On a hot afternoon in western North Carolina, the heavens opened and (to mix metaphors) it started raining “cats and dogs.” 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 s 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, he noted that for smooth cement channels (for which he performed experiments), C ≈ 2.5 × 10−3.

Question 1: (i) Estimate the speed v given that α ≈ 10° of arc.

(ii) Estimate the length of the crest-to-crest pattern.

Question 2: Linear stability analysis shows that the flow is stable (i.e., the amplitude of the waves does not increase with time) provided $v<2gh$. Is the flow stable?

Solution to Question 1: (i) α ≈ 10° of arc ≈ 0.17 rad, and the water flow was fast but not deep, and from the photograph I estimate h ≈ 2 mm = 2 × 10−3 m. Then $v≈[ 10×2×10−3×0.17/(2.5×10−3) ]12≈1m/s$.

(ii) The width of the Honda CRV is about 2 m, so the longer patterns at the top of the picture are ∼½ m: the nearer ones ∼0.1–0.2 m.

Solution to Question 2: $2gh≈210×2×10−3≈0.28m/s$, so the flow is not stable. Under these circumstances, the flow is unstable, but not completely chaotic or without structure, and manifests in the form of roll wave patterns, as here.

For more mathematical details, see Linear and Nonlinear Waves (1974) by G. B. Whitham (Section 3.2).