53. Equations 12 and 13 constitute a system of two nonlinear algebraic 

 equations in four unknowns: z(i,j+l), z(i+l,j), Q(1,J+1), and Q(i+1 ,j+l) . By 

 themselves, these two equations are not sufficient to evaluate all unknowns at 

 points {i,J+l) and (i+1 ,J+1) . Let Nl be the number of nodes in Channel 1. 

 However, the unknowns are common to any two neighboring cells. Because there 

 are Nl-1 cells between rows J and j+1 in Channel 1, two equations such as 

 Equations 12 and 13 can be written for each cell. The combination of all 

 cells provide 2 (Nl-1) equations for the evaluation of 2N1 unknowns. For a 

 single channel, two additional equations are needed to determine all the 

 unknowns, and they are provided by the external boundary conditions. For 

 networks consisting of interconnecting channels, each individual channel will 

 lack two additional equations. The additional equations are provided by 

 external boundary conditions where the channel meets the bay or the sea, and 

 by junction conditions where two or more channels meet. 



External boundary conditions 



54. In the inlet system shown in Figure 3, Node 1 of Channel 1 is an 

 external boundary node because it is not connected to another channel. End 

 nodes of Channels 4 and 5 are also external boundaries. If the water surface 

 elevation at an external boundary is known as a function of time, then 



zl;' - iz'){i' = (16) 



where (z')ib^ is the known water surface elevation at the external boundary 

 node ib at time step j+1 , and Equation 16 becomes available as one of the 

 supplementary equations. If the discharge is available at the external 

 boundary ib , then 



Qlt' - iO')i*i,' = (17) 



where (Q')ib^ is the known discharge at the external boundary ib at time step 

 j+1 , and Equation 17 becomes available as an alternative supplementary 

 equation provided by the external boundary. 



55. Other types of permitted boundary conditions providing the neces- 

 sary equations could be an equation expressing the discharge as a function of 

 water surface elevation, as in a weir or jetty, or an analytical expression 

 specifying water surface elevation or velocity as functions of time. In a 

 single channel, for any physical situation two supplementary equations such as 



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