and it follows that 



e = "2^(1 - e-K^)coswt [51] 



2A cos oit I 1 + e"^^ _ 1 - e"^^ "| j-^^l 



^ " " H ,^ .2 ^1 2KH " (KH)2 J ^ 



jt r 1 + 



r le ^ T L ^ 



?Ap"^^ r 



t, = =^-^ ( 1 - KH) sin ojt 



(1 -KH)2 J'^ --/RX2 ..KKl2 L 



[53] 



2 Vh/ J 



Plots of the above amplitudes and the heave phase angle are shown 

 in Figures 2 to 6 as functions of KH. Figure 2 shows the ratio of surge 

 amplitude to wave annplitude. For zero frequency this ratio is one and 

 for increasing frequencies it decreases monotonically to zero. Figure 3 

 shows the ratio of pitch angle to the maximum wave slope KA, multiplied 

 by the coefficient C = |^ + 6(ky/H) . This coefficient is equal to one if the 

 mass in the cylinder is uniformly distributed throughout its submerged 

 length. The ratio starts at one for zero frequency and decreases mono- 

 tonically to zero. Thus the pitch amplitude is always less than the wave 

 slope. Figure 4 shows the ratio of heave amplitude to wave height for 

 frequencies away from the vicinity of resonance. Near resonance, the 

 amplitude is shown in Figure 5 and the phase angle in Figure 6 for the 

 particular case R/H = 0. 1. The ratio of heave amplitude to wave ampli- 

 tude is unity for zero frequency, rises to a maximum of 



fdf— (0 



at the resonance frequency KH = 1, and then decreases monotonically to 

 zero. The phase angle is similar to conventional one-degree-of-freedom 



19 



