Finally, Kraijs and Sasaki (1979), following Liu and Dalrymple (1978), 

 include wave refraction in their analytic development. Inside the breaker 

 line they obtained the following result for the driving stress 



dS q J.- J' 



xy 5 2 tang . /h^s ^ r ,., .9 h ,^ 



"^ = 16 PY g lT3^8 ^^"% ^hT^ [(l-sm^a^ :-) - 



h b 



b b 



Note that neglecting wave setup, refraction and for small a, , equation (74) 

 reduces to that employed by Longuet-Higgins (eq. 48). These researchers 

 also modified the form of the bed shear stress and lateral mixing terms, so 

 their final results are deferred to the subsection on lateral mixing. 



b. Modified Bottom Shear Stress . The most important modifications to 

 the original model have been in relaxing the assumptions of a weak longshore 

 current and small wave incidence angle. This is because the longshore cur- 

 rent is inherently related to the bed shear-stress model employed. The major 

 weakness of Bowen's (1969a) model was the overly simplistic, linearized, shear- 

 stress term, x = pC^-v. All other models in Table 3 begin with a quadratic 

 form as given by equation (52) . 



Bottom shear stress is a vector oscillating in both direction and magni- 

 tude. The major difficulty is to find an expression for the effective bed 

 stress (and friction factors involved) in terms of a time-averaged current. 

 The coordinate axes and velocity vectors are shown in Figure 25. The instan- 

 taneous bottom sh^ar stress t is assumed to be in the direction of the re- 

 sultant velocity U of the vector sum of longshore current velocity v and bottom 

 wave orbital velocity u^ (eq. 52). 



tg = c^p|u|u = f^^i5P|u|u 



where U = v+-> U , and C^, f are combined current and wave friction factors. 

 B f cw 



The ratio of theoretical breaker current with no lateral mixing to maxi- 

 mum wave orbital velocity near the bed, v*/u^ has been shown to exceed unity 

 except for small wave angles . This ratio was used to argue the inconsistency 

 of the weak original model of Longuet-Higgins (1970) as discussed b^ Huntley 

 (1976a), Madsen, et al. (1978), and Liu and Dalrymple (1978). But v* is not 

 a physical velocity. Therefore Kraus and Sasaki (1979) used the ratxo v^^/ug^^ 

 instead, where v is the maximum current from experiments or theory. Surpris- 

 ingly, they showed that most laboratory experiments are invalid to test the 

 Longuet-Higgins (19 70) model since v /u -1-3, indicating strong longshore 

 currents are present. 



92 



