where the incident wave angle is no longer assumed small. Using the same 

 bed shear-stress formulation as before (eq. 54), except including wave setup, 

 and neglecting lateral turbulent mixing stresses, the following modified 

 equation is obtained 



-. 5 jY , tang sina ,^,. 



^ = 16^ ^ g^ 1+3y2/8 ~C- '^°^" ^^^^ 



for longshore currents. This reduces to equation (60a) inside the surf zone 

 when h->d. Equation (71) is essentially that given by Bakker (1970)^^, 

 Thornton (1969,1970a), and confirmed by Gourlay (1978) and others. At the 

 breakpoint, it becomes the modified reference velocity (using C, = "ghT" for 

 shallow linear waves) 



"a. 5 Y / 1 \^ tana . /■,^\ 



^b = 16^ ^ ^^\^ 1+3yV8 ^^"% ^°^"b ^^2^ 



For Y values from 0.5-1.2, the term [ (1/ (1+3y2/8) ] in equation (72) varies 

 from 0.91 to 0.65, so that the wave setup modification is not insignificant. 

 Komar (1975a, 1976b) obtained the additional term 1/(1 + y)'^ in equation (72) 

 that was shown to be inconsistent with previous assumptions for dSj^/dx and 

 Tg by Gourlay (1978). Komar (1975a, 1976b) also had the term (1+3y2/8)2 

 instead of the first power as in equation (72). The present report and 

 Gourlay (1978) have been unable to verify Komar 's form. Using this modified 

 v? in equation (64) gives for Longuet-Higgins, P-parameter (modified) 



P = ^ . , ff,,, (73) 



C Y 1 + 3y /8 



so that all of Longuet-Higgins '_ (1970) dimensionless results are still appli- 

 cable (eqs. 65, 66, and 67) if ^ and P are replaced by the modified versions 

 (eqs. 72 and 73). However, it must be remembered that this setup correction 

 assumes normal wave incidence and hence also neglects wave refraction effects 

 in the surf zone. 



All other modified models listed in Table 3 include some form of Snell's 

 law refraction in the surf zone to additionally modify the wave setup. As 

 shown earlier (eq. 40), the setup slope is no longer a constant proportion 

 to the beach slope and refraction results in less wave setup. For the numeri- 

 cal solution methods, the decoupled motion equations (28) and (42) are 

 solved together. The n solution from equation (28) includes effects of geo- 

 metry refraction in the dSj^^/dx term and is in turn used in equation (42) to 

 determine the longshore current profile. Numerical accuracy is important so 

 that careful numerical integration procedures are needed (e.g., see Jonsson, 

 Skovgaard, and Jacobsen, 1974). In all such models to date, current-refraction 

 effects on the calculated n values have not been incorporated to make the 

 equations a coupled set. 



2^Ibid. 



91 



