and 



/ 2 



■ik. = k = k / 1+f, (cos<|) -i sincj) ) . (A-44) 



Introducing these expressions in equation (A-41) and performing 

 the spatial average over an area of unit width and extending from 

 X = to X = £, the following equation is obtained: 



. . -3k.il 



4/ ^ . k a „ ^ 1 



J- / -, j~ 2 4j, 0+ r2 1-e -, ., ,^^ 



f, / 1+f, = ^- f ^ {^ :5T — 7-} . (A-45) 



b b 3Tr w -, , . 2 3 -2k. £ *- ^ 



(k h ) , 1 



o o 1-e 



Although not leading to an explicit equation for the friction 

 factor, f}^, this expression may be solved iteratively from knowledge 

 of f^^ and the wave characteristics. The bracketed term in equation 

 (A-45) arises from the spatial averaging process and becomes unity if 

 k.Jl << 1. Thus, in the immediate vicinity of x = we have 



Ar TT k a 



f, /l+f, = T- f ° ^ -, , (A-46) 



b b 3Tr w ,, , ,2 

 (k h ) 

 o 



which may be shown to lead to the same rate of amplitude attenuation 

 as the formula suggested by Putnam and Johnson (1949). 



c. Limitation of the Solution . To discuss the limitations of the 

 solutions obtained from the governing equations derived in the preceding 

 sections we consider the solution obtained for a progressive wave in 

 constant water depth, hQ, without bottom friction, i.e., f^^ = f, - <(>,= 0. 

 For this simple case we have from equations (A-37) and (A-42) , 



n = a cos (k x -wt) 

 + + ^ o 



U = T— v/ilT cos(k X -tot) . (A-47) 



+ h ^ o ^ 

 o 



The basic assumption made in the derivation of the equations 

 leading to this solution was that the vertical accelerations were 

 negligible. We may now reexamine this assumption by obtaining the 

 expression for the vertical velocity, W , from equation (A-1), 



W = a 0) (1 + r-^) sin(k X -tot) , (A-48) 







and the leading term arising from the vertical fluid acceleration in 

 equation (A-3) is 



13 



