X = 



v/(k - ma) 2 ) 2 + (coo) 2 



and 



k - mar 



(19) 



(20) 



Expressing equations (19) and (20) in dimensionless form enables a concise 

 graphical presentation. Defining the following quantities 



[k 

 a) = /— = natural frequency of undamped oscillation 

 V m 



c = 2mw = critical damping 



c 

 £ = — = damping factor 



and 



CO) C CO) CO 



k c„ k ul 



the dimensionless expression for the amplitude becomes 



Xk 



F.. 



1 



^Ht)T + h(t) 



(21) 



and the dimensionless phase angle is 



*(t) 



tan 9 = 2 — 



"(*)' 



(22) 



Equations (21) and (22), shown in Figure 6, indicate that the amplifica- 

 tion factor, Xk/F Q , and phase shift, 6, are functions only of the fre- 

 quency ratio, oj/u n , and damping factor, c / c c » The damping factor, c/c , 

 has a large influence on the amplitude and phase angle in the frequency region 

 near resonance (w/u> n = 1). For small frequency ratio (oj/oj < 0.2), the 

 response of the system is essential equal to the excitation force. However, 

 waves with periods at or near the resonance period of the structure (floating 

 breakwater) will generate an increase in the amplification factor (the exact 

 amount depending on the excitation period and degree of hydrodynamic damping). 

 Hence, the movement of a floating breakwater can be several times greater than 

 the incident wave amplitude. If the incident wave period is close to, but 

 less than, the resonant period of the structure, there will be a phase lag 

 between the wave and the structure motion that can approach 180°. This will 

 create a phase lag between the incident and regenerated wave motions which 

 increases the complexity of both the reflected and transmitted wave 

 characteristics. 



36 



