Woo Hflghl.H 



, Breaking Wove Hilghl, M, 



Figure 21. Schematic of wave setdown and setup due to normal wave incidence 

 on a plain beach. 



MWL change on a sloping bottom, g. If no net shear stresses are assumed at 

 the bottom or on the free surface, the momentum balance gives 



dS, 



dn 



-, — + pgh -j^ = 

 dx '^ dx 



(28) 



where h = d + n, and n is the MWL change above or below the Stillwater level 

 (SWL). If S-j^^ is known, equation (28) can be integrated to yield a MWL setdown 

 outside the breaker line and a MWL setup in the surf zone. Such an analysis 

 was first conducted by Longuet-Higgins and Stewart (1963, 1964) and also by 

 Bowen, Inman, and Simmons (1968) for plain sloping beaches and using S^^ from 

 linear theor}/ (eq. 23). However, equation (28) applies to an arbitrary 

 bottom profile (still assuming straight and parallel contours) as long as the 

 depth, d is monotonously decreasing. 



b. Wave Setdown . Seaward of the breakers, it is reasonable to neglect 

 wave reflections, percolation, bed-shear and internal turbulence dissipation 

 so that the waves propagate with constant energy flux. 



Ecn = Ecg = Constant. 



(29) 



Using equation (29), Longuet-Higgins and Stewart (1963) integrated equation 

 (28) over a plain sloping beach to obtain 



n = - 



1 H-^k 

 8 sinh(2kd) 



(30) 



73 



