Consider flow in a stream with depth $h_o=4m$ and Froude number $F_o=\frac{1}{8}.$ In what follows we will consider the effect of a gradual, local increase in elevation of the streambedfrom an otherwise $($relatively$)$ flat profile. The local increase will become progressive larger as we progress through the following problems $(a-c).$ Panels $(a-c)$ in the problem figure above correspond to the sub$-$problems $(a-c)$ below.

$($a$)$ First consider the relatively modest maximum bump height of $b_{\text{m}\text{a}\text{x}}=1m,$ illustrated in the top panel in the problem figure. Find the change in water surface elevation over the bump maximum, relative to a nearby upstream value. Do you expect the flow downstream of the bump to remain unperturbed, as illustrated in the top panel? In other words, what is the fluid velocity $u,$ upstream, over the bump maximum, and downstream?

$($b$)$ Suppose $b_{\text{m}\text{a}\text{x}}$ increases to $2\mathrm{.}5m,$ should we expect the surface elevation to change as illustrated? Support your response. What is the Froude number and water depth downstream: $F_{\text{e}\text{x}\text{i}\text{t}}$ and $h_{\text{e}\text{x}\text{i}\text{t}}?$ What is the Froude number of the flow over the bump maximum? Also, what is the fluid velocity $u,$ upstream, over the bump maximum, and downstream?

$($c$)$ Now suppose $b_{\text{m}\text{a}\text{x}}$ increases to $3m.$ Why is the original upstream flow$-$with $F_o=\frac{1}{8},h_o=4m-$no longer admissible? Find the new upstream conditions $F_n$ and $h_n.$ Again, what is the fluid velocity $u,$ upstream, over the bump maximum, and downstream?