PROBLEMS RELATING TO THE 



SHIP FORM OF MINIMUM 



WAVE RESISTANCE 



Hajime Maruo 



National University of Yokohama 



Yokohama, Japan 



INTRODUCTION 



The problem, to find a ship form which presents minimum resistance under 

 a certain condition imposed by the practical requirement, is one of the aims of 

 the ship designer. Experiments of methodical series of ship models have been 

 considered as the most reliable method to find the best form. On the other hand 

 recent developments in ship hydrodynamics urges the mathematical analysis of 

 components of ship resistance, and attempts were made to find out the ship form 

 of minimum resistance by means of the hydrodynamic theory. When the resist- 

 ance of a ship is separated into a component due to viscosity and that due to 

 wave-making, both of them have some correlation with the ship's form. As the 

 effect of the form upon the viscous resistance is not only the effect on the wetted 

 area but gives much influence to the boundary layer separation, our knowledge 

 is not enough to make a full analysis of the relationship between the viscous re- 

 sistance and the ship form. On the other hand, an analytical representation of 

 the wave resistance is made possible by virtue of the assumption of the inviscid 

 fluid and the technique of linearization of the fluid motion. The wave resistance 

 of a thin ship is given by celebrated Michell's integral. The problem of mini- 

 mizing Michell's integral has been a stimulating interest of theorists since 

 Weinblum first published his calculation in 1930 [1]. When the wave resistance 

 is represented by a functional of a function which gives the equation of the ship's 

 surface, to minimize the wave resistance becomes a purely mathematical prob- 

 lem, that is the calculus of variations. The method of solution employed by 

 Weinblum and his successors is a sort of approximation usually called Ritz's 

 method. It assumes a type of solution involving some unknown coefficients which 

 are determined by the condition of maxima or minima. It gives a reasonable 

 approximation provided the problem has a solution of the specified type. Doubts 

 were thrown with respect to the existence of the solution. The ship form of 

 minimum wave resistance or, exactly speaking, minimum Michell's integral, 

 can be expressed by a solution of an integral equation. Recently mathematical 

 investigations were made into the nature of the integral equation and the contro- 

 versy with respect to the existence of the solution seems to be nearly settled. 



Because of the fact that the numerical results for the above problem were 

 limited, attempts to apply the theory of minimum wave resistance to the practi- 

 cal ship design are quite scarce. However some shipbuilders have begun to 



1019 



