum wave resistance. 



SIDE CONDITIONS AND EXISTENCE OF THE SOLUTION 



The problem of minimum wave resistance can be considered only when 

 some side condition is imposed, because without any restriction, there is a so- 

 lution which makes no wave. It does not necessarily mean the trivial conclusion 

 that no ship makes no wave. In fact, a class of singularities was found which 

 are not accompanied with any wave term in the linearized potential of fluid mo- 

 tion. These wave-free singularities were found first by Krein [2] and later by 

 Bessho [3] independently. The latter considered an application to the practical 

 ship design problem. An important nature of the wave-free singularity is that 

 the total sum of the dipole singularities is zero. This fact can be interpreted to 

 Michell's theory that the linearized volume of the wave-free ship is zero. 

 Bessho's application of the wave-free singularity is the method of changing the 

 ship's form without any change in wave resistance. The theoretical result has 

 been proved by experiments. 



It can be easily understood that the restriction with respect to the volume 

 is one of the necessary side conditions mentioned before. However a constant 

 volume can not become a sufficient condition. The draft may become another 

 restriction, otherwise the volume can be placed infinitely downward, resulting 

 the wave resistance to be reduced to any extent. Therefore the problem of min- 

 imum wave resistance is usually considered under the conditions of constant 

 volume and constant draft. However the existence of the wave -free singularity 

 distribution invalidates the solution of the problem of this kind, because one can 

 obtain an infinite set of the solution by addition or subtraction of the wave -free 

 singularities. There is another difficulty. When the draft is fixed, or the lower 

 boundary of the singularity distribution is prescribed, the body can be reduced 

 to a fully submerged body, and Bessho [4] showed the wave resistance of sub- 

 merged singularity distribution to have no solution of minimum problem. Hence 

 the minimum problem of ships of given draft and constant displacement has no 

 solution. 



The minimum problem which has been usually considered is not such a gen- 

 eral one, but a problem to find out a longitudinal distribution of displacement 

 which makes the wave resistance minimum when the shape of the frame line is 

 given by a prescribed equation. 



Take the x-axis along the longitudinal axis of the ship, the y-axis athwart 

 ships and the z-axis draftwise downwards. Write the equation of the ship's sur- 

 face by the form like 



y - f(x,z) . (1) 



1020 



