rn 



u' 



dz 



J-h 







(h^ 



-n) 



0-2 



K = -^^ ^ , CA-16) 



the integrated momentum equation may be written: 



3^ {(h.n) u} . ^ {(h.n) k/} = -ChH-n) g |^ - -^ , (A-17) 



in which the shear stress on the free surface has been set equal to 

 zero. 



Using the results from linear wave theory (Eagleson and Dean, 

 1966), it may be shown that the value of Km as given by equation CA-16) 

 is 1.01 corresponding to a wave having h/L = 0.1. Thus, it is a good 

 approximation to take K^ in equation (A-17) equal to unity as is 

 normally done in open channel flow calculations. With K^ equal to unity 

 and incorporating the continuity equation, equation (A-11), the 

 momentum equation becomes: 



St * " 37= -g 57- ^Th^ • f^-^^^ 



The appropriate equations governing the propagation of long waves 

 over a rough bottom are therefore equations (A-11) and (A-18). However, 

 these equations cannot be solved until the shear stress in equation 

 (A-18) has been related to the kinematics of the problem. To this end 

 we may introduce the concept of the wave friction factor, f , as defined 

 by Jonsson (1966) , 



T^ = i P fju|u , (A-19) 



where |u| is the absolute value of the velocity. When this expression 

 is introduced in equation (A-18) the governing equations become: 



^ + g| (h+n) U} = (A-20) 



and 



— f lulu 



3U ,, 3U 3n 2 wl I ^. ^,^ 



3t ^ "J 37 = -g 37 - (h.n) ' ^^-21^ 



where the overbar notation has been dropped. 



108 



