$**$ Problem $6\mathrm{.}5$

Consider the Missouri River flow in Figure P$5\mathrm{.}2,$ controlled by clear water releases from a reservoir. Assume a constant Manning $n$ and a constant cross section without lateral inflow along the reach and determine the following: $($a$)$ the celerity $c$ versus $Q$ from the relation between the cross$-$section area and discharge; $($b$)$ the celerity of the wave generated

Unsteady Flow in Rivers

from a sudden increase in discharge by $10,000f\frac{t^3}{s}$; $($c$)$ estimate the relative magnitude of the terms of the Saint$-$Venant equation when the discharge increases from $25,000$ to $35,000f\frac{t^3}{s}$ in $1$day; $($d$)$ what approximation of the Saint$-$Venant equation is best suited for flood$-$wave routing?; and $($e$)$ what is the flow discharge at a cross section located $10$ miles upstream when the discharge is $Q=31,600f\frac{t^3}{s}$ and the water level rises at a rate of $4ft$ per day?