Economic Shape of Trapezoidal Open Channel Section with Freeboard

It can be proved mathematically that the maximum flow by the Manning formula is an open channel of trapezoidal cross section of fixed side slopes and with a fixed freeboard occurs r=(5/2)fx/q, in which r = the hydraulic radius of the wetted section; x = the bottom width of channel section; and q=2L-W, the difference between the total length of the side slopes, 2L, and the top width, W, of the excavated section; and f, herein called freeboard, is the vertical distance between the present or undisturbed ground surface and the proposed water surface in the finished channel. It is assumed to be below the original ground surface.