Kaplan 



Zi = - \ A33 dl . z + \ A^^i di . e 



(4) 



where ^j> and t^ are the bow and stern ^-coordinates respectively. 



In a similar manner, the lateral force (along y-direction) may 

 also be expressed by use of this same procedure, but certain addi- 

 tional factors enter in that case. These factors are the necessity 

 of including roll effects which influence the lateral velocity, cuid 

 also the fact that the representation of the lateral force is based upon 

 added mass terms that are evsiluated for motions relative to the free 

 surface level, rather than the body center of gravity position. Cor- 

 rections to refer the final forces to the center of gravity position are 

 made after finding the forces referred to the free-surface position. 

 The detailed procedures for determining these inertial force (and 

 moment) results, as well as all other forces of hydrodynamic , 

 hydrostatic, etc. nature are described in [ 6] , which is the basic 

 report on which the present section of this paper is based. In view 

 of this, only limited discussion of the remaining forces and moments 

 will be presented. 



The damping forces and moments are dissipative in nature, and 

 are primarily due to the generation of waves by the ship motions on 

 the surface, which continually transfer energy by propagating outward 

 to infinity. In accordance with the two-dimensional treatment used 

 for the analysis of inertial forces due to body motions, the same 

 concept is used in evaluating the local forces at a section of the ship 

 due to wave generation. With the ratio of the amplitude of the heave- 

 generated two-dimensional waves to the amplitude of heaving motion 

 of the ship section denoted by A^ , the vertical damping force per 

 unit vertical velocity of the ship section is expressed as 



N- =-eg!A^=pa.(^4f Afe (5) 





where "A^ ^^x Lewis-form sections are available as a function of 

 a)^B*/2g = ttB /X., for different beam-draft ratios and section coef- 

 ficients, where B* is the local beam and X the wave length. 



The vertical damping force at each section is 



^ = - n;,(z -ee), (6) 



and this is integrated over the ship length to determine the total 

 vertical damping force, given by 



1024 



