112 BENDS, TRANSITIONS, AND OBSTRUCTIONS 



[1001] 

 gr 



To integrate this equation the value of the velocity at all points 

 across the river must be stated in terms of r. Formulas may be obtained 

 under a variety of assumptions. 



A fairly good approximation is obtained by assuming V constant at 

 the average velocity, and assuming r to be constant at the value for the 

 center of the stream. Then the 



Difference in elevation of the two banks = -— - [1002] 



gR 



This will always give too small a value because the effect of the fila- 

 ments with the higher velocities more than offsets the effect of the 

 slower filaments, since the velocity enters to the second power in the 

 integral. 



If the actual velocity distribution across the stream is known, the 

 width may be divided into several sections, the difference in surface 

 elevation computed for each section using its appropriate velocity and 

 radius, and the total difference found as the sum of the differences for 

 the separate sections. 



If the velocity is zero at each bank and has a maximum value Vm at 

 the center, varying in between accotding to a parabolic curve : 



Total difference in surface elevation = 



The effect of the variable velocity distribution throughout a cross 

 section is, probably, to increase slightly the actual radius of curvature 

 followed by the moving water in going around a bend, especially for a 

 short curve. Thus, for a curve of 90 degrees the effective radius for 

 the whole stream is probably as great as the radius of the outer bank; 

 but for a curve of 180 degrees the effective or actual radius cannot much 

 exceed the average of the radii of the inner and outer banks. 



According to F. C. Scobey^ the lower elevation of the water at the 



^ " The Flow of Water in Flumes," by Fred C. Scobey, U. S. Department of Agri- 

 culture Technical Bulletin 393. 



