80 BACKWATER CURVES IN UNIFORM CHANNELS 



desirable. The first step is to determine the coefficients and exponents 

 in the following empirical equations: 



A3 



[704] 

 and 



A^C^R = Cby^ or ^ = c^/ [705] 



n 



in which A, T, and R represent the area, surface width, and hydraulic 

 radius when the depth of flow is >' ; C and n are the Kutter and Manning 

 coefficients most appropriate at the depth y; and Ca, c^, a, and b are 

 the empirical coefficients and exponents to be determined. The 

 choice of equations (705) depends upon whether the Kutter or the 

 Manning formula is to be used in evaluating the eff'ect of friction. 



Bakhmeteflf calls the function ACVr or — AR^'^ the " convey- 



n 



ance " of the channel at depth y. This quantity has only to be mul- 

 tiplied by the square root of the slope to give the discharge for uniform 

 flow at the given depth. Bakhmeteff discovered that for both arti- 

 ficial and natural channels the square of this function is almost invari- 

 ably susceptible of accurate approximation by a monomial exponential 

 function of the depth.* 



For rectangular or triangular channels, the coefficient and exponent 

 in equation (704) may be determined by simple algebra. For other 

 channels, corresponding values of -4^/7" and y over the range of depths 

 of the backwater curve are plotted logarithmically, with values of 

 log y as abscissas. A straight line is drawn fitting the points as well 

 as possible. Measurement of its slope will give the numerical value 

 of the exponent a. The coefficient Ca is obtained by substituting a pair 

 of simultaneous values for a point on the line, or by extending the line 

 to its intersection with the line y equals unity. A similar procedure is 

 employed for the determination of the coefficient Cb and the exponent b. 



Another method for determining a and b is simpler, and often just as 

 accurate. Let the subscripts 1 and 2 refer to values of the variables 

 at the lower and upper end of the range of depths. Then 



and 



b = 

 * Ibid. 



log {A,VT2} - log (AiVri) 

 log yo - log yi 



log {A2'R2"') - log jA^'Ri'") 

 log 3^2 - log 3^1 



