16e 



Added Resistance of Ships in Waves 

 dx (1 - sin /3) Ai[x,^U (-1 + sin /S) Ai(x, ^] 



+ 2A2 (X, 77-/3) 



dz. 



It can immediately be concluded that in this approximation the longitudinal 

 force component depends on the radiation potential (P^ only; in contrast with the 

 transverse component, which also depends on the diffraction potential cp^. It is 

 consistent with the theory developed here to use for 4>^ the values which can be 

 calculated with the two-dimensional strip theory. 



In the next part of the paper we shall restrict ourselves to the longitudinal 

 force component, since this component is important for the computation of the 

 added resistance of a ship in waves. 



After substitution of the various expressions the formula for F^ reads 



cos /3 



8^2 J_ 1 



^ |P(x)| ^dx , 



(3) 



where 



P( 



-1 



-'C( X) 



ifz + (i-fz) 



k( z+iy) , 

 ? ^ ^ d2 



and a is the dimensionless amplitude of the ship motion at the point x . Let the 

 heave amplitude be given by ze^^' and the pitch angle amplitude by "^e^^^. In- 

 troducing the dimensionless quantities 



z = z/h , 4' - ^0/277h 



we obtain 



7t2 T 2 



a = z^ + — ^2.1,2 



277L 



x'^ ^p + —5; — x z cos (e 1 - £3) • 



The expression for |P(x)| ^ can be reduced to a more simple form 



-'C(x) 



The behaviour of 0^ ^t infinite distance of the two-dimensional body can be 

 written as 



0^ = Ae'^^^'^'y')"^^ . 

 Applying Green's theorem in the y- z plane leads to 



641 



