Question $1(35$ marks$)$

A rectangular channel is $5m$ wide with two reaches with slopes of $0\mathrm{.}01$ for the first reach and then a sudden change to a slope of $0\mathrm{.}001$ for the second reach. This channel carries a flow of $80\frac{m^3}{s}$ and has a Manning's roughness $($n$)$ of $0\mathrm{.}015.$

a$)$ What is the at normal depth $y_n$ at each slope of the channel?

b$)$ What is the critical depth of this channel $y_c?$

$3$

c$)$ Prove there is a hydraulic jump, explain in terms of the definition of hydraulic jumps, $y_n,y_c$ and $y.$

d$)$ What is the conjugate depth of the hydraulic jump?

$6$

e$)$ Does the hydraulic jump appear on the first or second reach? What type of water profile would be present?

$2$