| λ = 2π √ | T | , |
| gρ |
and the minimum value is
| v = 4√ | 4 | T g | . |
| ρ |
For waves whose length from crest to crest is greater than λ, the principal force concerned in the motion is that of gravitation. For waves whose length is less than λ 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 ψ, 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
dψ = 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 2θ, the velocity of the body is w sec θ, 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 = λ/τ we have