direction. If c be the velocity of propagation, we get from (27), since n = cm = c. 2π/λ,

If we consider only the series which is propagated in the positive direction, we may take for the same reason as before

which gives

my

-my

φ = A (e""" + €) sin (mx nt).........(30);

my

L = g (h − y) + nA (eTM" + eTM") cos (mx — nt)................. (31),

and for the equation to the free surface

mh

g (y− h) = nA (€TM3 + € ̃13) cos (m.x – nt)......... (32).

Equations (21), (22), (23) may be got from (30), (31), (32) by writing y+h for y, Ae for A, and then making h infinite. When is very small compared with h, the formula (29) reduces itself to (24): when on the contrary λ is very great it reduces itself to (7). It should be observed however that this mode of proving equation (7) for very long waves supposes a section of the surface of the fluid to be the curve of sines, whereas the equation has been already obtained independently of any such restriction. The motion of the individual particles may be determined, just as before, from (30). We get

Hence the particles describe elliptic orbits, the major axes of which are horizontal, and the motion in the ellipses is the same as in the case of a body describing an ellipse under the action of a force tending to the centre. The ratio of the minor to the major axis is that of 1 my to 1+ε -2my which diminishes from the surface downwards, and vanishes at the bottom, where the ellipses pass into right lines.

my

-my

-mh

The ratio of the horizontal displacement at the depth hy to that at the surface is equal to the ratio of e' +6 to m + € The ratio of the vertical displacements is that of "-" to -€. The former of these ratios is greater, and the latter

my

-my

less than that of (hy) to 1. Hence, for a given length of wave, the horizontal displacements decrease less, and the vertical displacements more rapidly from the surface downwards when the depth of the fluid is finite, than when it is infinitely great.

In a paper "On the Theory of Oscillatory Waves*" I have considered these waves as mathematically defined by the character of uniform propagation in a mass of fluid otherwise at rest, so that the waves are such as could be propagated into a portion of fluid which had no previous motion, or excited in such a portion by means of forces applied to the surface. It follows from the latter character, by virtue of the theorem proved in Note IV, that udx + vdy is an exact differential. This definition is equally applicable whether the motion be or be not very small; but in the present note I have supposed the species of wave considered to be defined by the character of periodicity, which perhaps forms a somewhat simpler definition when the motion is small. In the paper just mentioned I have proceeded to a second approximation, and in the particular case of an infinite depth to a third approximation. The most interesting result, perhaps, of the second approximation is, that the ridges are steeper and narrower than the troughs, a character of these waves which must have struck everybody who has been in the habit of watching the waves of the sea, or even the ripples on a pool or canal. It appears also from the second approximation that in addition to their oscillatory motion the particles have a progressive motion in the direction of propagation, which decreases rapidly from the surface downwards. The factor expressing the rate of decrease in the case in which the fluid is very deep is e, y being the depth of the particle considered below the surface. The velocity of propagation is the same as to a first approximation, as might have been seen a priori, because changing the sign of the coefficient denoted by A in equations (21) and (30) comes to the same thing as shifting the origin of a through a space equal to λ, which does not alter the physical circumstances of the motion; so that the expression for the velocity of propagation cannot contain any odd powers of A. The third approximation in the case of an infinite depth gives an increase in the velocity of propagation depending upon the height of the waves. The velocity is found to be equal to

*

Cambridge Philosophical Transactions, Vol. VIII. p. 441. [Ante, Vol. 1. p. 197.]

c。 (1+2π2α2x2), c, being the velocity given by (24), and a the height of the waves above the mean surface, or rather the coefficient of the first term in the equation to the surface.

A comparison of theory and observation with regard to the velocity of propagation of waves of this last sort may be seen at pages 271 and 274 of Mr Russell's second report. The following table gives a comparison between theory and experiment in the case of some observations made by Capt. Stanley, R.N. The observations were communicated to the British Association at its late meeting at Swansea*.

In the following table

A is the length of a wave, in fathoms;

B is the velocity of propagation deduced from the observations, expressed in knots per hour;

C is the velocity given by the formula (24), the observations being no doubt made in deep water;

D is the difference between the numbers given in columns B and C.

In calculating the numbers in table C, I have taken g=322 feet, and expressed the velocity in knots of 1000 fathoms or 6000 feett.

The mean of the numbers in column D is 1.94, nearly, which is about the one-eleventh of the mean of those in column C. The quantity 1.94 appears to be less than the most probable error of any one observation, judging by the details of the experiments; but as all the errors lie in one direction, it is probable that the

* Report for 1848, Part II. p. 38.

+ I have taken a knot to be 1000 fathoms rather than 2040 yards, because the former value appears to have been used in calculating the numbers in column B.

formula (24) gives a velocity a little too small to agree with observations under the circumstances of the experiments. The height of the waves from crest to trough is given in experiments No. 1, 2, 3, 6, 7, by numbers of feet ranging from 17 to 22. I have calculated the theoretical correction for the velocity of propagation depending upon the height of the waves, and found it to be 5 or 6 of a knot, by which the numbers in column Cought to be increased. But on the other hand, according to theory, the particles at the surface have a progressive motion of twice that amount; so that if the ship's velocity, as measured by the logline, were the velocity relatively to the surface of the water, her velocity would be under-estimated to the amount of 1 or 12 knot, which would have to be added to the numbers in column B, or which is the same subtracted from those in column C, in order to compare theory and experiment; so that on the whole 5 or 6 would have to be subtracted from the numbers in column C. But on account of the depth to which the ship sinks in the sea, and the rapid decrease of the factory from the surface downwards, the correction 1 or 12 for the "heave of the sea*" would be too great; and therefore, on the whole, the numbers in column. C may be allowed to stand. If the numbers given in Capt. Stanley's column, headed "Speed of Ship" already contain some such correction, the numbers in column C must be increased, and therefore those in column D diminished, by 5 or 6.

It has been supposed in the theoretical investigation that the surface of the fluid was subject to a uniform pressure. But in the experiments the wind was blowing strong enough to propel the ship at the rate of from 5 to 78 knots an hour. There is nothing improbable in the supposition that the wind might have slightly increased the velocity of propagation of the waves.

There is one other instance of wave motion which may be noticed before we conclude. Suppose that two series of oscillatory waves, of equal magnitude, are propagated in opposite directions. The value of which belongs to the compound motion will be

(εmy + € ̃my) [A cos (mx − nt) + A cos (mx + nt + a)],

* I have been told by a naval friend that an allowance for the "heave of the sea" is sometimes actually made. As well as I recollect, this allowance might have been about 10 knots a day for waves of the magnitude of those here considered.

the squares of small quantities being neglected, as throughout this note.

Since

cos (mx - nt) + cos (mx+nt + a) = 2 cos (mx + x) cos (nt + §a),

we get by writing

as to get rid of a,

4 for A, and altering the origins of x and t, so

-my

$ = A (€TM3 + € ̃TM) cos mx cos nt......... (34).

This is in fact one of the elementary forms already considered, from which two series of progressive oscillatory waves were derived by merely replacing products of sines and cosines by sums and differences. Any one of these four elementary forms corresponds to the same kind of motion as any other, since any two may be derived from each other by merely altering the origins of x and t; and therefore it will be sufficient to consider that which has just been written. We get from (34)

We have also for the equation to the free surface

(35).

(εmy + € ̃m) cos mx sin nt...... (36).

Equations (35) shew that for an infinite series of planes for which mx = 0,π, = ± 2π, &c., i. e. x = 0,= ± 1λ, = ± λ, &c., there is no horizontal motion, whatever be the value of t; and for planes midway between these the motion is entirely horizontal. When t=0, (36) shews that the surface is horizontal; the particles are then moving with their greatest velocity. As t increases, the surface becomes elevated (4 being supposed positive) from x = 0 to x, and depressed from x = λ to x = λ, which suffi入 ciently defines the form of the whole, since the planes whose equations are x = 0, x = λ, are planes of symmetry. When nt, the elevation or depression is the greatest; the whole fluid is then for an instant at rest, after which the direction of motion of each particle is reversed. When nt becomes equal to π, the surface again becomes horizontal; but the direction of each particle's motion is just the reverse of what it was at first, the magnitude of the velocity being the same. The previous motion. of the fluid is now repeated in a reverse direction, those portions of the surface which were elevated becoming depressed, and vice versa. When nt = 2π, everything is the same as at first.

S. II.

16