ILLUSTRATIVE PROBLEM 81 



Determining a and b by this procedure corresponds to fitting the straight 

 Hne through the initial and final points on the logarithmic plot. It has 

 the disadvantage of not revealing the degree of approximation for the 

 intermediate points, but it is much quicker than the graphical method, 

 and can safely be used when experience indicates that the logarithmic 

 plot will differ but little from a straight line. After a and b have been 

 determined by this method, the coefficients Ca and Cb are found by sub- 

 stitution of a pair of simultaneous values, preferably for an inter- 

 mediate point. 



A suggestion which seems to offer an obvious improvement in com- 

 putation procedure is that the desired exponents could be obtained by 

 simple mathematical processes from empirical exponential relations for 

 the surface width and the wetted perimeter. The relation for the 

 surface width would only have to be integrated to obtain the relation 

 for the area, which could be divided by the relation for the wetted 

 perimeter to obtain an expression for the hydraulic radius, and so on. 

 These operations could be performed more easily upon the simple 

 empirical equations than upon the original data. Unfortunately, 

 exponential values obtained by this process are likely to be very inac- 

 curate. The logarithmic plots of the surface width and the wetted 

 perimeter do not approximate straight lines very well, and the errors 

 produced by performing mathematical operations upon empirical rela- 

 tionships are likely to be surprisingly large. On the other hand, the 

 data for equations (704) and (705) can almost always be closely approx- 

 imated by a straight line on the ogarithmic plot. The errors intro- 

 duced in the final integration of the backwater function may be kept 

 to a minimum by determining the necessary empirical relations directly 

 in the form of equations (704) and (705). 



Illustrative Problem 



Determine values of the coefficients and exponents in equations (704) and 

 (705) for a trapezoidal channel with bottom width of 100 feet and side slopes 

 of 1 on 1, to apply over a range of depths between 10 and 25 feet. Let n = 0.022. 



The computations are based on the values for the same cross section which 

 were computed in Table 701. 



Depth A^/T 



