(1) In general. P ^ 0.4 . 



Pi 

 V = AX + B,X 0<X<1 (inside breakers) (65a) 



where 



V = B2X X>1 (outside breakers) (65b) 



A = ^ (66a) 



(1 - f P) 

 1 4 ^ 16 "^ p 



^ + 'fn" ■ (66b) 



P2 = - 4 - 16 + p ^^^^^ 



P2-1 



B, = A (66d) 



1 P1-P2 



2 



P1-P2 



A (66e) j 



\ 



so that all the constants (A, p^ , p„, B , and B„) in equation (65) depend upon j 

 P. L I L I I 



(2) Singularity, P = 0.4 . The coefficient A becomes infinite for , 

 P = 0.4. For this singularity, Horikawa (1978a) gives the solution ( 



1 



V = "ll X - y XlnX 0<X<1 (inside breakers) (67a) 



V = -i| X~ ^ X>1 (outside breakers) (67b) 



The family of solutions for some represen}:ative P values is shown in ! 

 Figure 24 (after Longuet-Higgins, 1972a). Taking larger P values gives more 

 lateral turbulent mixing to smooth and spread the theoretical longshore cur- 

 rent profile across the surf zone and beyond the breakers. Using lower P J 

 values shifts the maximum longshore current toward the breakers. The theory i 

 requires estimates for three parameters (y, Cf, and N) for a solution on a J 

 given plane beach slope. It neglects wave setdown and setup effects in mean 

 water depth but includes the maximum setup location as the new shoreline. 



The radiation stress theory of longshore currents as summarized by equa- 

 tions (65), (66), and (67) provided the needed breakthrough in 1970. Since j 



88 



