^s = ^c + (^ '^ + ("Bn>/^)' • ^("'^ - ^^^ 



(77) 



The expression E(in) is a complete elliptic integral of the second kind, i.e., 



rTr/2 



E(in) 



(1 - msin^y) dy 



(78a) 



with the parameter m given by 



a2 

 Bm 



^m + ^' 



(78b) 



For weak currents (v <<u„ ), equations (76) and (77) reduce to 

 m Bm 



^By =7Vl"Bml 



(79) 



which is identical to equation (51) used by Longuet-Higgins (1970). The 

 scientists' friction coefficient C^ could then be defined as fw/2 for the 

 weak current-small angle theory. This form is also identical to that employed 

 by Thornton (1969) who similarly began from the wave-current approach of 

 Jonsson (1966a)2^. Equation (77) requires determination of both f^ and f^ 

 coefficients. For steady free surface flows, f^ depends on both water depth 

 and bottom roughness. The experimental determination of fw in oscillatory 

 water tunnels is still in progress. Details are beyond the intended scope of 

 this review. The nonlinear nature of both t„ and f prevents simple analytic 

 formulation of the current profile. Jonsson, -^Skovgaard and Jacobsen (1974) 

 also included a different lateral mixing formulation (see below) . The com- 

 plete final results are discussed in Chapter 4. 



Liu and Dalrymple (1978) gave a complete analysis for the. strong current 

 large-angle model with small angles as a special case. When U/il >>1 they 



found 



^By 



f [v2 

 cw 



l2 

 Bm 



(80) 



but emphasized that to be applicable, Ug must also be very small if v dimin- 

 ishes for small a. They neglected lateral turbulent mixing stresses in all 

 longshore current formulations. Their work is primarily of interest for the 

 large-angle modifications on longshore current theory as described below. 



2 7lbid. 



94 



