$5\mathrm{.}2$A rectangular channel, $5$ m wide, carries a flow of $20$ m$3/$s$.$ The slope of the channel S$0=0\mathrm{.}005$ m$/$m and Manning$$s n is $0\mathrm{.}030.$ The flow encounters a step of height $0\mathrm{.}50$ m$.$ The depths far upstream and downstream of the step are equal to the normal depth, y$0.$

a$)$ Calculate y$0.$

b$)$Determine if the reach is mild, steep, or critical.

c$)$Sketch all profiles both upstream and downstream of the obstructions.

d$)$On your sketch, indicate the upstream and downstream depths of each profile you draw and provide supporting calculations.

e$)$Sketch the cycle of points on the E$$y diagram for each of the upstream and downstream depths identified in part $($d$)$ and label sections of the cycle that correspond to any surface water profiles, jumps, or in$-$stream obstructions.

$5\mathrm{.}5$For the step and flow conditions specified in Problem $5\mathrm{.}2,$ design an equivalent:

Sluice gate opening

Constriction