How fast does the water travel down river when the discharge gates of a large dam are opened? Can the discharge-wave travel downstream faster than the water? Is this likely to occur for a real river and an actual sudden discharge from a large reservoir?
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$\begingroup$ A hint for one of your questions: if you've got a stone and you throw it in the river, will the ripples move downstream faster than the river flow? $\endgroup$– Carl WitthoftJan 3, 2014 at 19:57
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$\begingroup$ @CarlWitthoft Do we allow for hydraulic jumps? $\endgroup$– Isopycnal OscillationJan 3, 2014 at 20:00
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$\begingroup$ @IsopycnalOscillation Dunno--ask the OP :-) . $\endgroup$– Carl WitthoftJan 3, 2014 at 20:07
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$\begingroup$ @CarlWitthoft For OP's question, regarding a sudden release of a large volume of water such as in a large dam discharge, I believe that in the initial stages the fluid velocity would exceed the wave speed, much like in a hydraulic jump, as the orbital velocity of fluid particles is very large (wave breaking). For later stages, the wave speed would just depend on the depth through the linear wave speed relation for shallow water waves. $c = \sqrt{gD}$ where $D$ is the depth and $g$ is the gravitational acceleration. $\endgroup$– Isopycnal OscillationJan 3, 2014 at 20:15
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1$\begingroup$ Yes, they would be very different. Look up Stokes Drift - en.wikipedia.org/wiki/Stokes_drift $\endgroup$– Isopycnal OscillationJan 3, 2014 at 21:05
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1 Answer
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The Manning Formula is an empirical equation that describes uniform open channel flow. It depends upon several factors, including roughness and sinuosity of the river channel.
This paper: The Colorado River in Grand Canyon; how fast does it flow? describes some dye experiments performed in the Colorado River in the 1990s. Several releases (at different volumetric rates) were made from lake Powell and the time to reach different distances downriver were measured. These experiments showed that the water velocity down river was higher for the higher volume releases (~3 miles/hr for ~30,000 cubic feet/sec.) The discharge waves traveled faster than the water, traveling a distance of ~235 miles in ~1.6 days rather ~4.3 days.