For a steady flow, the rating curve is unique for a non-erodible section where the flow is uniform, but it is a loop for an erodible section when the flow is non-uniform. For an unsteady flow, the rating curve does exist, but it is more complicated, Relationships among water level, flow velocity, and discharge are all affected by flood hydrograph of upstream or tide from downstream. A more general relationship between water level and flow velocity (or discharge) with flood hydrograph from upstream and tide from downstream for time-dependent is derived analytically from diffusion equation and continuity equation. The upstream and downstream boundary conditions are expressed in terms of harmonic functions rather than a step function. The analytical solutions are compared with the numerical results obtained by using finite difference model with implicit scheme based on the complete Sanit-Venant equations for unsteady flow in open channel. It is found from the study that: the peak flow times at different locations are shifted due to the kinematic wave velocity; therefore, the rating curves at different locations are spread, not complete a loop like peacock tail feather.
The rating curves for the subcritical flows are below the line of , and the supercritical flow rating curves are above . The dimensionless amplitude due to downstream tide is still a function of time not only function of position. The comparison between the analytical results and numerical results are in good agreement, not only for the weighting factor, Pt = 0.70, but also for Pt = 0.50. This analytical model can be used without any sophisticated computing machine; in fact, a simple desk calculator and a table of error function are sufficient in carrying out the computation based on the analytical solution.