substituting equation (29) into equation (26) gives 



C 2 = it (^) (30) 



2tt V L / 



which gives small -amplitude, shallow-water wave celerity as 



C = /gd (31) 



Because z < d, it will always be true that z/L ■*■ whenever d/L -» 0. 

 Therefore, 



• u 2lTZ ^ 2ttz f „-, 



smn >- (32) 



L L 



. , 2nd 2ird czx\ 



sinn y L~>~>J 



L L 



cosh III. - i (34) 



L 



cosh ► 1 (35) 



L 



Considering that 



, 2n(z + d) , 2ttz , 2tTd . , 2ttz . , 2nd ,_,.. 



cosh ±J1S± ^L = cosh cosh + sinh sinh — — - (3d) 



L L L L L 



and substituting equations (32) to (35) into equation (36) , 



, 2n(z + d) 

 cosh — 



1 + ^ /2ju^ (37) 



The disturbed water surface elevation, n, at any point in relation to 

 its undisturbed location is given by 



n = a cos [2n (* - ^)] (38) 



Substituting equations (33) , (37) , and (38) into equation (27) , 



M 



(t 1 ) 



/2nz\ / 2ird \ 



n n V L / V L / 



u = 2tt — — (39) 



T 



Noting that z/L ■> 0, 



u = 2- ± (40) 



d T 



40 



