Example. $($Exam $2022)$

A long rectangular channel of width $2\mathrm{.}5$ m $,$ slope $0\mathrm{.}004$ and Manning's roughness coefficient $\backslash ($ n$=0\mathrm{.}022\backslash$mathrm$\{$~m$\}^\{-1/3\}\backslash$mathrm$\{$~s$\}\backslash )$ carries water at $\backslash (4\backslash$mathrm$\{$~m$\}^\{3\}\backslash$mathrm$\{$~s$\}^\{-1\}\backslash ).$ Temporary works narrow the channel at one location to $1\mathrm{.}1$ m for a short distance.

$($a$)$ Find the normal depth in the main channel and show that the slope is hydraulically mild.

$($b$)$ Show that a hydraulic transition takes place at the narrow point and find the depth just downstream of the narrowed section, confirming that supercritical flow is possible here.

$($c$)$ Use two steps in the gradually$-$varied$-$flow equation to estimate the distance from the end of the narrow section to the downstream hydraulic jump.