b. Resisting Stress, x . For the time-average bottom shear stress, t 



Longuet-Higgins (1970) derived 

 2 



'B .Vl^ml^ (51) 



where 



C = a boundary resistance coefficient due to both waves and longshore 

 current, v 

 I u^ I = the absolute value, maximum wave orbital velocity near the bottom 

 for sinusoidal motion 



Longuet-Higgins (1970) obtained equation (51) from the usual quadratic stress 

 law in real time for the vector value of x 



D 



"B=ifwcPl^l^ = VI^bI^ (52) 



where u^ is the wave orbital particle velocity just above the bottom boundary. 



To go from equation (52) for x to the time-average value in equation (51) 

 is not trival. Longuet-Higgins (19/0) made the additional assumptions that 



(1) the longshore current velocity v is small in comparison with the 

 wave orbital velocity, u^, and 



(2) the wave incident angle a is very small in the surf zone so that 

 V is roughly normal to ti . 



Taking the time-averaged mean value of |u | over one wave period and assuming 

 Cj represents a constant mean value over this same period, the component of Xg 

 in the y-direction becomes that given by equation (51) . Assumption (1) above 

 essentially makes this a linearized bed stress term or a weak current theory. 

 Removal of assumptions (1) and (2) has been a significant achievement in 

 modifying the original theory as described below. 



Further approximations are needed to put equation (51) in a more usable 

 form. Using linear theory to obtain u^ in shallow water and taking y = H/h 

 give 



ug^ = I y/ih (53) 



so that equation (51) becomes if wave setup is again neglected 



Xg = ^C^g'V(tane)'^x'^v (54) 



This form clearly shows how the longshore current v is introduced and how the 

 bottom shear stress varies as x"^. 



84 



