A different approach was suggested by Bijker (1966) , modified by Swart 

 (1974)23 and fully discussed by Bijker and v.d. Graaff (1978)^^. This theory 

 is unique from those above in that the location above the bed where the wave 

 orbital velocity and current velocity are specified is explicitedly defined. 

 The elevation chosen was equal to the laminar sublayer thickness which is 

 dependent upon the bed roughness height. In this way, modern turbulent boun- 

 dary layer theory is incorporated into the bed shear-stress model. Expres- 

 sions for the shear stress at this elevation used the resultant velocity U as 

 defined in Figure 25. They then obtained the result 



: = 2pgv2 

 By rZr 



C-^T 

 c 



T/4 



-T/4 



{ [1 + £;— =— sincjt sina], 



[1 + {f^ sinujt) + 2? -1^ sincjt sin0]'^}dt 



V V 



(83) 



where 



p = a dimensionless parameter found from wave orbital velocity experi- 

 ments (e.g., Jonsson, 1966a)2^ relating the wave velocity at the 

 bottom to that at the reference elevation 

 K = the von Karman constant ^ 0.4 

 C = the Chezy friction coefficient 

 f„ = the wave friction coefficient from t 



T = the wave period 



w i> 



Bijker (1966) numerically integrated equation (78) for a range of realistic 

 values of v, iL , £; and a. Curve fitting the results for |ctj< 20° gave 



TBy = ^'[0.75 + 0.45 (5^)'"''] 



(84) 



For weak currents and small incident wave angles equation (80) can be inte- 

 grated directly to yield a form identical with equation (51) taking 



'^2iF 



^f = 



(85) 



A numerical example is shown in Figure 27 adapted from Bijker and v.d. 

 Graaff (1978). Here, turbulent mixing is neglected along with wave setup 

 effects. The result labeled vj is for weak current small- angle model but 

 each velocity is calculated from a local friction term. This makes the profile 



23lbid. 

 26ibid. 

 2® JONSSON, I.G. , "Wave Boimdary Layers and Friction Factors," Prooeedings^ 



loth Coastal Engineering Conferenoe (Tokyo), Vol. I, Ch. 10, 1966b, pp. 



127-148 (not in bibliography). 



97 



