integrated over tlie surface has a component in phase with acceleration 

 and a component in phase with velocity, The component in phase with 

 acceleration is normally referred to as the added mass, while the compo- 

 nent in phase with velocity is the damping. 



The hydrodynamic coefficients shown in this section are derived in 

 greater detail in Appendix C. 



c. Mooring Forces. At the time the spring constants for the mooring 



lines are computed, mooring force coefficients are also calculated. These 



are : 



AF 



— = change in mooring line force per unit displacement in 



°'i sway, heave, or roll when i = 1, 2, or 3, respectively. 



The forces in the mooring lines may then be computed once the motions 

 have been found. 



A F 

 Mooring Force = E (= — ) a ■ (t) 



i=l 



Aa . 



The description of the linear system is now complete. The block diagram 

 in Figure 3 shows the relationships among the calculations which are 

 required. 



2. Nonlinear Theoretical Model . 



Measurements taken at the Tenakee, Alaska, floating breakwater be- 

 fore this research program was begun indicated the presence of a long- 

 period oscillatory motion of the breakwater. These long-period motions 

 were manifested most clearly in the measured mooring line forces. Look- 

 ing at these, one can visually observe an oscillation with a period of 

 about 60 seconds superimposed over the expected shorter period oscilla- 

 tions. Figure 4 shows the results of a spectral analysis of the seaward 

 mooring line data after a low-pass filter has been applied (the tech- 

 nique for performing the spectral analysis is given in Section III of 

 this report) . 



The linear theoretical model permits the system to respond only at 

 the frequency of the incident wave. In order to explain the presence of 

 these long-period oscillations, nonlinearities must be included in the 

 analysis. To perform a mathematically complete analysis including all 

 nonlinear effects is beyond the present state of the art. However, in 

 the case of the floating breakwater, one can show that if two incident 

 waves are considered and second-order terms are retained, then an excit- 

 ing force is present at the difference between the frequencies of the 

 incident waves. The complete derivation in Appendix e shows that the 

 nonlinear pressure may be expressed as: 



19 



