Page:EB1911 - Volume 05.djvu/287

For waves whose length is less than $\lambda$ the principal force concerned is that of surface-tension. Lord Kelvin proposed to distinguish the latter kind of waves by the name of ripples.

When a small body is partly immersed in a liquid originally at rest, and moves horizontally with constant velocity ${\mbox{V}}$, waves are propagated through the liquid with various velocities according to their respective wave-lengths. In front of the body the relative velocity of the fluid and the body varies from ${\mbox{V}}$ where the fluid is at rest, to zero at the cutwater on the front surface of the body. The waves produced by the body will travel forwards faster than the body till they reach a distance from it at which the relative velocity of the body and the fluid is equal to the velocity of propagation corresponding to the wave-length. The waves then travel along with the body at a constant distance in front of it. Hence at a certain distance in front of the body there is a series of waves which are stationary with respect to the body. Of these, the waves of minimum velocity form a stationary wave nearest to the front of the body. Between the body and this first wave the surface is comparatively smooth. Then comes the stationary wave of minimum velocity, which is the most marked of the series. In front of this is a double series of stationary waves, the gravitation waves forming a series increasing in wave-length with their distance in front of the body, and the surface-tension waves or ripples diminishing in wave-length with their distance from the body, and both sets of waves rapidly diminishing in amplitude with their distance from the body.

If the current-function of the water referred to the body considered as origin is $\psi$, then the equation of the form of the crest of a wave of velocity $w$, the crest of which travels along with the body, is

$d\psi =w\,ds$

where $ds$ is an element of the length of the crest. To integrate this equation for a solid of given form is probably difficult, but it is easy to see that at some distance on either side of the body, where the liquid is sensibly at rest, the crest of the wave will approximate to an asymptote inclined to the path of the body at an angle whose sine is $w/{\mbox{V}}$, where $w$ is the velocity of the wave and ${\mbox{V}}$ is that of the body.

The crests of the different kinds of waves will therefore appear to diverge as they get farther from the body, and the waves themselves will be less and less perceptible. But those whose wave-length is near to that of the wave of minimum velocity will diverge less than any of the others, so that the most marked feature at a distance from the body will be the two long lines of ripples of minimum velocity. If the angle between these is $2\theta$, the velocity of the body is $w\sec \theta$, where $w$ for water is about 23 centimetres per second.

[Lord Kelvin’s formula (1) may be applied to find the surface-tension of a clean or contaminated liquid from observations upon the length of waves of known periodic time, travelling over the surface. If $v=\lambda /\tau$ we have

${\mbox{T}}={\frac {\rho \lambda ^{3}}{2\pi \tau ^{2}}}\coth {\frac {2\pi h}{\lambda }}{\frac {g\lambda ^{2}\rho }{4\pi ^{2}}},$

(2)

$h$ denoting the depth of the liquid. In observations upon ripples the factor involving $h$ may usually be omitted, and thus in the case of water $\rho =1$)

${\mbox{T}}={\frac {\lambda ^{3}}{2\pi \tau ^{2}}}{\frac {g\lambda ^{2}}{4\pi ^{2}}}$

(3)

simply. The method has the advantage of independence of what may occur at places where the liquid is in contact with solid bodies.

The waves may be generated by electrically maintained tuning-forks from which dippers touch the surface; but special arrangements are needed for rendering them visible. The obstacles are (1) the smallness of the waves, and (2) the changes which occur at speeds too rapid for the eye to follow. The second obstacle is surmounted by the aid of the stroboscopic method of observation, the light being intermittent in the period of vibration, so that practically only one phase is seen. In order to render visible the small waves employed, and which we may regard as deviations of a plane surface from its true figure, the method by which Foucault tested reflectors is suitable. The following results have been obtained

Clean	74.0
Greasy to the point where camphor motions nearly cease	53.0
Saturated with olive oil	41.0
Saturated with sodium oleate	25.0

(Phil. Mag. November 1890) for the tensions of various water-surfaces at 18° C., reckoned in C.G.S. measure.

The tension for clean water thus found is considerably lower than that (81) adopted by Quincke, but it seems to be entitled to confidence, and at any rate the deficiency is not due to contamination of the surface.

A calculation analogous to that of Lord Kelvin may be applied to find the frequency of small transverse vibrations of a cylinder of liquid under the action of the capillary force. Taking the case where the motion is strictly in two dimensions, we may write as the polar equation of the surface at time $t$

$r=a+\alpha _{n}\cos n\theta \cos pt,$

(4)

where $p$ is given by

$p^{2}=(m^{3}-n){\frac {\mbox{T}}{\rho a^{3}}}$

(5)

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 ${\tfrac {1}{100}}$ 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 $p/2\Pi$, and a the radius of the sphere, we have

$p^{2}=n(n-1)(n+2){\frac {\mbox{T}}{\rho a^{3}}},$

(6)

$n$ 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 $p=2\Pi$, ${\mbox{T}}=81$, $\rho =1$, $n=2$. Thus $a=2.54{\mbox{ cms.}}$, 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:—