d^2y 4[pi]^2
p = - T ---- = ---------- Ty.
dx^2 [lambda]^2
This pressure must be added to the pressure due to gravity g [rho] y.
Hence the waves will be propagated as if the intensity of gravity had
been
4[pi]^2 T
f = g + ---------- -----
[lambda]^2 [rho]
instead of g. Now it is shown in hydrodynamics that the velocity of
propagation of waves in deep water is that acquired by a heavy body
falling through half the radius of the circle whose circumference is the
wave-length, or
f[lambda] g[lambda] 2[pi]T
v^2 = --------- = --------- + -------------. (1)
2[pi] 2[pi] [rho][lambda]
This velocity is a minimum when
______
/ T
[lambda] = 2 [pi] / ------,
\/ g[rho]
and the minimum value is
_______
4 / Tg
v = / 4 -----.
\/ [rho]
For waves whose length from crest to crest is greater than [lambda], the
principal force concerned in the motion is that of gravitation. 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 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 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/V, where w is the velocity of the wave and 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
[rho][lambda]^3 2[pi]h g[lambda]^2[rho]
T = --------------- - coth-------- - ---------------- (2)
2[pi][tau]^2 [lambda] 4[pi]^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)
[lambda]^3 g[lambda]^2
T = ------------ - ------------ (3)
2[pi][tau]^2 4[pi]^2
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 deg. 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 + a_n cos n[theta] cos pt, (4)
where p is given by
T
p^2 = (n^3 - n)--------. (5)
[rho]a^3
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 Dupre 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
T
p^2 = n(n - 1)(n + 2)--------, (6)
[rho]a^3
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], T = 81, [rho] = 1, n =
2. Thus a = 2.54 cms., or one inch very nearly.]