Water Waves Radiated by Coherent Vortices in a Submerged Jet

The physical problem - We consider a jet of water discharging from a nozzle submerged at a depth D below the free surface of a large body of water at rest, as in the experimental study of Bernal and Madnia (1989). The free surface is taken as the plane Z = 0 and the Z-axis points upward. The X-axis is chosen aligned with the jet and pointing in the same direction. The origin of the X-axis is taken directly above the nozzle, which thus is located at the point (X = 0, Y = 0, Z = -D). The jet is unstable to axially symmetric disturbances, which result in the formation of a periodic train of fairly coherent ring-like vortices. These vortices are created at a frequency co, and are convected downstream with a speed U roughly 70% of the jet exit velocity. The vortices grow in amplitude until nonlinear motions destroy their coherence, within 5 to 8 diameters of the jet exit, as is described in several experimental studies noted in Bernal and Madnia (1989), notably Yule (1978), Zaman and Hussain (1980), Hussain and Zaman (1980), Crow and Champagne (1971), and Hussain and Zaman (1981). An understanding of the free-surface disturbance created by a coherent periodic system of traveling vortices and observed in the experimental study of Bernal and Madnia (1989) is sought.