at a station distant from the origin are recognized; the first corresponds
to the arrival of condensational waves, the second to that of
distortional waves, and the third to that of the Rayleigh waves (see
Elasticity).
The theory of waves diverging from a centre in an unlimited crystalline medium has been investigated with a view to optical theory by G. Green (1839), A. L . Cauchy (1830), E. B . Chrisroffel (1877) and others.
The surface which represents the wave-front consists of three sheets, each of which is propagated with its own special velocity.
It is hardly worth while to attempt an account here of the singularities of this surface, or of the simplifications which occur for various types of crystalline symmetry, as the subject has lost much of its physical interest now that the elastic-solid theory of light is practically abandoned.
§ 3. Water-Waves. Theory of "Long" Waves,
The simplest type of water-waves is that in which the motion of the particles is mainly horizontal, and therefore (as will appear) sensibly the same for all particles in a vertical line. The most conspicuous example is that of the forced oscillations produced by the action of the sun and moon on the waters of the ocean, and it has therefore been proposed to designate by the term " tidal " all cases of wave-motion, whatever their scale, which have the above characteristic property.
Beginning with motion in two dimensions, let us suppose that the axis of X is drawn horizontally, and that of y vertically upwards. If we neglect the vertical acceleration, the pressure at any point will have the statical value due to the depth below the instantaneous position of the free surface, and the horizontal pressure-gradient dpidx will therefore be independent of y. It follows that all particles which at any instant lie in a plane perpendicular to Ox will retain this relative configuration throughout the motion. The equation of horizontal motion, on the hypothesis that the velocity (m) is infinitely small, will be
du
dp
dit
^= -51= -^'^
(')
where rj denotes the surface-elevation at the point *. Again, the
equation of continuity, viz..
du1dv
iu+a-y=°
(2)
gives
C'du,
du
if the origin be taken at the bottom, the depth being assumed to be
uniform. At the surface we have y = h+i], and v = dn/dt, subject to
an error of the second order in the disturbance. To this degree of
approximation we have then
(4)
at
~
"dx-
(5)
If we eliminate u between (i) and (4) we obtain ap~'^dx
with
c'=gh
(6)
The solution is as in § I, and represents two wave-systems travelling with the constant velocity V(g/i). which is that which would be acquired by a particle falling freely through a space equal to half the depth.
Two distinct assumptions have been made in the foregoing investigation.
The meaning of these is most easily understood if we consider the case of a simple-harmonic train of waves in which V=0 cosset- x), u=^cosk(, ct-x), (7)
where fe is a constant such that 2!r/;fe is the wave-length X, The first assumption, viz. that the vertical acceleration may be neglected in comparison with the horizontal, is fulfilled if kh be small, i.e if the wave-length be large compared with the depth. It is in this sense that the theory is regarded as applicable only to " long " waves.
The second assumption, which neglects terms of the second order in forming the equation (i), implies that the ratio n/h of the surface elevation to the depth of the fluid must be small. The formulae (7) indicate also that Ln a progressive wave a particle moves forwards or backwards according as the water-surface above it is elevated or depressed relatively to the mean level. It may also be proved that the expressions
T = hphfu'-dx, V = gf>Jrfdx,
(8)
for the kinetic and potential energies per unit breadth are equal in the case of a progressive wave. It will be noticed that there is a very close correspondence between the theory of " long " water-waves and that of plane waves of sound, e.g. the ratio nlh corresponds exactly to the " condensation "
in the case of air-waves.
The theory can be adapted, with very slight adjustment, to the case of waves propagated along a canal of imy uniform section, provided the breadth, as well as the depth. be srnall compared with the wave-length. The principal change is that in (6) h must be understood to denote the mean depth. The theor); was further e;<tended by G. Green (1837) and by Lord Rayleigh to the case where the dimensions of the cross-section are variable. If the variation be sufficiently gradual there is no sensible reflection, a progressive wave travelling always with the velocity appropriate to the local mean depth. There is, however, a variation of amplitude; the constancy of the energy, combined with the equation of continuity, require that the elevation 17 in any particular part of the wave should vary as b-ih-i, where 6 is the breadth of the water surface and h is the mean depth. Owing to its mathematical simplicity the theory of long waves in canals has been largely used to illustrate the dynamical theory of the tides.
In the case of forced waves in a uniform canal, the equation (l) is replaced by
du
dij, ..
where X represents the extraneous force. In the case of an equatorial canal surrounding the earth, the disturbing action of the moon, supposed (for simplicity) to revolve in a circular orbit in the plane of the equator, is represented by X = -^sin2(a<-t -f-|-0,
(10)
where a is the earth's radius, H is the total range of the tide on the " equilibrium theory, " and a is the angular velocity of the moon relative to the rotating earth. The corresponding solution of the equations (4) and (9) is
iHira
X.
(n)
The coefficient in the former of these equations is negative unless the ratio h/a exceed a^a/g, which is about 1/31 1. Hence unless the
depth of our imagined canal be much greater than such depths as are actually met with in the sea the tides in it would be inverted. I.e. there would be low water beneath the moon and at the antipodal point, and high water on the meridian distant 90° from the moon. This is an instance of a familiar result in the theory of vibrations, viz. that in a forced oscillation of a body under a periodic force the phase is opposite to that of the force if the imposed frequency exceed that of the corresponding free vibration (see Mechanics). In the present case the period of the free oscillation in an equatorial canal 11,250 ft. deep would be about 30 hours. When the ratio j)//i of the elevation to the depth is no longer treated as infinitely small, it is found that a progressive wave system must undergo a continual change of type as it proceeds, even in a uniform canal. It was shown by Sir G. B . Airy (1845) that the more elevated portions of the wave travel with the greater velocities, the expression for the velocity of propagation being approximately. Hence the slopes will become continually steeper in front and more gradual behind, until a stage is reached at which the vertical acceleration is no longer negligible, and the theory ceases to apply. The process is exemplified by sea-waves running inwards in shallow water near the shore. The theory of forced
periodic waves of finite (as distinguished from infinitely small) amplitude was also discussed by Airy. It has an application in
tidal theory, in the explanation of " over tides " and " compound tides " (see Tide).
§ 4. Surface-Waves.
This is the most familieir type of water-waves, but the theory is not altogether elementary. We will suppose in the first instance that the motion is in two dimensions x, y, horizontal and vertical respectively. The velocity-potential (see HYDROiiECHANics) must satisfy the equation
(0
and must make d<t>/dy=o at the bottom, which is supposed to be plane and horizontal. The pressure-equation is, if we neglect the square of the velocity,
-
37 -«>'+ const.
(2)
Hence, if the origin be taken in the undisturbed surface, we may write, for the surface-elevation, "•m, ..
gL3i.
(3)
with the same approximation.
We have also the geometrical
condition
dr,
d±
dl° ldyy^o
(4)
The general solution of these equations is somewhat complicated.
