where p is given by
| p² = (n³ − n) | T | . (5) |
| ρa³ |
If n = 1, the section remains circular, there is no force of restitution, and p = 0. The principal vibration, in which the section becomes elliptical, corresponds to n = 2.
Vibrations of this kind are observed whenever liquid issues from an elliptical or other non-circular hole, or even when it is poured from the lip of an ordinary jug; and they are superposed upon the general progressive motion. Since the phase of vibration depends upon the time elapsed, it is always the same at the same point in space, and thus the motion is steady in the hydrodynamical sense, and the boundary of the jet is a fixed surface. In so far as the vibrations may be regarded as isochronous, the distance between consecutive corresponding points of the recurrent figure, or, as it may be called, the wave-length of the figure, is directly proportional to the velocity of the jet, i.e. to the square root of the head. But as the head increases, so do the lateral velocities which go to form the transverse vibrations. A departure from the law of isochronism may then be expected to develop itself.
The transverse vibrations of non-circular jets allow us to solve a problem which at first sight would appear to be of great difficulty. According to Marangoni the diminished surface-tension of soapy water is due to the formation of a film. The formation cannot be instantaneous, and if we could measure the tension of a surface not more than 1⁄100 of a second old, we might expect to find it undisturbed, or nearly so, from that proper to pure water. In order to carry out the experiment the jet is caused to issue from an elliptical orifice in a thin plate, about 2 mm. by 1 mm., under a head of 15 cm. A comparison under similar circumstances shows that there is hardly any difference in the wave-lengths of the patterns obtained with pure and with soapy water, from which we conclude that at this initial stage, the surface-tensions are the same. As early as 1869 Dupré had arrived at a similar conclusion from experiments upon the vertical rise of fine jets.
A formula, similar to (5), may be given for the frequencies of vibration of a spherical mass of liquid under capillary force. If, as before, the frequency be p/2Π, and a the radius of the sphere, we have
| p² = n(n − 1) (n + 2) | T | , (6) |
| ρa³ |
n denoting the order of the spherical harmonic by which the deviation from a spherical figure is expressed. To find the radius of the sphere of water which vibrates seconds, put p = 2Π, T = 81, ρ = 1, n = 2. Thus a = 2.54 cms., or one inch very nearly.]
Tables of Surface-tension
In the following tables the units of length, mass and time are the centimetre, the gramme and the second, and the unit of force is that which if it acted on one gramme for one second would communicate to it a velocity of one centimetre per second:—