Q$2)$ In a rectangular channel with a bottom width of $B=4m,$ the critical water depth is determined as $y_{cr}=1m.$ The manning roughness parameter is $n=0\mathrm{.}015.$ The bottom slope of the channel takes three different values, as shown in Figure $1$: $S_{01}=0\mathrm{.}0009,S_{o2}=0\mathrm{.}09,$ and $S_{0}3=0\mathrm{.}0081.$

a$)$ Calculate the critical channel bottom slope.

b$)$ Calculate the uniform water depths and determine only the type of flows on three of the slopes.

c$)$ Show the water surface profiles on the slopes.

Calculate the water depth at a distance of $x=200m$ upstream of the first slope $S_{o1}=0\mathrm{.}0009($in three steps$).$