Qualitative agreement among the curves appears to be good. The deviation 

 between the computed nonlinear force and either the linear or the second-order 

 force represents aspects of force which could not be predicted from linear or 

 second-order theory. 



Another comparison of the computed force with the force predicted by 

 second-order theory is obtained from a Fourier analysis of the curve of force 

 versus time for the last period of forced motion. According to second-order 

 theory, the total dimensional force on the body when its center (xQ,yQ) 

 oscillates vertically according to the formula y = hg sin (at) is given by 



F = 2 pg A (0.97 Tr/4) + 2 pg b^ e F^ sin (at + 6i) 

 + 2 pg b^ e^ F20 + 2 pg b^ e^ F21 sin (2at + 62) (28) 



where e = hg/b is the perturbation parameter. The first term represents the 

 hydrostatic force on the body at its neutral water position. The second term 

 represents the first-order force on the body, and the last two terms represent 

 second-order modifications to the first-order force on the body. A Fourier 

 analysis of the computed nonlinear force versus time curve will produce 

 coefficients of the Fourier expansion in the orthogonal functions 

 {1, sin (nt), cos (nt) }. The coefficients are obtained from the usual 

 integrals by numerical integration. In the case of the computed nonlinear 

 force, the center of the body (xQ,yo) oscillates vertically according to the 

 formula yg = hg cos (at) instead of y = hg sin (at). After considering this 

 difference, the nonlinear results can be used to arrive at the five parameters 

 describing the force In Equation (28). The procedure, of course, assumes that 

 the contribution to the force from third-order effects is insignificant. 

 Table 1 shows how the computed results compare with those of Lee [17], and 



21 



