Water flows through a rectangular channel that is $3\mathrm{.}00m$ wide, having a slope of $0\mathrm{.}50\%,$ and Manning's $n$ value of $0\mathrm{.}035.$ A vertical gate is placed into the channel, across the full width of the channel, and water pools upstream of the gate to a depth of $4\mathrm{.}00m.$ The gate is opened to a height of $0\mathrm{.}50m,$ and water continues to contract in depth downstream of the gate, with $C_c=0\mathrm{.}60.$

Downstream of the gate a hydraulic jump forms. Determine:

a$)$ The discharge coefficient that applies to the gate, $Cd$

b$)$ The flow rate $Q$ that passes through the gate, given its geometry and the flow conditions described.

c$)$ The normal depth that should exist in the channel, given its geometry, slope, roughness, and the calculated flow rate.

d$)$ The upstream conjugate depth that exists just prior to the hydraulic jump, assuming that the downstream conjugate depth is the channel's normal depth calculated in part $c.$

e$)$ What is the water surface profile classification that exists upstream of the gate?

f$)$ What is the water surface profile classification that exists downstream of the gate, prior to the hydraulic jump?