Question $(1)$

a$)$ A $2-$h unit hydrograph is given by:

$U(t)=0\mathrm{.}5c\frac{m}{h}$ for $0t2h$

$U(t)=0$ for $t3h$

Determine:

$($i$)$ the S$-$curve corresponding to the given $2-$h UH

$($ii$)$ Using the S$-$curve developed above, determine the $4-$h unit hydrograph

b$)$ The rainfall excess of a storm is modelled as:

$I(t)=6c\frac{m}{s}$ for $0=4h$

The corresponding direct runoff hydrograph is expressed in terms of depth over unit catchment area per hour $(c\frac{m}{h})$ as:

$Q(t)=6\mathrm{.}0tc\frac{m}{h},$ for $0t4h$

$Q(t)=0,$ for $t>8$

where $t$ is in hours. Determine:

$($i$)4-$h unit hydrograph of the catchment and corresponding S$-$curve of the catchment

$($ii$)3-$h unit hydrogen of the catchment.