^ = -2^ 



"(2-^ - 1) j^-|- 



(116) 



where c is cnoidal theory celerity, and n is wave surface variation based 

 on second-order cnoidal theory. The results again give less wave setdown 

 than linear theory and are closer to experiments in this regard as discussed 

 in Chapter 4 where other even closer approximations are also shown. 



Maximum wave setup, ri , is determined by Jonsson and Buhelt (1978), 

 using a series of solitary waves just outside the breaker line to calculate 

 the mean water depth at the breakers, 



h, = 0.344y" ^2(H /L )"^3h (117) 



D O O O 



When inserted into equation (37) using H, = yh, they obtained the result 



T^ = 0.107y^2 (h /l )~/3 (118) 



H o o 



o 



This expression includes setdown effects and holds for any beach profile 

 where depth decreases continuously. These results are for normal wave in- 

 cidence and have been found to explain how beach slope and wave steepness 

 influence maximum setup in experimental data (Ch. 4). 



b. Uniform Longshore Current Profile . James (1973, 1974b) was pri- 

 marily interested in how a nonlinear wave theory for the radiation stresses 

 would affect the longshore current profile. The same wave theories as re- 

 viewed for wave setup were employed. He took the full nonlinear bed fric- 

 tion expression (eq. 52) and Longuet-Higgins (1970) expression for lateral 

 mixing stress (eq. 55) in his numerical solution methods. All results are 

 for plane, flat beaches with spilling breakers and y = 0'85. A comparison 

 between the linear and nonlinear theories is shown in Figure 37 (after 

 Gourlay, 1978) where it is important to recognize that the nonlinear solu- 

 tions include both the nonlinear bed stress and wave setup effects while the 

 linear solution does not (original model, Longuet-Higgins, 1970). With this 

 in mind, it is observed that the linear theory required significantly larger 

 bottom friction values (to match experimental surf zone currents) and is more 

 sensitive to eddy viscosity variations than the nonlinear theory. 



James (1974b) concluded that the results for longshore current in the 

 surf zone are of the same order of magnitude for both linear and nonlinear 

 theories . Perhaps this is the reason his research remains the only one known 

 of this nature. All such efforts require numerical solution methods which, 

 in light of their use for two-dimensional solutions, is not an additional 

 problem. Dutch researchers are planning to vary the wave theory in the 

 radiation stress terms of their numerical model (Vreugdenhil, 1980). Their 

 future results and others of a similar nature will be of considerable interest. 



132 



