Consider waves made by dropping objects (one after another

Chapter 10, Problem 10.6

(choose chapter or problem)

Consider waves made by dropping objects (one after another from a fixed location) into a stream of depth y that is moving with speed V as shown in Fig. P10.6 (see Video V10.5). The circular wave crests that are produced travel with speed \(c = (gy)^{1/2}\) relative to the moving water. Thus, as the circular waves are washed downstream, their diameters increase and the center of each circle is fixed relative to the moving water. (a) Show that if the flow is supercritical, lines tangent to the waves generate a wedge of half-angle \(\alpha/2 =\arcsin(1/\mathrm{Fr})\), where \(\mathrm{Fr} =V/(gy)^{1/2}\) is the Froude number. (b) Discuss what happens to the wave pattern when the flow is subcritical, Fr < 1.