in which 



y = 7- 



(103) 



Introducing equation (86) this may be written 



U = A C0£ :; 



' ' ' ' s h 



1/2 

 Jj(2Vy'^^) 



f y 



1/2 



(104) 



in which 



"V = k a A-i tan2(|) 



OS 



(105) 



Inserting equation (104) in equation (101) the average rate of 

 energy dissipation per unit surface area of the rough slope may be 

 written : 



2 



^n " h P-^9 ^ ^c tan3 y 

 ^ h " ^ ^0 



1/2 

 J^(24'y'/^) 



Ty 



1/2 



dy} tan2(|) . (106) 



If the average rate of energy dissipation is evaluated using the 

 nonlinear expression for the bottom shear stress (eq. 82) one obtains 



, COsB rl 



F 1 -c s s 



E^ = TT pf —. 



D 2 w £ cos 



T 







'^"~ d |u|u2 dt} , 



s ^0 



(107) 



where U again is periodic and given by equation (84) with u given by 

 equation (91). Performing the time averaging and introducing |u| as 

 given by equation (104) the average rate of energy dissipation per unit 

 area pf the slope becomes 



F - 2 . IaI^ 3„ 3 ^^ 



D 5-n w , 3 s I ^ 



h ■' 



J^(2fy^/2)| ^ 



'i'y 



1/2 



dy 



(108) 



Using Lorentz' principle of equivalent work by equating equations 

 (106) and (108) results in the following expression for the friction 

 angle (j) : 



56 



