On a hot afternoon in western North Carolina, the heavens opened and (to mix metaphors) it started raining “cats and dogs.”1 The rain was torrential, and soon a river of water was flowing downhill into the parking lot. In rivers and dam spillways, these are known as flood waves. The picture shows a particular type of flood waves known as roll waves—shock-like patterns separated by smooth profiles. In a steady-state situation, the drag forces on the water in the channel are balanced by the down-slope gravitational force. The dimensionless friction coefficient is C, the stream depth is h, v is the average speed of the water, g is the gravitational acceleration, and α is the angle of the slope (assumed to be small so that sin αα). In a steady state, Cv2 = ghα.

In an early paper by Harold Jeffreys,2...

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