E3 Questions on Gradually varied flow - including integration of the backwater curve.

A rectangular channel with a bottom width of $4.0m$ and a bottom slope of $0.0008$ has a discharge of $1.50m^{3}/s$. In a gradually varied flow in this channel, the depth at a certain location is found to be $0.30m$ assuming $n=0.016$, determining the type of GVF profile, the critical depth, and the normal depth.
(Answer: M2, $y_{c}=0.24m$, $y_{n}=0.43m$)

The figure below shows a backwater curve in a long rectangular channel. Determine using a numerical integration method, the profile for the following high flow conditions: $Q=10m^{3}/s$, $b=3m$, $n=0.022$, and a bed slope of $0.001$. Take the depth just upstream of the dam as the control point equal to $5m$. At what distance is the water level not affected by the dam? perform your integration using a) 2-steps and b) 10-steps
(Answer: $y_{n}=2.437m$.
Using $y_{0}$ (Euler) 2-step $x=-3490m$, 10-step $x=-4724m$.
Using $y_{1/2}$ 2-step $x=-4480m$, 10-step $x=-5906m$.
Using a forth-order Runge Kutta method gave 2-step $x=-3893m$, 10-step $x=-5461m$)

Using the figure below, determine the profile for the channel conditions using a step length of $\Delta x=100m$. $Q=600m^{3}/s$, $n=0.04$, the bed slope of the rectangular channel is $S_{o}=0.002$ and has a width of $B=50m$. The sill height of the weir is $2.5m$ and the water depth over the weir is $4m$. Compare results from each method.