to obtain consistent functions k(g>*, y*) which allow us to calculate the influence of 

 all important form variations. 



4. We assume the curves of exciting force coefficients E z , Ey as known, 

 although explicit data for the diffraction effect are not yet available. For the present 

 purpose the Smith effect and the relative motion correction can be included; in principle 

 the second correction requires the use of Equation (2) . 



The functions E allow us to deal with any wave component constituting the 

 irregular seaway, hence by superposition with the total seaway. 



The curves E z , Ey show that it is possible to consider A* = 1 as a significant 

 point for determining the character of heave and pitch for two reasons : 



1. firstly, the point A* = 1 lies in the final region of large wave effects; 



2. secondly, the maximum steepness of the Ey curve occurs actually in the range 

 0.9 <; A* ^ 2 — » 2.5, although magnitude and slope strongly depend upon the shape 

 of the waterline (the coefficient a). However, quantitative data so obtained cannot be 

 widely generalized. 



The analysis of ship motions can be conducted as follows [51,5]: 



too 



We start with the relation A = — — (1 + F& *) and plot a diagram of the 



function Wo * + Fto * 2 . 



Resonance conditions are found from 



v* = co * + Fco * 2 , (24) 



yielding the critical value for Wo * or 



2tt 



\* B (F) = -. (25) 



coo* 2 



For F — 



v* = coo*, (26) 



2tT T*2 



so that X**(0) = — = . (27) 



?*2 2tt 



\*R (0) = A* B is a characteristic value for a given ship, v * = constant, indicating 

 the wave length ratio at which synchronism occurs in the hove-to condition [51]. 

 Assuming 



1.2 < T* < 2.5, or 5.2 > v* > 2.5, (28) 



one obtains 



0.23 < X** < 1. (29) 



The reasoning can be easily extended to cope with the general case. Resolving 

 (21) with respect to F one obtains the critical Froude number 



j/*A* /~\* 



F R = */ — (30) 



2ir i 2-7T 



Curves can be drafted to show the dependence of the critical Froude number F R upon 

 A* with v* as parameter. The strong variability of F R with v* is obvious. 



The curves F R (A*, v*) divide the subcritical from the supercritical range. 



To investigate the behavior of a given ship, one proceeds as follows: 



1. k(g>*) and p, can be plotted versus w * and A. 



2. Exciting functions £\ and E^ are calculated and represented in the form 

 E] y \ = E(\*) =£[«*]. 



84 



