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## Q. 15.6

If a person throws a pebble into a 10 ft deep river ﬂowing at 1 ft/s (Figure 15.23), will the resulting ripples travel upstream? At what speed will the ripples travel in each direction?

## Verified Solution

For a small pebble, we can assume the waves (i.e., ripples) are of small amplitude. If that is the case, the value of the Froude number allows us to determine if a ripple can propagate upstream. A simple calculation yields $Fr=V/\sqrt{gy}=1\ \mathrm{ft} /s/ \sqrt{(32.2\ \mathrm{ft} /s^2)(10\ \mathrm{ft} )}=0.056$. Since Fr < 1, the ripples can travel upstream. For small amplitude waves, the wave speed in stationary ﬂuid of this depth is

$c_0=\sqrt{gy}=\sqrt{(32.2\ \mathrm{ft} /s^2)(10\ \mathrm{ft} )}=17.9\ \mathrm{ft} /s$

Thus, the ripples will propagate upstream on this river at c = c0 − V = 17.9 – 1 = 16.9 ft/s and downstream at c = c+ − V = 17.9 + 1 = 18.9 ft/s.