By taking the finite width of such a fixed rigid structure to an infi- 

 nitely small limit, Ursell (1947) developed a theory for the partial transmis- 

 sion and partial reflection of waves in deep water with the barrier extending 

 from the water surface to some depth, D, such that 



K 1 (2irD/L i ) 



yTr 2 l2(2TTD/L i ) + k2(2ttD/L 1 ) 2 



(12) 



where I (2iTD/L i ) and K (2ttD/L ;L ) are modified Bessel functions. Experimental 

 studies evaluated the goodness of fit of this theory; these data are shown in 

 Figure 3. Wiegel (1959a) investigated this conceptual model with a considera- 

 tion of the wave power transmission (the time rate of energy propagation), and 

 he determined that the ratio of the transmitted wave power, P fc , to the 

 incident wave power, P., is 



4tt(D + d)/L n . sinh 4it(D + d)/L n . 



sinh 4ird/L- sinh 4ir/L- 



1 i (13) 



4ird/L n - 

 1 + 



sinh 4ird/L-£ 



The transmission coefficient, C t , is the square root of ? t /?^. Experiments 

 performed by Wiegel (1959a) (Fig. 4) demonstrated that this theory might be 

 useful for first approximations but that improvements in the theory were 

 needed. A consistent trend of decreasing transmission coefficient with 

 increasing wave steepness was evident from these laboratory measurements (all 

 other things remaining constant). The power transmission theory predicts the 

 transmission coefficient more closely for the deepwater wave (d/L = 0.68) than 

 for the other conditions tested (Wiegel, 1960). While these early attempts to 

 describe the physical process are simple in theory, they offer initial first 

 approximations and provide a foundation on which more elaborate numerical 

 techniques can be used. 



2. Dynamics of Elastically Moored Floating Breakwaters . 



The response behavior of a single degree of freedom, linearly damped, 

 vibrating spring-mass system is analogous to the response of a floating 

 structure to waves (Harleman and Shapiro, 1958; Dean and Harleman, 1966; 

 Sorensen, 1978). When a linear system of one degree of freedom is excited, 

 its response will depend on the type of excitation and the damping which is 

 present. The equation of motion of the system (Fig. 5) , developed from 

 Newton's second law of motion, will generally be of the form 



9 2 x 



+ F . + kx = F(t) (14) 



3t 



where 



m = mass of the structure 



k = restoring stiffness coefficient 



x = direction of motion 



F(t) = excitation as a function of time 



F, = damping force 



33 



