A smooth hump with height $z=0\mathrm{.}25m$ is positioned on the bed of an inclined rectangular

channel, as schematically represented in Figure $1($not to scale$).$ The channel has width $B=$

$2$ m $,$ slope $S_0=0\mathrm{.}003$ and Manning $n=0\mathrm{.}03,$ and the discharge in the channel is $Q=1\frac{m^3}{s}.$

The flow at position $1($just upstream of the hump$)$ is subcritical and the flow at position $2$

$($over the hump$)$ is critical. Downstream the hump, the channel is long enough, so normal

flow conditions $($uniform flow$)$ are achieved at position $4.$ For the calculations restricted only

around the hump $($i$.$e$.,$ from position $1$ to position $3),$ related to tasks $($a$),($b$),($c$)$ and $($d$)$

below, you can assume, that the pressure distributions are hydrostatic at positions $1$ and $3$;

that local or friction losses can be neglected; and that the channel slope has no effect $($i$.$e$.,$

the elevations of the channel bed at positions $1$ and $3$ can be considered approximately the

same$).$ Complete the following tasks:

Note: For any tasks involving drawing, you must draw on you answer booklet $($with the white

pages$)$; any drawings you make in the exam paper will not be considered.

$($a$)$ Calculate the flow depth at position $2($i$.$e$.,$ the critical flow depth $y_c).$

$($b$)$ Calculate the flow depth $y_1$ at position $1.$

$($c$)$ Calculate the flow depth $y_3$ at position $3.$

$($d$)$ Calculate the horizontal force that the water exerts on the hump and indicate the

direction of this force.

$($e$)$ Calculate the normal flow depth $y_0$ at position $4$ and classify the slope of the channel.

$($f$)$ Copy Figure $1$ in your answer booklet and draw the following $($only schematically, not

required to be to scale$)$:

$($i$)$ The free$-$surface profile along the channel, from position $1$ to position $4,$

including any relevant features;

$($ii$)$ Indicate and label all regions and$/$or positions of critical, supercritical and

subcritical flow;

$($iii$)$ Indicate and label all regions of uniform, gradually varying and rapidly varying

flow. Figure $1.$ Schematic representation $($not to scale$)$ of the side view of the channel considered

in Question $5.$