1 }^ J' 

 = - pC g\(tan6) ^ (58b) 



.*2,.„.^% 



p = Npg'^(tanB) '2 (58c) 



to give the second- order differential equation for v 



% d^v ,5 -% dv -h- -3/2 



p- "- —^ + J px ^ -^ - qx^ = -rx ^ (59) 



dx2 ^ dx 



with the right-hand side vanishing outside the surf zone. Thus v(x) is the 

 solution of two second-order linear, ordinary differential equations which 

 must match in magnitude and gradient of v(x) at the breaker line and vanish 

 at x=0 and the ocean boundary. 



(1) Neglecting Lateral Turbulent Mixing . Taking p=0 greatly 

 simplifies equation (59) to give as an approximate solution 



~* r - 5tt Y , „,o,sina,- ... 

 V* = — X = 77" 7^- g(tanB)^(— :; — )x mside surf zone 

 q Id L^ C 



(60a) 



V* = outside surf zone (60b) 



On a plane beach with y, C , and (sina/c) , all constants in the surf zone, 

 the longshore current profile is triangular in shape, reaching a peak at the 

 breaker line and dropping to zero outside the breakers (see Fig. 24, P = 0). 



(2) Reference Longshore Current Velocity . Again neglecting later- 

 al turbulent mixing stresses, the longshore current velocity at the breaker 

 line V* can be used as_a reference velocity. It becomes from equation (60a) 

 after again taking h, = d, = tangx, 



f b 



Other forms result from taking C, = /gd^ in equation (61) . In reality, the 

 velocity discontinuity at the breaker line cannot occur so that the lateral 

 turbulent mixing smooths the velocity profile. The solution with lateral 

 mixing is simplified using dimensionless variables with v|^ as the reference 

 velocity. 



e. Dimensionless Longshore Currents . Longuet-Higgins (1970) introduced 

 the dimensionless variables, X and V as 



X = ^ and V = ^ (62) 



\ ^b 



into equation (59) to obtain 



86 



