where u = ■=r~ = (gk tanh kh) ^ = a mean frequency 

 o 

 T = the deepwater mean wave period (i.e., peak period for narrow spec- 

 trums) 



and H = the mean wave height 



r- \ 



H-^H^ evfi-^ ) (127) 



rms f 



rms 



with H = the fictitious root mean square wave height 

 rms 

 R, = the local breaker height 



erf = the error function. 



Small errors (<10 percent 'in deep water) result if H is replaced by H in 

 equation (126). Using a bed shear-stress model of the form 



^By ' ^fPV (128) 



in the longshore momentum balance equation (42) without lateral mixing gave 



_ T _ dS 



V = + - ° - sinh kh — ^ (129) 



2 c^ H dx 



with X again defined a positive in the ocean direction. When the wave setup 

 is included as expressed in terms of the bottom slope for a plane beach 

 (i.e., see eq. 68) this expression for v becomes 



T _ dS dS -1 



V = — ^ — tan3 sinh kh -~^(1 + -~- -—■) (130) 



2pc^H ^^ Pg^ ^^ 



where the wave setup has yet to be determined. 



Battjes (1974a) defined a normalized current velocity 



c.T 

 f o - 



(131) 



tang-rrH 

 o 



to give 



_ L h _ dS dS -1 



V* = -2— sinh k h -^ (pgh + — ~) (132) 



H H ^^ ^^ 



o 



which required numerical evaluation. As a check, the dimensional expression 

 in equation (130) for a narrow spectrum in shallow water reduced to the 

 modified Longuet-Higgins model (eq. 71) including setup. 



142 



