$15\mathrm{.}23.$ A broad$-$crested weir is located in a $40-$m$-$wide rectangular channel, as shown in Figure P$15\mathrm{.}23.$ The channel conveys a discharge of $500\frac{m^3}{s}$ under a bed slope of $1\mathrm{.}8{10}^{-3}.$ The Manning roughness

$s_0-{1\mathrm{.}0}^{{?}^{?}}10-\frac{111}{11}$

Figure P$15\mathrm{.}23$ A broad$-$crested weir for Problem $15\mathrm{.}23.$

coefficient, $n,$ is $0\mathrm{.}032.$ The sill height of the weir is $2\mathrm{.}0m.$ The weir discharge is given as

$Q=1\mathrm{.}705C_{d}B{h}^{\frac{3}{2}}$

where $B$ is the weir width, $C_d$ is the discharge coeffcient $=0\mathrm{.}86,$ and $b$ is the water head above the weir. Determine the length of the backwater curve using the step$-$by$-$step method.