errors), and boundary geometry for the proper simulation of flow vorticity, 

 circulations, eddies, etc. in numerical models are still an active research 

 area. The modeling effort described by Vreugdenhil (1980) is by far the best 

 effort to model nature because it includes all the physically important terms 

 and simulates them numerically with the most accuracy. 



VI. NONLINEAR AND IRREGULAR WAVES 



All the analytic and numerical methods described previously in this 

 chapter were with radiation stresses computed from linear wave theory for 

 regular sinusoidal waves. The stress components are then simply given by 

 equations (23), (24), and (26) or first-order theory. Because of the rela- 

 tively crude assumptions needed for modeling the surf zone energy dissipa- 

 tion, it could be argued that higher order radiation stress terms were not 

 warranted. On the other hand, since irregular waves naturally occur and 

 break at different offshore locations, the stochastic approach to longshore 

 current modeling may be more realistic. Both nonlinear and irregular wave 

 theories of MWL change and longshore currents are reviewed in this section. 

 Differences and similarities with the linear regular wave theories are noted. 



1. Nonlinear Waves . 



a. MWL Change . James (1973, 1974a) used third-order Stokes theory in 

 deep water and a modified (Iwagaki, 1968) '*-^ cnoidal wave theory nearshore to 

 compute the higher order radiation stress needed to define wave setdown and 

 setup for spilling breakers on plane, gentle slopes. Wave setdown was less 

 than that found by linear theory especially near the breaking point. Theo- 

 retical wave setup is'also less and the gradient is not a constant proportion 

 of the beach slope as in linear theory. Numerical methods are employed to 

 integrate the resulting ordinary differential equations. 



The theory for cnoidal waves over a gently sloping bottom (Svendsen, 

 1974)^^ is used by Svendsen and Hansen (1976) to derive analytic expressions 

 for wave setdown. Near breaking both nonlinearity and vertical acceleration 

 effects must be included in the wave theory. Using the actual bottom velocity 

 in cnoidal waves given by 



Jl 



^.,aa-^).icd^ (115) 



dX 



setdown became 



'*^ IWAGAKI, Y., "Hyperbolic Waves and Their Shoaling," Coastal Engineeving 

 in Japan^ Vol. II, 1968, pp. 1-12 (not in bibliography). 



'*^ SVENDSEN, I. A., "Cnoidal Waves Over a Gently Sloping Bottom," ISVA, 

 Series Paper No. 6, Technical University of Denmark, Lyngby, 1974, i 

 (not in bibliography) . 



131 



