o 



-~> 



a 4 



u 

 o 



tJ 3 



B. 











1 1 



^c/c c =o 





















Jc/c 



\ 1 



; = 0.I25 































c 

 o 







/« j\H~ c/c c=0.20 









<a 







JJJ ' \\l 1 









•H 







^| 1 V\^C/C C =0.50 









•H 





^H-J ,% 



1 











1 







V 















c/c c 



0-. 



c/c 



c = c 



125 



' < c7c c = Q20 



J-~r- 





















< C/C C =6.5 

 '"NC/(V=I 













































































r&* 



^ 



V 





















Frequency Rotio , w/u 



cy Roll 



b. 



, «/«„ 



Figure 6. Amplification factor and phase shift for single degree of 

 freedom, forced harmonic oscillation system with various 

 linear damping (after Harleman and Shapiro, 1958). 



The term rigidly restrained indicates that a structure under consideration 

 is moored in such a way that the natural frequency of the structure-mooring 

 system is high with respect to the forcing frequency (the frequency ratio, 

 w/u) n , is small). The problems associated with mooring a floating breakwater 

 on a water surface that fluctuates greatly with season is analogous to the 

 difficulties involved with mooring a small boat on a body of water which 

 experiences a large tidal range. Raichlen (1968) provided insight into the 

 boat moaring problem when he analyzed the surging motion of several classes of 

 small boats (20 to 40 feet long) subjected to uniform periodic standing waves 

 with crests normal to the longitudinal axis of the moored boats, with two 

 bowlines and two stern lines. The free oscillation surge periods for three 

 mooring line conditions (0, 4, and 8 inches slack) for different initial 

 displacements were measured and compared quite well with theory (Fig. 7). All 

 other things being equal, the forcing function is directly proportional to the 

 wave amplitude, and it is evident that either a boat or floating breakwater 

 moored with slack lines at one tide stage may have taut lines at another; 

 hence, its response will vary with stage. Raichlen (1968) determined that 

 some smaller boats have larger natural periods of response than larger boats 

 because the larger boats have stiffer mooring lines compared with their 

 weight. 



3. Theoretical Model Developments . 



a. Model by Adee, Martin, Richey, and Christensen . The fundamental equa- 

 tions of motion have been utilized by Adee (1974, 1975a, 1975b, 1976a, 1976b), 

 Adee and Martin (1974), and Adee, Richey, and Christensen (1976) to theoreti- 

 cally predict the complex performance of a floating breakwater. The results 

 of these investigations are such that, in order to perform the calculated 

 estimations, the user need only know the incident wave frequencies, the 

 contour of the breakwater cross section, and the physical properties of the 

 breakwater (mass, moment of inertia, and restoring-f orce coefficients). The 

 prediction model was developed from two-dimensional, linearized solutions of 

 the hydrodynamical equations formulated in terms of a boundary value problem 

 for the velocity potential. This theoretical model for predicting the dynamic 

 behavior characteristics of a floating breakwater estimates (a) total trans- 

 mitted and reflected waves and their components, (b) wave forces on the 



37 



