136 SLOWLY VARIED FLOW 



the work of computation to an absolute minimum. The values of 

 inflow are set on an easily made slide rule and the corresponding values 

 of outflow are read directly, with no intermediate computations. 



Assume that the inflow i is constant during a step A^, and that the 

 average outflow o is equal to the mean of the outflow rates Oi and 02 at 

 the l^^eginning and end of the step. Then, by comparison with equation 



: ,,' i-At= 'Z ^ -At-^- AS [1107] 



f his can be written 



At = — • At + S2 — Si 



or 



i- At-\- (5i - Joi • At) = (52 + J02 • At) [1108] 



If AHs taken as a constant, then 

 i • At is a function of the inflow, 

 Si — 2^1 • AHs a function of the water-surface elevation at the 



beginning of the step, and 

 S2 + 2^2 • Ai is a function of the water-surface elevation at the end 



of the step. 



The water-surface elevation at the beginning of the step is, of course, 

 the same as the water-surface elevation at the end of the previous step. 

 In order to start the routing computations, then, the initial water- 

 surface elevation must be known. As a matter of fact, it is not essential 

 that the initial elevation be known precisely, for errors due to assuming 

 an incorrect value will disappear rapidly as the routing computations 

 proceed. 



The procedure of the routing computations is now evident. Knowing, 

 from the preliminary computations, the relations S = fs{E) and = 

 fi{E), and also the elevation at the beginning of the first step, the value 

 of {S — ^0 • At) at the beginning of that step may be computed. From 

 the given inflow hydrograph, the value oii - At for the step is computed. 

 Adding this to (5 — Jo • At) gives the value of (5 + Jo • At) at the 

 end of the step, from which the value of the elevation at the end of the 

 step can be computed. The process is repeated for each of the succeed- 

 ing steps. The computations are facilitated by preparing beforehand 

 tables or charts showing the relationships 



S-lo'At=ME) 

 and 



S-\-^o-At=f7{E) 



