Wave setup reached a steady-state (non-oscillating) value after approximately 

 7-8 min (Figure 12), the velocity in the offshore direction reached a steady 

 zero value (velocity should be exactly zero) after approximately 7-8 min 

 (Figure 13), and the longshore current reached a steady value after 11-12 min 

 (Figure 14). These tests demonstrated how well the radiation boundary condi- 

 tion worked. 

 Model tests 



44. The wave-induced current and setup model was tested by considering 

 plane beach problems where either laboratory data or analytic solutions were 

 available. The first test considered wave setup and was run for a case of 

 normal wave incidence on a plane smooth laboratory beach. The laboratory 

 tests were conducted by Bowen, et al. (1968). Waves with a period of 1.14 sec 

 and deepwater wave height of 6.45 cm were propagated on a 1 on 12 beach slope. 

 The numerical model used a 50 by 3 variable rectangular grid which represented 

 the entire 40-m-long laboratory flume. Walls were used for the lateral 

 boundaries as well as the offshore boundary in order to correspond to the 

 laboratory situation. The solution allowed for the effect of setup on the 

 wave heights in the surf zone. The solution was an iterative process since 

 the setup depends upon wave heights which in turn depend upon setup. As the 

 solution proceeded, the wave heights for cells in the surf zone were computed 

 for each time-step by using H = y(h + r\) , where n is the setup and y is 

 a breaking index. The radiation stresses were changed accordingly. As sug- 

 gested by Bowen, et al. (1968), a y of 1.15 was used. A spin-up time of 

 lOAt was used. 



45. A comparison of the steady-state setup values from the numerical 

 model (after 150 At) with those observed by Bowen, et al. (1968) is shown 

 in Figure 15. There is excellent agreement in the offshore region. In the 

 surf zone, the numerical model predicts higher setups than observed. This is 

 not surprising since the numerical model did not allow flooding and runup. 

 The slope of the mean water line in the surf zone is very similar in the 

 laboratory measurements and numerical model results (Figure 15). 



46. A second test case considered was that of waves approaching a plane 

 beach at an angle. The deepwater angle of approach 9 was 20 deg, the deep- 

 water wave height was 10 ft, and the period was 12 sec. A plane beach with a 

 constant bottom slope of 1 on 30 was used. A drag coefficient c of 0.01 and 

 a breaking index y of 0.82 were used. Lateral mixing and advection were 



36 



