Table 4. Here again, equation (89) is employed to find H outside the breaker 

 line. With these formulations analytic solutions of the basis momentum 

 balance equations (28) and (42) are still possible by the decoupling concepts 

 employed previously. However, it must be emphasized that only normal wave 

 incidence effects on wave setup are incorporated in their procedure. This is 

 done for mathematical convenience and produces different results than ob- 

 tained by Liu and Dalrjnnple (1978) when lateral mixing is neglected. 



(1) No Lateral Mixing Stress . Setting equation (74) equal to 

 equation (88), taking a modified reference velocity v, from equation (72) 

 with cosa, assumed near unity, Kraus and Sasaki (1979) obtained the dimension- 

 less longshore current velocity as 



7r Y \, sin^a, , 



V = ^ ^ [ (l-sinV X) ' T-^ X (l-sin2a, X) "] (93) 



^b (l+sin^a^X) ^ ^ ^ 



b 



within the surf zone 0< (X = •^)<1. Outside, V = for X> 1. Results 



from equation (90) are shown in Figure 28. As %-^o (Note that a^ = o means 

 theory reduces to original order (zero order) solution given by Longuet-Higgins, 

 1970, the triangular solution is regained. Increasing ex, causes the rela- 

 tive profile to be lowered across the surf zone. Figure 28 is directly com- 

 parable with the results of Liu and Dalrymple (1978) in Figure 26 since both 

 normalized currents by equation (72) as reference velocity, vj. Although the 

 general shapes are similar, the results of Kraus and Sasaki (1979) are gener- 

 ally 50 percent greater near the breaker line (a, < 30°). The only difference 

 is in treatment of wave setup wherein Liu and Dalrymple (1978) included full 

 setup effect for large wave incidence. Hydraulically, this trend for free 

 surface flows is in the right direction. As shown by equation (40) oblique 

 angle wave setup is less than that caused by normal incident waves. For con- 

 stant bed resistance and friction slope, velocity decreases as water depth 

 decreases. The only surprise is in the magnitude of the difference. The 

 full implications of the influence of wave refraction and wave setup on the 

 longshore current theory need further research. Kraus and Sasaki (1979) ar- 

 gued that wave setup in the field was more complex than given by equations (35) 

 or (40) to justify their result. Also, they point out that including the 

 cosa, term in v* would reduce the magnitude of their results . 



(2) General Solution With Lateral Mixing . The y-direction momentum 

 balance equation (42) with lateral mixing is solved by expanding v in a power 

 series. _ The imknown coefficients are determined from the boundary conditions, 

 namely_v and dv/dx continuous at the breaker line and finite within the bounds 

 where x-^o and infinity. In the usual dimensionless terms, the longshore cur- 

 rent profile is 



V = 



E (A X + B xP)X^, within surf zone 0<X<1 

 n n 



(94) 

 Z C X , outside the surf zone X>1 



n=o 



103 



