where n_ o ^"*^ "^t 2 ^^^ ^^^ values of n at the crest (i.e., cos 9 = 1, 

 cos 29 = i) and trougli (i.e., cos 9 = -1, cos 29 = 1) according to second- 

 order theory. 



Figure 2-8 shows the surface profile n as a function of 9. 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 eleva- 

 tion 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„ 1 = value of u at crest according to first-order theory, 

 c , i 



Uj^ , = value of u at trough according to first-order theory, 



u - = value of u at a crest according to second-order theory, 



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



According to first-order theory, a crest occurs at z = H/2, cos 9=1 and a 

 trough at z = -H/2, cos 9 = -1. Equation (2-13) therefore implies 



with 



and 



with 



According to second-order theory, a crest occurs at z = n^, 2 ~ 0.602 meter 

 (2.48 feet), cos 9 = cos 29 = 1 and a trough at z = n^ 2* "^ -0.398 meter 

 (1.52 feet), cos 9 = -1, cos 29 = 1. Equation (2-51) there'fore implies 



HgT cosh[2Tr(z + d)/L] 



c,2 2L cosh(2Trd/L) 



3 /TfH\2 cosh[4Tr(z + d)/L] 



4 \ L/ sinh'*(2iTd/L) 



2-39 



