BACKWATER CURVES 61 



theorem. Before proceeding with the analysis of these curves it is 

 necessary to list the notation to be used. 



y = depth of water flowing in the channel at any section. 



X = horizontal distance along the channel from some arbitrary 

 origin to the section where the depth is y. 



V = average velocity of the flowing water, considered positive when 

 toward the right. 



Q = volume of water flowing per foot width of channel. Q is con- 

 stant, for the flow is steady. 

 yc = depth of critical flow. For a rectangular channel gyc^ = Q^. 



(See Chapter II.) 

 Su) = slope of the water surface. The slope is customarily considered 

 to be positive for a downward slope in the direction of flow. 

 This is illustrated by the Chezy formula, in which the slope 

 must be positive. Though mathematically awkward, this 

 sign convention has roots so deep in hydraulic practice that 

 it seems best to continue its use, even where to do so intro- 

 duces difficulties, 

 ^o = slope of the channel bottom, considered positive if downward 

 in the direction of flow. 



dv 



— = slope of the water surface relative to the bottom. In order to 



^ conform with the usual convention of the calculus, this is 



taken as positive if the depth increases in the direction of flow. 



For flow without friction to satisfy Bernoulli's theorem, the slope of 

 the water surface must equal the rate of change of velocity head with 

 respect to distance along the stream, or 



