where a is a small quantity, the axis of z being that of symmetry. The wave-length of the disturbance may be called λ, and is connected with k by the equation k = 2π/λ. The capillary tension endeavours to contract the surface of the fluid; so that the stability, or instability, of the cylindrical form of equilibrium depends upon whether the surface (enclosing a given volume) be greater or less respectively after the displacement than before. It has been proved by Plateau (vide supra) that the surface is greater than before displacement if ka > 1, that is, if λ < 2πa; but less if ka < 1, or λ > 2πa. Accordingly, the equilibrium is stable if λ be less than the circumference; but unstable if λ be greater than the circumference of the cylinder. Disturbances of the former kind lead to vibrations of harmonic type, whose amplitudes always remain small; but disturbances, whose wave-length exceeds the circumference, result in a greater and greater departure from the cylindrical figure. The analytical expression for the motion in the latter case involves exponential terms, one of which (except in case of a particular relation between the initial displacements and velocities) increases rapidly, being equally multiplied in equal times. The coefficient (q) of the time in the exponential term (eqt) may be considered to measure the degree of dynamical instability; its reciprocal 1/q is the time in which the disturbance is multiplied in the ratio 1 : e.
The degree of instability, as measured by q, is not to be determined from statical considerations only; otherwise there would be no limit to the increasing efficiency of the longer wave-lengths. The joint operation of superficial tension and inertia in fixing the wave-length of maximum instability was first considered by Lord Rayleigh in a paper (Math. Soc. Proc., November 1878) on the “Instability of Jets.” It appears that the value of q may be expressed in the form
| q = √ ( | T | ) · F(ka), (2) |
| ρa³ |
where, as before, T is the superficial tension, ρ the density, and F is given by the following table: —
| k²a². | F(ka). | k²a². | F(ka). |
| .05 | .1536 | .4 | .3382 |
| .1 | .2108 | .5 | .3432 |
| .2 | .2794 | .6 | .3344 |
| .3 | .3182 | .8 | .2701 |
| .9 | .2015 |
The greatest value of F thus corresponds, not to a zero value of k²a², but approximately to k²a² = .4858, or to λ = 4.508 × 2a. Hence the maximum instability occurs when the wave-length of disturbance is about half as great again as that at which instability first commences.
Taking for water, in C.G.S. units, T = 81, ρ = 1, we get for the case of maximum instability
| q-1 = | a3/2 | = .115d3/2 (3), |
| 81 × .343 |
if d be the diameter of the cylinder. Thus, if d = 1, q-1 = .115; or for a diameter of one centimetre the disturbance is multiplied 2.7 times in about one-ninth of a second. If the disturbance be multiplied 1000 fold in time, t, qt = 3loge 10 = 6.9, so that t = .79d3/2. For example, if the diameter be one millimetre, the disturbance is multiplied 1000 fold in about one-fortieth of a second. In view of these estimates the rapid disintegration of a fine jet of water will not cause surprise.
The relative importance of two harmonic disturbances depends upon their initial magnitudes, and upon the rate at which they grow. When the initial values are very small, the latter consideration is much the more important; for, if the disturbances be represented by α1eq1t, α2eq2t, in which q1 exceeds q2, their ratio is (α1/α2)e−(q1 − q2)t; and this ratio decreases without limit with the time, whatever be the initial (finite) ratio α2 : α1. If the initial disturbances are small enough, that one is ultimately preponderant for which the measure of instability is greatest. The smaller the causes by which the original equilibrium is upset, the more will the cylindrical mass tend to divide itself regularly into portions whose length is equal to 4.5 times the diameter. But a disturbance of less favourable wave-length may gain the preponderance in case its magnitude be sufficient to produce disintegration in a less time than that required by the other disturbances present.