Integrating,
x = k logε {y + √ (y2 − k2)} + constant;
and, since y = b/2 when x = 0,
x = k logε [{y + √ (y2 − k2)} / {1⁄2 b + √ (1⁄4 b2 − k2) }].
Assuming values for y, the values of x can be found and the curve drawn.
The figure has been drawn for a channel the minimum section of which is a half hexagon of 4 ft. depth. Hence k = 2; b = 9.2; the rapid flattening of the side slopes is remarkable.
Steady Motion of Water in Open Channels of Varying Cross Section and Slope
§ 115. In every stream the discharge of which is constant, or may be regarded as constant for the time considered, the velocity at different places depends on the slope of the bed. Except at certain exceptional points the velocity will be greater as the slope of the bed is greater, and, as the velocity and cross section of the stream vary inversely, the section of the stream will be least where the velocity and slope are greatest. If in a stream of tolerably uniform slope an obstruction such as a weir is built, that will cause an alteration of flow similar to that of an alteration of the slope of the bed for a greater or less distance above the weir, and the originally uniform cross section of the stream will become a varied one. In such cases it is often of much practical importance to determine the longitudinal section of the stream.
The cases now considered will be those in which the changes of velocity and cross section are gradual and not abrupt, and in which the only internal work which needs to be taken into account is that due to the friction of the stream bed, as in cases of uniform motion. Further, the motion will be supposed to be steady, the mean velocity at each given cross section remaining constant, though it varies from section to section along the course of the stream.
| Fig. 118. |