Ship Form of Minimum Wave Resistance 



It has been shown that the best form does not exist under a single side con- 

 dition of constant volume unless the elementary ship with prescribed vertical 

 distribution is assumed. However Krein pointed out that a solution could exist 

 if another side condition such as a fixed area of the wetted surface would be 

 added. From the mathematical point of view, the solution under this dual condi- 

 tion is equivalent to the ship form for which the sum of the wave resistance and 

 the skin friction becomes minimum. Lin, Webster and Wehausen [12] computed 

 the ship form of minimum total resistance, which was assumed as the sum of 

 Michell's integral and the frictional resistance according to Schoenherr's mean 

 line. Their results are quite plausible except undulating lines which seem to be 

 a consequence of an improper choice of the series used for the expansion of the 

 solution. 



According to Froude's hypothesis, the frictional resistance of a ship is 

 equivalent to the frictional resistance of a flat plate of same length and same 

 area. However the frictional resistance of a curved surface is an integration of 

 the longitudinal component of the tangential stress. If the local frictional coef- 

 ficient Cf at a point where the normal to the surface makes an angle a to the 

 longitudinal axis, x say, the total friction is given by 



J pU^Jj C[ sin adS . (28) 



When the surface is expressed by an equation, y = f(x, z) , one can put 



Bx 



Therefore the frictional resistance becomes 



R, = pU^JJc; 1^7(17 dxdz. (29) 



Taking the mean value of the local friction, one may write 



R, = PU^C Jf,^T(|f dxd. (30) 



where c^ is regarded as the frictional resistance coefficient of the ship. The 



area 



^l^^^' 



S„ = 2 I I VI + (^1 dxdz (^^^ 



1027 



