Muskingum-Cunge Weighting Factor and Storage Constant

Plain Muskingum needs a gauge record to calibrate its weighting factor. The Muskingum-Cunge method works $X$ out from the channel itself: from the discharge $Q$, the top width $T$, the bed slope $S_0$, the flood wave speed $c$ and the routing distance step $\Delta x$. The storage constant comes with it, $K = \Delta x/c$, the time the wave takes to cover that step. The two go into the Muskingum routing step, which makes this the method for an ungauged stream.

The fraction inside the bracket compares the discharge with $T S_0 c \Delta x$, which is itself a flow. When the two are equal, $X = 0$. When $Q$ is smaller, $X$ is positive and approaches 0.5 as $Q$ shrinks. When $Q$ is larger, $X$ goes negative. So $X$ always comes out below 0.5, and, unlike the Muskingum range of 0 to 0.5, a negative value is a legitimate result here. The page's example is the equal case: a top width of 40 m, bed slope 0.001, wave speed 2 m/s and step 1,000 m make the product 80 m³/s, so $Q = 80$ gives $X = 0$ and $Q = 40$ gives $X = 0.25$.

The relations are equations 17-20 and 17-22 of the Natural Resources Conservation Service's National Engineering Handbook, Part 630, chapter 17. Its Example 17-4 uses them on a real reach: with $Q = 684.18$ ft³/s, $T = 44.6$ ft, $S_0 = 0.00091$ and $c = 4.297$ ft/s, the distance step is 2,735 ft, longer than the 2,600 ft reach, so the step is the reach and $X = -0.254$. In Example 17-5 the long floodplain reach is cut in half, to 1,300 ft, because the distance step of 1,736 ft is shorter than the reach, and $X = -0.590$.

Use one reference discharge, usually the peak of the inflow hydrograph, and take the top width and the wave speed at that flow. The wave speed is the input to watch: it comes from the channel's rating, and a rounding in it moves $X$; in Example 17-5 the printed value is $-0.590$, while the rounded $c = 1.779$ gives $-0.5894$. The same section gives the distance step as $\Delta x = \tfrac{1}{2}[c\,\Delta t + Q/(T S_0 c)]$ (equation 17-21), and a reach longer than that step is cut into equal shares.