(=H"" sin2 8. cos (σt +20 +€),

Lastly, we have the semi-diurnal tides, for which

.(15)*

where σ differs but little from 2w. These include the 'lunar semi-diurnal' (σ = 28°·984), the 'solar semi-diurnal' (σ = 30°), and the 'luni-solar semi-diurnal' (σ = 2w = 30°·082).

For a complete enumeration of the more important partial tides, and for the values of the coefficients H', H", H"" in the several cases, we must refer to the investigations of Darwin, already cited. In the Harmonic Analysis of Tidal Observations, which is the special object of these investigations, the only result of dynamical theory which is made use of is the general principle that the tidal elevation at any place must be equal to the sum of a series of simple-harmonic functions of the time, whose periods are the same as those of the several terms in the development of the disturbing potential, and are therefore known à priori. The amplitudes and phases of the various partial tides, for any particular port, are then determined by comparison with tidal observations extending over a sufficiently long period†. We thus obtain a practically complete expression which can be used for the systematic prediction of the tides at the port in question.

f. One point of special interest in the Harmonic Analysis is the determination of the long-period tides. It has been already stated that under the influence of dissipative forces these must tend to approximate more or less closely to their equilibrium values. Unfortunately, the only long-period tide, whose coefficient can be inferred with any certainty from the observations, is the lunar fortnightly, and it is at least doubtful whether the dissipative forces are sufficient to produce in this case any great effect in the direction indicated. Hence the observed fact that the fortnightly tide has less than its equilibrium value does not entitle us to make any inference as to elastic yielding of the solid body of the earth to the tidal distorting forces exerted by the moon ‡.

* It is evident that over a small area, near the poles, which may be treated as sensibly plane, the formulae (14) and (15) make

5x rcos (ot++), and respectively, where r, w are plane polar co-ordinates. in Arts. 211, 212.

r2 cos (ot+2+€),

These forms have been used by anticipation

† It is of interest to note, in connection with Art. 187, that the tide-gauges, being situated in relatively shallow water, are sensibly affected by certain tides of the second order, which therefore have to be taken account of in the general scheme of Harmonic Analysis.

Darwin, l.c. ante p. 323. See, however, the paper by Rayleigh cited on p. 343 ante.

CHAPTER IX

SURFACE WAVES

227. WE have now to investigate, as far as possible, the laws of wavemotion in liquids when the vertical acceleration is no longer neglected. The most important case not covered by the preceding theory is that of waves on relatively deep water, where, as will be seen, the agitation rapidly diminishes in amplitude as we pass downwards from the surface; but it will be understood that there is a continuous transition to the state of things investigated in the preceding chapter, where the horizontal motion of the fluid was sensibly the same from top to bottom.

We begin with the oscillations of a horizontal sheet of water, and we will confine ourselves in the first instance to cases where the motion is in two dimensions, of which one (x) is horizontal, and the other (y) vertical. The elevations and depressions of the free surface will then present the appearance of a series of parallel straight ridges and furrows, perpendicular to the plane xy.

The motion, being assumed to have been generated originally from rest by the action of ordinary forces, will necessarily be irrotational, and the velocity-potential will satisfy the equation

To find the condition which must be satisfied at the free surface (p = const.), let the origin O be taken at the undisturbed level, and let Oy be drawn vertically upwards. The motion being assumed to be infinitely small, we find, putting = gy in the formula (4) of Art. 20, and neglecting the square of the velocity (q),

η

Hence if n denote the elevation of the surface at time t above the point (x, 0), we shall have, since the pressure there is uniform,

provided the function F (t), and the additive constant, be supposed merged in the value of dø/dt. Subject to an error of the order already neglected, this may be written.

106
g Lat

.(5)

Since the normal to the free surface makes an infinitely small angle (on/ox) with the vertical, the condition that the normal component of the fluid velocity at the free surface must be equal to the normal velocity of the surface itself gives, with sufficient approximation,

This is in fact what the general surface condition (Art. 9 (3)) becomes, if we put F (x, y, z, t) = y — ŋ, and neglect small quantities of the second order. Eliminating between (5) and (6), we obtain the condition

η

to be satisfied when y = 0. This is equivalent to Dp/Dt = 0.

.(7)

In the case of simple-harmonic motion, the time-factor being eit+e), this condition becomes

дф 02 = 9 ay

.(8)

.

228. Let us apply this to the free oscillations of a sheet of water, or a straight canal, of uniform depth h, and let us suppose for the present that there are no limits to the fluid in the direction of x, the fixed boundaries, if any, being vertical planes parallel to xy.

Since the conditions are uniform in respect to x, the simplest supposition. we can make is that is a simple-harmonic function of x; the most general case consistent with the above assumptions can be derived from this by superposition, in virtue of Fourier's Theorem.

where P is a function of y only. The equation (1) of Art. 227 gives

.(1)

(2)

(3)

The condition of no vertical motion at the bottom is dø/dy = 0 for y

The value of o is then determined by Art. 227 (8), which gives

σ2 = gk tanh kh.

Substituting from (4) in Art. 227 (5), we find

=

or, writing

and retaining only the real part of the expression,

η

= a cos kx. sin (ot + €).

.(7)

This represents a system of 'standing waves,' of wave-length λ = 2π/k, and vertical amplitude a. The relation between the period (2π/o) and the wave-length is given by (5). Some numerical examples of this dependence are given on p. 357.

In terms of a we have

ф

ga cosh k (y + h)
σ cosh kh

cos kx. cos (ot + €),

(8)

and it is easily seen from Art. 62 that the corresponding value of the stream

If x, y be the .co-ordinates of a particle relative to its mean position (x, y), we have

if we neglect the differences between the component velocities at the points (x, y) and (x + x, y + y), as being small quantities of the second order. Substituting from (8), and integrating with respect to t, we find

where a slight reduction has been effected by means of (5). The motion of each particle is rectilinear, and simple-harmonic, the direction of motion varying from vertical, beneath the crests and hollows (kx = m), to horizontal, beneath the nodes (kx = (m + 1) π). As we pass downwards from the surface

L. H.

23.

to the bottom the amplitude of the vertical motion diminishes from a cos kx to 0, whilst that of the horizontal motion diminishes in the ratio cosh kh: 1.

When the wave-length is very small compared with the depth, kh is large, and therefore tanh kh= 1*. The formulae (11) then reduce to

with

X = aeky sin kx. sin (σt + €), y = aeky cos kx. sin (ot + e), ..(12)
o2 = gk.
(13)

...

The motion now diminishes rapidly from the surface downwards; thus at a depth of a wave-length the diminution of amplitude is in the ratio e-2 or 1/535. The forms of the lines of (oscillatory) motion (const.), for this case, are shewn in the annexed figure.

In the above investigation the fluid is supposed to extend to infinity in the direction of x, and there is consequently no restriction to the value of k. The formulae also give, however, the longitudinal oscillations in a canal of finite length, provided k have the proper values. If the fluid be bounded by the vertical planes x 0, x = 1 (say), the condition a/dx - 1 (say), the condition a/dx = 0 is satisfied at both ends provided sin kl 1, 2, 3,.... The wave-lengths of the normal modes are therefore given by the formula λ = 21/m. Cf. Art. 178.

=

0, or kl

mπ, where m

=

229. The investigation of the preceding Art. relates to the case of 'standing' waves; it naturally claimed the first place, as a straightforward application of the usual method of treating the free oscillations of a system about a state of equilibrium.

In the case, however, of a sheet of water, or a canal, of uniform depth, extending horizontally to infinity in both directions, we can, by superposition of two systems of standing waves of the same wavelength, obtain a system of progressive waves which advance unchanged with constant velocity. For this, it is necessary that the crests and troughs of one component system should coincide (horizontally) with the nodes of the other, that the amplitudes of the two systems should be equal, and that their phases should differ by a quarter-period.

* This case may of course be more easily investigated independent.