wave components, CT, and O^ , is represented by the sum of the re- 

 sponse due to O", alone, which we laay call Xj" f and the re- 

 sponse due to g-j^ alone, X';'*' • However, as pointed out by 

 Wehausen (1971) it is only in the special case that Mij (-07) = /Mij(0"i) 

 and Ntj (-^1 ) ■= Mii(0"i.) that this could in fact be the case since 

 it is only then that ^xj'** V- ^j'^ ) could represent a solution 

 to an equation of the form of (1) . In spite of this, such equa- 

 tions have been used with some success to describe the motion of 

 a vessel in random waves (see, e.g. Fuchs and Mac Camy (1953), 

 Fuchs (1954), St. Denis and Pierson (1953) in which the values of 

 /V ■ ■ and A7<-j were taken as constant at some average value. It 

 appears, however, that it is currently common practice to utilize 

 the superposition discussed above regardless of the difficulty 

 associated with frequency dependent coefficients of mass and damp- 

 ing. Wehausen (1971) has discussed a further method of treating 

 the linearized motion of floating bodies in random seas when the 

 added mass and damping coefficients are frequency dependent as, 

 of course, is the case for large-displacement floating bodies. 

 The procedure outlined is based on the initial-value problem 

 approach . 



3.1 Nonlinear Effects 



The theory of ship motions is based on linear hydrodynamics. 

 This means that the amplitude of waves and the response to these 

 waves is considered to be small and, therefore, all terms aris- 

 ing in the analysis which are proportional to such amplitudes to 

 the second-power and above are neglected. 



If one considers two wave components of frequency 0~, and CTt 

 for instance, and retains terms through the second-order in the 

 small amplitude parameter mentioned above, certain "second-order" 

 forces and moments arise. These forces are oscillatory and com- 

 ponents at frequency Co", +0"i.^ as well as at CCT,-0\) arise. The 

 force occurring at frequency (CT, ^ <T^) is of considerably smaller 

 magnitude than the first-order wave-induced loads and occurs with- 

 in or above the frequency range of the first-order forces. Thus, 

 these small amplitude, high-frequency forces are of little con- 

 sequence and are neglected. 



The force (or moment) component at frequency (oj- CTl) falls 

 below the first-order excitation force frequency range, and al- 

 though the forces are small compared to the first-order forces, 

 these second-order forces can have significant effects due to 

 the fact that the resonance frequency of the mooring system may 

 lie near these frequencies. 



Newman (1974) has developed an approximate method for eval- 

 uating the slowly-varying second-order drift-forces in random 

 waves which is based solely on the mean drift-force. The mean 

 drift-force is simply the diagonal terms in a square matrix of 

 coef f ic ients desc r ibing the low-frequency force components associa- 

 ted with all combinations of the components associated with a 

 wave spectrum. Newman's approximation renders the evaluation of 

 slowly-varying second-order forces on floating bodies within the 

 realm of the computationally feasible for practical floating body 

 situations. Without the use of this approximation the computa- 

 tional chore would be formidable for three-dimensional bodies. 



4. EVALUATION OF THE MEAN DRIFT-LOADS 



There are two methods for evaluation of mean drift forces 



50 



