4.II.16G

A long straight canal has rectangular cross-section with a horizontal bottom and width $w(x)$ that varies slowly with distance $x$ downstream. Far upstream, $w$ has a constant value $W$, the water depth has a constant value $H$, and the velocity has a constant value $U$. Assuming that the water velocity is steady and uniform across the channel, use mass conservation and Bernoulli's theorem, which should be stated carefully, to show that the water depth $h(x)$ satisfies

$\left(\frac{W}{w}\right)^{2}=\left(1+\frac{2}{F}\right)\left(\frac{h}{H}\right)^{2}-\frac{2}{F}\left(\frac{h}{H}\right)^{3} \text { where } F=\frac{U^{2}}{g H}$

Deduce that for a given value of $F$, a flow of this kind can exist only if $w(x)$ is everywhere greater than or equal to a critical value $w_{c}$, which is to be determined in terms of $F$.

Suppose that $w(x)>w_{c}$ everywhere. At locations where the channel width exceeds $W$, determine graphically, or otherwise, under what circumstances the water depth exceeds $H .$

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