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