Flood Wave Celerity from the Rating Curve

A flood does not travel at the speed of the water in the river. The rise in flow moves as a wave, and this page gives the speed of that wave, $c$, from the channel's rating relation between the discharge $Q$ and the flow area $A$. If the rating is a power curve, $Q = x A^{m}$, the wave speed is the slope of that curve, $c = dQ/dA$, and the slope works out to $m$ times the average velocity: $c = m Q/A = m V$. So the exponent $m$ says how many times faster than the water the wave runs.

In the National Engineering Handbook of the Natural Resources Conservation Service (Part 630, chapter 17), the exponent comes from the rating itself: it is the slope of the rating on log-log paper, found from rating table data as in Table 17-13, and it is not allowed to fall below 1.0. For a wide channel with Manning's equation, $m = 5/3$, so a flood wave travels two-thirds faster than the water it is made of. The page's example uses that case: 10 m³/s over 5 m² is water moving at 2 m/s, so the wave runs at 10/3 m/s, or 3.3333 m/s.

The handbook's worked examples use the same relation at a reference discharge. In Example 17-4, $m = 1.3873$, $Q = 684.18$ ft³/s and $A = 220.9$ ft² give $c = 4.297$ ft/s, and read backwards the exponent is 1.3874 against the printed 1.3873. In Example 17-5, $m = 1.0276$, so 3,385.4 ft³/s over 1,956 ft² gives a wave at 1.779 ft/s, only a little faster than the water. In Table 17-13 the rating has $m = 1.5$ for discharges up to 500 ft³/s, and 200 ft³/s goes with 110 ft², so $c = 1.5 \times 200 / 110 = 2.727$ ft/s.

Two things limit how far to trust the result. The relation $c = m V$ holds only as far as the power-law rating $Q = x A^{m}$ holds, so use the exponent over the flows its rating covers; Table 17-13 gives its $m$ for discharges up to 500 ft³/s. And $Q$ and $A$ must belong to the same flow, the reference stage, since $c$ is the slope of the rating at that point. The wave speed is what the Muskingum-Cunge method needs to find the storage constant and the weighting factor of a reach.