Figure $7\mathrm{.}2$ shows a rectangular channel, contracting smoothly from a width of $27\mathrm{.}8$m to $18\mathrm{.}2$ m$.$ With a flow rate of $39\mathrm{.}9$m$^(3)$s$^(-1),$ the water velocity upstream from the contraction is $0\mathrm{.}9$ ms$^(-1).$ Neglecting energy losses, determine the depth and velocity in the contraction. Find the depth in the contraction if the energy loss is $0\mathrm{.}13$m$.$

Water depth at section $1$

Critical depth at section $1$

Critical depth at section $2$

Water depth at section $2$ without energy loss

Water depth at section $2$ with energy loss