then, many modifications attempting to improve the generality and accuracy of 

 the original model have appeared in the literature. These are summarized in 

 the next section along with the original contributions of Bowen (1969a) and 

 Thornton (1969). 



3. Modified Models . 



The modified theories for longshore currents since 1970 are listed in 

 Table 3. These theories still retain linear' wave theory to calculate the 

 radiation stresses for regular waves. Nonlinear and irregular wave theories 

 are discussed in Section VI. In some cases, closed- form analytic solutions 

 are no longer possible due to the beach profile or bed shear-stress formulation 

 employed. Rather than discuss each model separately, the major modification 

 areas involved are reviewed. In most cases, actual theoretical results will 

 be discussed in Chapter 4 and compared with experimental measurements. 



a. Beach Profile and Wave Setup . Those theories that are for a beach 

 profile with straight and parallel bottom contours but with depth in the surf 

 zone that is monotonously decreasing (arbitrary profile) require numerical 

 integration methods for a solution. Thornton (1969) was the first to provide 

 such an analysis. 



Wave setdown effects on the MWL are neglected by all theories except the 

 numerical integrations by Jonsson, Skovgaard, and Jacobsen (1974) and Skovgaard, 

 Jonsson, and Olsen (1978). However, the influence of wave-induced setup has 

 been incorporated in most subsequent theories because of its influence upon y. 



Bowen (1969a) and Komar (1975) simply took the resulting wave setup 

 slope equation for normal wave incidence on a plane beach (see eq . 35) which 

 for the x-coordinate system becomes 



1 = - [-; ^7-7-] tang = -Ktan3 (^8) 



Letting 



1 + 8/(3y ) 



d(d4n) dd , dn ^ . ^^ , 



7= — = -p^ + -^=- = tang - Ktanf 



dx dx dx 



or 



m = (l-K)tanB ^^^ (69) 



1+3y2/8 



then the following modified equations result, neglecting wave setdown. 



For the driving stress term, equation (49) now becomes 



dS 5 



xy 5 ? tang . u\ '2 /Sina, ^,„^ 



T=^ = YZ PY (gh) ( — - — )cosa (70) 



dx 16 



1+3y-^/8 



89 



