of the effort was the wave field determinations for both refraction and dif- 

 fraction (Liu and Mei, 1976a). Wave-current interactions, convective accelera- 

 tions, and lateral mixing stresses were neglected, as were wind and flooding 

 effects. However, the governing equations remained nonlinear in the surf 

 zone since the solution permits a large wave setup relative to the Stillwater 

 depth. Bottom shear stresses are calculated from the weak current small- 

 angle model. 



The solution was facilitated by use of the transport stream function and 

 a coordinate transformation to get finer resolution nearshore. The finite- 

 differencing techniques employed were equivalent to a Gauss-Seidel relaxation 

 technique. Iteration procedures are needed to first calculate the stream 

 functions, then n and a new shoreline, and then to repeat this process to 

 convergence of a specified amount. Example results for one test case of the 

 offshore breakwater are shown in Figure 34. Besides needed improvements in 

 the many empirical approximations, it was also concluded that future efforts 

 must include the convective accelerations, lateral turbulent mixing stresses, 

 and wave-current interactions. Future improvement would also come from non- 

 linear refraction and diffraction models, inclusion of breakwater energy ab- 

 sorption properties, and location of the breaker line. Controlled laboratory 

 experiments were requested to confirm the theory since incomplete experimental 

 data (such as Gourlay, 1978) were then unavailable. 



Mei and Angelides (1977) applied the model to study currents on a beach 

 of constant slope around a circular island. Liu and Lennon (1978) at Cornell 

 University employed the finite-element (weighted residual) method to essentially 

 model the same set of simplified equations with neglected terms as discussed 

 above. In addition, the equations were linearized in the alongshore direction 

 by assuming n small compared to local water depth in this direction. No itera- 

 tion procedure was consequently required. The finite-element method was 

 claimed to be "...more efficient and powerful than the existing finite-dif- 

 ference models." This statement is not substantiated, however. 



c. University of Delaware . The shortcomings of the model developed by 

 Noda, et al. (1974) have been addressed in a series of developments by 

 Birkemeier and Dalrymple (1975, 1976) and Ebersole and Dalrymple (1979, 1980). 

 The 1976 model included a different wave breaking ratio y, wave setup effects, 

 the Longuet-Higgins (1970) bed-shear model, coastline flooding, and wind- 

 shear shears. The final motion equations employed were in Eulerian form and 

 included the unsteadiness but neglected the convective accelerations and 

 lateral stress components. A staggered, explicit finite-difference schere 

 was devised to solve the system of equations. Wave height fields were found 

 using the refraction program of Noda, et al. (19 74) modified to include cal- 

 culation of the time dependency by a finite-difference routine. This, in turn, 

 gave the radiation stresses from the linear wave theory for all depths. The 

 major difficulty was the large number of time steps dictated by the stability 

 requirements for two-dimensional expllcity schemes. No accuracy analysis of 

 the finite-difference method utilized is presented. Only a limited number of 

 tests were performed. Most were for steady-state conditions so that the local 

 acceleration terms became iteration steps in the steady-state solutions. A 

 one-dimensional unsteady wave setup computation demonstrated how setup lagged 

 the deepwater wave height due to traveltime of the wave toward shore. 



125 



