66 BACKWATER CURVES— INTRODUCTORY 



By geometry S^ = Sq — dy/dx. Substituting this value of Sy, in the 

 above equation 



dy d /V^\ 



Equation (606) is the general differential equation for all of the back- 

 water curves shown in Fig. 602. Substituting the value of S/ found by 

 eliminating Q^/C^ between equations (604) and (605) and the value of 



d (V^\ 



I — I from equation (601), we obtain 

 \2g/ 



dx 



S -— = - 

 dx 



(cTfAiP^ 



which is the differential equation for the backwater curves in an infi- 

 nitely wide rectangular channel, friction evaluated by Chezy's formula. 

 It has the integral solution 



ynZ 

 x = — 



'JO 



/I C^Xri z^ + z + l 1 _i22 + l"| 



[608] 



in which z stands for y/yn- Tables of the function contained within the 

 brackets were computed by Bresse for positive values of z. Table 601 

 gives values of this function for positive and negative values of z.^ 



Cases of the backwater curves. All possible kinds of backwater 

 curves are shown in Fig. 602. The curves are plotted to an exaggerated 

 vertical scale, and have been shown dotted near the critical depth as a 

 reminder that this portion of the curve does not possess the same 

 degree of accuracy as the rest of the curve, owing to neglect of the 

 vertical components of velocity. In several cases the curve has either 

 a beginning or an abrupt ending at the critical depth. In plotting the 

 curves by the use of Table 601, M2 and M3 will appear as one con- 

 tinuous curve, and similarly with SI and S2. This is because mathe- 

 matically they are represented by the same continuous function. In 

 steady flow in a uniform channel at constant grade it would be impossible 

 for any backwater curve to cross the critical depth. The water surface 

 could cross the critical depth only through a hydraulic jump. 



Many of the peculiar features of the curves shown in Fig. 602 may 



' In evaluating Part 2 of the table, a constant, 7r/3-\/3, was added to ^[z] to prevent 

 the function from changing sign. Similarly, the constant tt/v 3 was added in com- 

 puting the values in Part 3. Since no backwater curve utilizes more than one part 

 of the table, the introduction of a constant in this manner does not affect the use 

 of the table, except to make errors less likely. 



