Yim 



Fig. 2 - Schematic diagram for a surface 

 ship and the coordinate system 



with a given vertical distribution of volume. In their solution, they all found 

 either some singularities in the functions representing hull shapes at the ends 

 of ships, or bulblike forms around the bows and the sterns. Wehausen, Webster, 

 and Lin (1962) treated the optimum forebodies of ships with a given afterbody as 

 well as three-dimensional symmetric ships without any restriction on the verti- 

 cal distribution of volume. However they took the ship surface area into account 

 to minimize the wave and friction resistance, and they too found big bulblike 

 forms near the bottom of bows for higher Froude numbers. 



Havelock's wave resistance formula (1934) from the regular waves due to 

 the singularity distribution on the center plane of a ship is essentially the same 

 as Michell's, as long as the linear relation of the ship hull form with the singu- 

 larity distribution 



, s V df , , 



m(x, z) = -r— -5— (X, z) 



27T dx ^ ' 



(1) 



is used, where m(x, z) is the source strength and f(x, z) is the ship hull form. 



Inui (1957) calculated an exact hull form (body streamlines of a double 

 model) from a given source distribution for zero Froude number (flat free sur- 

 face), and he used this hull form for his model experiment to test waves and the 

 wave resistance. He compared his experimental results with his calculated 

 wave heights and the wave resistance due to the source distributions. He found 

 that the calculation agrees better with his experiment on his model than the 

 corresponding Michell's model satisfying (1). The way Karp, Lurye, and Kotik 

 (1958) interpreted their result to a ship form of infinite draft is similar to the 

 idea of Inui's which we have just described. The singular behavior of Michell's 

 ship hull can be easily treated by reinterpreting Michell's ship hull as the dis- 

 tribution of various singularities like sources or doublets either distributed or 

 concentrated. 



1068 



