I-Min Yang 



My fourth comment is that the equation of motion reveals an 

 excitation that consists of a single component related to wave displa- 

 cement. This is correct only so long as the excitation is linear (in 

 which case its amplitude is frequency dependent), but not when the 

 excitation is non-linear (in which case its amplitude is a function of 

 both frequency and amplitude of motion). 



My fifth comment relates to the conclusion that the accuracy 

 of the approximate approach is well within the limits of practical en- 

 gineering usefulness. But comparison is against another computa- 

 tional method and not against measured reality, and such a compa- 

 rison leads to an appreciation of the validity of the approximation, 

 not of the basic method. 



The paper raises a large number of stimulating questions but 

 it provides a paucity of answers to them, and my recommendation is 

 quite simple. I suggest that the author complete his paper and add 

 thereto whatever experimental or trial data he can adduce in support 

 of the technique described. Only then will it be possible to appreciate 

 the paper and to comment constructively on its intrinsic merits, 

 which at present stand unrevealed. 



REPLY TO DISCUSSION 



I-Min Yang 



Tetra Tech, Inc. 

 Pasadena^ California^ U.S.A. 



From the comments made by the discusser, it seems to me 

 that he has some misunderstanding about the definition of nonlinear 

 differential equations. Hence he can not find any originality in this 

 paper. In equation (l), the independent variable is the time t , and 

 the dependent variables are the components of the displacement vec- 

 tor x. Since it is assumed that f(x) is a nonlinear vector function 

 of x , the differential equation (l) is therefore nonlinear. The fre- 

 quency a) in this equation is just a parameter, that is, for a particu- 

 lar case, it is a constant. The hydrodynamic coefficients (virtual 

 masses and damping coefficients), and amplitudes and phase angles 

 of wave forces depend on co and the dock in a very complicated man- 

 ner, but for a fixed go , they are just constants. Thus they have no- 

 thing to do with the nonlinearity of differential equations. (For 



680 



