A concrete channel, manning roughness n $=0\mathrm{.}013,$ rectangular section, basal width b $=3\mathrm{.}5[$m$]$ and slope i $=0\mathrm{.}0009$ carries a flow of $12[$m$3/$s$].$ In its course it meets an abrupt widening with a downstream gradient of dimension a$1$ and new basal width b $=4[$m$].$ At a distance L downstream of the abrupt widening there is an abrupt flow addition of $5[$m$3/$s$]$ from a side channel. At a distance of $200[$m$]$ downstream of the inflow there is a new abrupt enlargement only with a downstream gradient of dimension a$2=0\mathrm{.}5[$m$].$

$($a$)$ What is the minimum length L for normal runoff head to occur in that reach?

$($b$)$ Calculate the maximum allowable step height a$1$ so as not to cause a free fall, verify the answer.

answer.

$($c$)$ Select a value of a$1$ lower than the maximum head a$1$ obtained in $($b$)$ and continue working upstream.

$($d$)$ Determine the complete hydraulic axis.