(18)
Fig. 4.
—
iSinS-,
wave-length X at points to the right of O is uniform, being that
proper to a wave- velocity c, viz. X = 27rcVg. The disturbance is
therefore followed by a train of waves of approximately simple harmonic
profile, of the length indicated. An approximate calculation
shows that, e.xcept in the immediate neighbourhood of the
source of disturbance, the surface-elevation is given by
2P0. er
"=^1?='"!?.
•
•
•
where x is now measured from O, and Poi^fpdx) represents the
integral of the disturbing surface pressure over the (infinitely small)
breadth of the band
on which it acts. The
case of a diffused
pressure can be iny.
ferred by integration.
The annexed figure
gives a representation
of a particular case,
obtained by a more
exact process.
The
pressure is here supposed
uniformly distributed over a band of breadth AB.
A similar argument can be applied to the case of finite depth (h).
but since the wave-velocity cannot exceed ij{2gh) the results are
modified if the velocity c of the travelling pressure exceeds this
limit. There is then no train of waves generated, the disturbance
of level being purely local.
It hardly needs stating that the investigation
applies also to the case of a stationary surface disturbance
on a running stream, and that similar results follow when the
disturbance consists in an equality of the bottom. In both cases we
have a train of standing waves on the down-stream side, of length
corresponding to a wave-velocity equal to that of the stream.
The effect of a disturbance confined to the neighbourhood of a
paint of the surface (of deep water) was also included in the investigations
of Cauchy and Poisson already referred to.
The
formula analogous to (12), in the case of a local impulse, is
•
•
•
(19)
where r denotes distance from the source.
The interpretation is
similar to that of the two-dimensional case, except that the amplitude
of the annular waves diminishes outwards, as was to be expected, in
a higher ratio.
The effect of a pressure-point travelling in a straight line over
the surface of deep water is interesting, as helping us to account
in some degree for the peculiar system of waves which is seen to
accompany a ship. The configuration of the wave-system is shown
by means of the lines of equal phase in the annexed diagram, due to
V. W . Ekman (1906), which
differs from the drawing originally
given by Lord Kelvin (1887)
in that it indicates the difference
of phase between the
transverse and diverging waves
at the common boundary of the
two series. The two systems of
waves are due to the fact that
at any given instant there are
too previous positions of the
moving pressure-point
which
have transmitted vibrations of
After V Walfrid Ekman, 0» S/aKoiwrjr stationary phase to any given
Waves, n Runmns Water.
p^j^^^ p ^.^^.^ ^^^ ^^^^^ ^^, j^^
riG. 5.
figure. When the depth is finite
the configuration is modified, and if it be less than c'/g, where c is
the velocity of the disturbance, the transversal waves disappear.
The investigations referred to have a bearing on the wave-resistance
of ships.
This is accounted for by the energy of the new wave groups
which are continually being started and left behind. Some
experiments on torpedo boats moving in shallow water have indicated
a falling off in resistance due to the absence of transversal
waves just referred to.
For the effect of surface-tension and the
theory of " ripples " see Capillary Action.
§ 5. Surface-Waves of Finite Height.
The foregoing results are based on the iissumption that the
amplitude may be treated as infinitely small. Various interesting
investigations have been made in which this restriction is, more or
less, abandoned, but we are far from possessing a complete theory.
A system of exact equations giving a possible type of wave motion
on deep water was obtained by F. J . v . Gerstner in 1802, and
rediscovered by W. J. M. Rankine in 1863. The orbits of the
particles, in this type, are accurately circular, being defined by the
equations
x=a- -k -^e'sink{a-ct), y = b — k -^e'cosk(, a-ct),
(i) where (a, b) is the mean position of the particle, k = 2Trl; and the wave-velocity is ^ = V(g/*) = V(gX/2^). -.
(2)
xxvin 8
The lines of equal pressure, among which is included of course the
surface-profile, are trochoidal curves. The extreme form of waveprofiie
is the cycloid, with the cusps turned upwards. The mathe-FiG.
6.
matica! elegance and simplicity of the formulae (i) are unfortunately
counterbalanced by the fact that the consequent motion of the
fluid elements proves to be " rotational " (see Hydromechanics),
and therefore not such as could be generated in a previously quiescent
liquid by any system of forces applied to the surface.
Sir G. Stokes, in a series of papers, applied himself to the determination
o! the possible " irrotational " wave-forms of finite height
which satisfy the conditions of uniform propagation without change
of type. The equation of the profile, in the case of infinite depth, is
obtained in the form of a l-ourier series, thus
y = aco5kx+ka'^cos2kx+ik^a'cosikx+ . . .,
(3) the corresponding wave-velocity being appro.ximately ^-vm+m' - ^^) where X=27r/fe. The equation (3), so far as we have given the development, agrees with that of a trochoid (fig. 7). As in the case of Gerstner's waves the outline is sharper near the crests and flatter
in the troughs than in the case of the simple-Fig. 7. harmonic curve, and these features become accentuated as the ratio of the amplitude to the wave-length increases. It has been shown by Stokes that the extreme form of irrotational waves differs from that of the rotational Gerstner waves in that the crests form a blunt angle of 120°. Ac- cording to the calculations of J. H . Michell (1893), the height is then about one-seventh of the wave-length, and the wave-velocity exceeds that of very low waves of the same length in the ratio 6:5. It is to be noticed further that in these waves of permanent type the motion of the water-particles is not purely oscillatory, there being on the whole a gradual drift at the surface in the direction of propagation. These various conclusions appear to agree in a general way with what is obscr'ed in the case of sea-waves. in the case of finite depth the calculations are more difficult, and we can only here notice the limiting type which is obtained when the wave-length is supposed very great
compared with the depth (h). We have then practically the " solitary wave " to which attention was first directed by J. Pjc 8 Scott Russell (1844) from observation. The theory has been worked out by J. Boussinesq (1871) and Lord Rayleigh. The surface-elevation is given by V^asecWhMb), (5) provided 6' =/!=(/! -fa)/3o.
(6) and the velocity of propagation is c = ^lg{h+a)] (7) In the extreme form a = h and the crest forms an angle of 120°. It appears that a solitary wave of depression, of permanent type, is impossible. Bibliography. — Experimental researches: E. H . u . W . Weber, Wellenlehre (Leipzig, 1825); J. Scott Russell, " Report on Waves, " Brit. Assoc. Rep. (1844). Theoreticalv, 'orks:S. D . Poisson, " Memoire sur la theorie du son, " /. de I'ecole polyl. 7 (1807); " Mem. sur la theorie des ondes, " Mem. de I'acad. roy. des sc. I (1816); A. Cauchy. " Mel?i. sur la theorie des ondes, " Mem. de I'acad. roy. des sc. (1827); Sir G. B . Airy, " Tides and Waves, " Er.cyd. Metrop. (1845). Many classical investigations are now most conveniently accessible 2a
