and from Table C-2 



/27rd\ 

 cosh t 1 = 1.2040 , 



sinh I 1 = 0.6705 , 



/47rd\ 

 cosh = 1.8991 , 



L 



ttH^ cosh(2rrd/L) 



2 + cosh 



■ 47rd\ 



= 0.48 



8L sinh' (27rd/L) 



Therefore 



7? = 2 COS0 + 0.48 cos 2d , 



Tj, 2 = 2.48 ft., 



r?^ 2 = ~ 1-52 ft. 



where n^ 2 ^"'^ ^t 2 ^^^ ^^^ values of n at the crest 

 (i.e. cos 9=1, cos 26 = 1) and trough (i.e. cos 6 = - 1, 

 cos 26 = 1) according to second-order theory. 



Figure 2-8 shows the surface profile n as a function of 6. The 

 second-order profile is more peaked at the crest and flatter at the 

 trough than the first-order profile. The height of the crest above 

 SWL is greater than one-half the wave height; consequently the 

 distance below the SWL of the trough is less than one-half the 

 height. Moreover, for linear theory, the elevation of the water 

 surface above the SWL is equal to the elevation below the SWL; 

 however, for second-order theory there is more height above SWL 

 than below. 



(b) For convenience, let 



u^ 7 = value of u at crest according to first-order theory, 



^t 1 ~ v^l"® °^ " at a trough according to first-order theory, 



u„ 2 ~ value of u at a crest according to second-order theory, 



u+ 9 = value of u at a trough according to second-order theory. 



2-41 



