$3.)$ a$)$ A rectangular cross$-$section channel system with a constant flow rate shown in the figure below consists of four different channel slopes.

a$1.$ Determine the flow regimes at the different sections of the channel and draw the water surface profiles along the channel.

a$2.$ Indicate the possible observable types of water surface profiles along the channel on the figure.

Open Channel profile

b$.$ For a gradually varied flow of type $A_3,$ determine the sign of the slope $\frac{dy}{dx}$ at various locations along the channel using the dynamic equation of the gradually varied flow $\frac{dy}{dx}=\frac{I_o-J}{1-N_{Fr}^2}.$ Evaluate the limiting values mathematically. Draw the water surface profile on a figure according to your mathematical evaluation. $N_{Fr}$ is the Froude number.$)$