Direct Step Method

Please observe the profile below for a stream that flows into a reservoir

formed behind a dam.

The stream cross section can be approximated to a trapezoidal channel

section with a base width $B_w=20$ feet. The side slopes are such that $H$:$V=$

$z$:$1=1$:$1($that is$,z=1)$ on the left bank and right bank too. The bed of the

stream is steep and has a slope of $S_0=0\mathrm{.}013.$ The natural stream's Manning

roughness coefficient is $n=0\mathrm{.}028.$ The flow in the stream is $Q=552cfs,$

A dam causes the backwater to rise in the stream. Hence the water jumps

from a depth of $y_1$ to $y_2,$ dissipating energy in a hydraulic jump $($H$.$J$.).$ Then

the water gradually becomes deeper in the reservoir up to a maximum depth

of $y_4$ behind the upstream face of the dam. The water surface in the

reservoir is a gradually varied flow $($GVF$)$ profile known as an $S_1$ curve. You

are asked to use the Direct Step Method to compute the length of the $S_1$ curve. To save computation time at the expense of accuracy, please work on

two steps only:

Find the distance $D{x}_1$ from $y_2=5\mathrm{.}9$ feet $($ft$)$ to $y_3=7\mathrm{.}7ft$

Find the distance $D{x}_2$ from $y_3$ to $y_4=9\mathrm{.}2ft$

Finally, find the total length $($L$)$ of the $S_1$ curve.