CASES OF THE BACKWATER CURVES 69 



equal to jn- This means that except in channels having critical slope, 

 all curves must cross the critical depth vertically. If y equals zero, 

 dy/dx is a positive quantity which is equal to Sq if yc equals yn- This 

 means that the curves which approach the bottom intersect it at a 

 definite angle. If y approaches infinity, dy/dx approaches ^o, which 

 means that the water surface becomes horizontal as the depth of flow 

 becomes very great. Finally, when yn equals yc, dy/dx must equal Sq 

 unless y equals y„, when it may equal zero. 



An Ml curve is produced when the lower end of a long flume having a 

 mild grade is submerged in a reservoir to a greater depth than the 

 normal depth of flow in the flume. 



An M2 curve results when the bottom of the flume at its lower end is 

 submerged in the reservoir to a depth less than the normal depth. If 

 the depth of submergence is greater than the critical depth, then as 

 much of the M2 curve will form as lies above the water surface in the 

 reservoir. If the amount of submergence is less than the critical depth 

 the M2 curve should end abruptly with its lower end tangent to a vertical 

 line and at a height equal to the critical depth. Actually, vertical 

 components of velocity will become an appreciable factor, and near the 

 end it will merge into the local phenomenon known as the drop-off. 



The M3 curve is very peculiar in that its lower left-hand end starts 

 from the bottom of the channel at an acute angle and its upper right- 

 hand end terminates abruptly, tangent to a vertical line. Thus the 

 curve is restricted in length to definite limits in both directions. On 

 this account it can exist only under favorable circumstances. 



For example, if water issued at high velocity from a reservoir through 

 a submerged gate opening, it could follow this curve, provided that the 

 water could be conducted away by some change in channel conditions, 

 before the right-hand end of the curve should be reached. Such a 

 change might be a sufficient increase in the grade of the channel. The 

 curve could also be terminated abruptly at the right-hand end by a 

 hydraulic jump. The location of the jump may be found by the 

 method described in Chapter III, for rectangular channels, or that of 

 Chapter IV, for other channels. Still another method of terminating 

 the M3 curve would be by one of the methods of changing from low 

 stage to high stage without the hydraulic jump, as described in Chap- 

 ter II. 



The higher the initial velocity of the issuing water, the farther down 

 towards the left the M3 curve would begin. At the end of the curve, 

 where it intersects the bottom of the channel, the velocity of the water 

 would be infinite, so this point represents a limit that could never be 

 reached physically. 



