fixed ratio throughout the entire surf zone. This assumption is also important 

 in longshore current theory. A complete review of surf zone empiricism will 

 be presented in the section. 



Now if it is again assumed (Longuet-Higgins and Stewart, 1964) that 

 linear wave theory is applicable to compute S , that shallow-water conditions 

 prevail so S = 3/2E, and that y is constant xn the surf zone, then 



S =^pgY2(n+d)2 (34) 



XX 16 ° 



Using equation (34) in the momentum balance equation (28) , where h = (n + d) 

 is retained in the second term, Bowen, Inman, and Simmons (1968) showed that 

 for a plane beach of slope, tan B 



dS ^ 1 + 8/(3y^) ^^^ ^ ^^^^ 



For a given constant index y» this meant that the mean water surface slope 

 (setup) was proportional to the beach slope as illustrated in Figure 21. 



Integration of equation (35) to find n on a plain beach reduces to a 

 simple trigonometric analysis. All that must be specified is the magnitude 

 of the breaker index Y> the location_of the breakpoint, and the magnitude of 

 the wave setdown at the breakp^oint, n, • Again, using shallow-water theory 

 and equations (30) and (33), n, becomes 



\ = - I6 ^^ ^''^ 



For the maximum wave setup n at the shoreline, Battjes (1974a, b) used equation 

 (36) and simple geometry to show that 



\ = 1^ ^«b (^^> 



This means that the MWL at the shoreline is predicted to rise about 25 percent 

 of the breaker wave height due to wave setup. Horizontal distances to locate 

 setup values of interest can easily be determined from the geometry involved. 

 The key assumptions are use of linear theory for S and a constant y in the 

 surf zone. 



(2) Reflective- Type Beaches . Plunging- type breakers are found on 

 steep reflective beaches with narrow surf zones. Most laboratory beaches are 

 of this type. Two theories have been advanced for setup primarily based on 



75 



