A channel has a flat, horizontal bottom. Its straight, vertical bounding walls have a variable, and somewhat adjustable, width $w$ between those walls. We consider flow through a section that starts at constant width $w_o$ and then undergoes a smooth and gradual variation in stream width, first decreasing to $w_{\text{m}\text{i}\text{n}}$ and then gradually increasing back to the original width $w_o,$ with continuation as a straight channel of that width $w_o.$ Adopt a one$-$dimensional inviscid model for flow through this transition, letting $h$ be water depth and $u$ be velocity; they are $h_o$ and $u_o$ in the starting constant$-$width part of the channel.

$($a$)$ Show that the Froude number $F$ must vary with $w$ during the flow through the transition in such a way that

$w\frac{F}{(1+\frac{F^2}{2}{)}^{\frac{3}{2}}}=$ constant.

$($b$)$ Show that this expression predicts the smallest possible value of $w($which we call $w_{\text{c}\text{r}\text{i}\text{t}})$ when $F=1.$ Evaluate $w_{\text{c}\text{r}\text{i}\text{t}}$ in terms of $w_o$ and the Froude number $F_O$ of the flow before the channel constriction begins. $[$Ans$.$: $w_{\text{c}\text{r}\text{i}\text{t}}=(\frac{3}{2}{)}^{\frac{3}{2}}{w}_o\frac{F_o}{(1+\frac{F_o^2}{2}{)}^{\frac{3}{2}}}]$

$($c$)$ Suppose that $w_{\text{m}\text{i}\text{n}}>w_{\text{c}\text{r}\text{i}\text{t}},$ so that a solution exists with the assumed $F_O.$ If will $F,h,$ and $u$ increase or decrease from $F_o,h_o$ and $u_o$ as the thinnest cross section is approached? What if $F_O>1?$

$($d$)$ Now suppose that the walls of the channel are moved so as to make $w_{\text{m}\text{i}\text{n}}1?Is$ there a unique solution $in$ that case downstream $of$ the constriction? How about upstream?