These swellings separate more widely as they descend with increased rapidity; but falling through great heights, the whole may finally be dissipated in a mist.
Note.—The annular swellings contain air and arise from a periodical succession of pulsations near the orifice, which must be produced by very small oscillations of the entire mass of the liquid, so that the velocity of the flow is periodically variable. The sucking, whistling noise which is often heard in the descent of water through an orifice is caused by air drawn in by the whirling motion. [See Fig. 103].
If an orifice in a vessel looks downward, and the column of liquid over it be short, this will simply drop out by its own weight, starting at a velocity of o. But if a considerable depth of liquid be above, its gravity produces a corresponding pressure on its base, or on that liquid which is near it; so that, if a plug be removed from an orifice in or close to the base, the liquid starts at once into rapid motion.
Fig. 99.
Each particle of a jet A issuing from the side of a vessel moves horizontally with the velocity above mentioned, but it is at once drawn downward by the force of gravity in the same manner as a bullet fired from a gun, with its axis horizontal. It is well known that the bullet describes a parabola with a vertical axis, the vertex being the muzzle of the gun. Now, since each particle of the jet moves in the same curve, this jet C takes the parabolic form. In every parabola there is a certain point called the focus, and the distance from the vertex to the focus fixes the magnitude of a parabola in much the same manner as the distance from the center to the circumference fixes the magnitude of a circle.
Now it can be proved that the focus B is as much below as the surface of the water is above the orifice. Accordingly, if water issues through orifices which are small in comparison with the contents of the vessel, the jets from orifices at different depths below the surface take different forms, as shown at D. If these curves are traced on paper held behind the jet, then, knowing the horizontal distance and the vertical height, it is easy to demonstrate that the jet forms a parabola.
Quantity of Efflux.—If we suppose the bottom of a vessel containing water to be thin, and the orifice to be a small circle whose area is A [(see Fig. 100)] where A B represents an orifice in the bottom of a vessel.