n = 0.45, nk £ = 0.2, 0.4, 0.6, and 0.8, and f = 5, one possible choice 

 of S is to take it equal to unity, i.e., using the graphs corresponding 

 to n//s = 0.45 with nk I and £/S = f to obtain values of R and T. An 

 extreme alternate choice would be to a ssume S = 1.67, i.e., using the 

 graphs prepared for n/v^ = 0.45/ A. 67 = 0.35 with the values of nk^i 

 and f/S = f/1.67 = 3.0. It was found this way that the estimates of R 

 and T varied at most by 0.01 with the above choices of S. This may be 

 taken as an indication of the insignificant importance of the value 

 assigned to the coefficient S. 



Thus, it is concluded that the value assigned to the coefficient S 

 is of little consequence and that we may safely take S = 1.0. However, 

 this result may be utilized to simplify the presentation of results. 

 Thus_, rather than presenting a series of graphs for different values of 

 n//S, one set of graphs, for ejcample corresponding to n/Zs^ = 0.45, 

 suffices. The factor S^ is without physical significance and is deter- 

 mined by requiring that the value of n/v^^ = 0.45 for a given structure 

 for which n, the porosity, is known. Thus, if n is known the value of 

 S^ is obtained from 



2 



S = r " 1 (28) 



^* ^0.45^ ' 



and Figures 2 and 3 may be used with nk I and f/S^ to obtain estimates 

 of R and T. 



a. Simplified Solution for Structures of Small Width . For many 

 breakwaters the width, I, is of the same order of magnitude as the 

 depth of water, h^. Thus, for relatively long incident waves, k^^ 

 and consequently k^Jl may be assumed to be small. Thus, with the 

 assumption of k^jj, << 1, the general formulas for the reflection and 

 transmission coefficients given in the previous section may be simplified 

 considerably. 



The nature of the simplification is expressed by expanding the 

 exponentials in terms of their Taylor series, i.e., 



e^^^^ = 1 +_ ik£ + 0(k£)^ (29) 



2 

 and adopting 0(k£) as the degree of accuracy of the simplified expres- 



Introducing the expansion given by equation (29) in equation (22) 

 yields : 



24 



