A long straight canal has rectangular cross-section with a horizontal bottom and width that varies slowly with distance downstream. Far upstream, has a constant value , the water depth has a constant value , and the velocity has a constant value . 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 satisfies
Deduce that for a given value of , a flow of this kind can exist only if is everywhere greater than or equal to a critical value , which is to be determined in terms of .
Suppose that everywhere. At locations where the channel width exceeds , determine graphically, or otherwise, under what circumstances the water depth exceeds