This is a small fraction only when M, the modulus, is small, but the cnoidal waves then resemble sinusoidal waves; and it is obvious that in this case the equation of their surface may be developed in a rapidly convergent Fourier-series, of which Sir G. Stokes has given the first two terms.

After some more discussion about these cnoidal waves, concerning their velocity of propagation and the motion of the particles of fluid below their surface, we proceed to a closer examination of the deformation of long waves. To this effect

we apply the equation for

δη
at

to various types of non-stationary

waves, and it will appear that, though sinusoidal waves become steeper in front when advancing, other types of waves may behave otherwise.

In our investigations (in accordance with the method used by Lord Rayleigh, Phil. Mag. 1876, vol. i. p. 257, whose paper has been of great influence on our researches), we start from the supposition that the horizontal and vertical u and v of the fluid may be expressed by rapidly convergent series of the form

u=ƒ+yf1+y2f2+...
v=y&1+y2$2+...

where y represents the height of a particle above the bottom of the canal, and where f,fi,... P1, P2,... are functions of a and t. Of course the validity of this assumption must be proved later on by the fact that series of this description can be found satisfying all the conditions of the problem.

From one of these conditions, viz., the incompressibility of Du dv

น

the liquid, which is expressed by + =0, we may дх dy

and from another, viz., the absence of rotation in the fluid,

In this manner we obtain the following set of equations :—

and, moreover, if o be the velocity potential and the stream

function :

which set of equations satisfies for the interior of the fluid all the conditions of the problem, whilst at the same time it is easy to see that for long waves these series are rapidly convergent. Indeed, for such waves the state of motion changes slowly with a, and therefore the successive differentialquotients with respect to this variable of all functions referring, as ƒ does, to the state of motion, must rapidly

decrease.

Passing now to the conditions at the boundary, let p1 (a constant) be the atmospheric pressure, pi' the pressure at a point below the surface where the capillary forces cease to act, and T the surface-tension. We then have, distinguishing here and elsewhere by the suffix (1) those quantities which refer to the surface,

; дх

but, according to a well-known equation of hydrodynamics,

P=

2

1/af

+

2 xət

8 ax2

1 d1ƒ 1/0 24x

1 af 2 1 aƒ d3ƒ 1

+

1 dƒ 1 d2ƒ d1ƒ df 720 x 48 x 2 • Əx1 72 x 120 გ გენ 720 Ət

+

By differentiation with respect to a equation (5) may be

Moreover, a second equation must hold good at the surface,

In order to satisfy equations (6) and (7) by the method of successive approximations, we put y1=l+n, f=%+ß, where and go are supposed to be constants, and 7 and 8 small functions depending upon a and t. Dealing, then, with the fact that for long waves, whose wave-length is great in comparison with the depth of the canal, every new differentiation with respect to a gives rise to continually smaller quantities, these equations become as a first approximation:

where a is an arbitrary constant which we will suppose to be

small.

428

Drs. Korteweg and de Vries on the

It is obvious that this solution coincides with the one usually given for the case of long waves of arbitrary shape made stationary by attributing to the fluid a velocity equal and opposite to that of the waves, on the assumption that the velocity in a vertical direction may be neglected and that the horizontal velocity may be considered uniform across each section of the canal.

But, if we wish to proceed to a second approximation, we have to put

j=yo− 2 (n+a+y)

(9)

where is small compared with ŋ and a. On substituting this in (6) and (7) and on writing out the result, rejecting all terms* which are small compared with any one of the remaining terms, we find respectively :—

In eliminating

ax

348

=

dt

27

where

from these equations, we have at last

dn _ 3q, d(‡n2 + zxn+}oden)

(12)

(13)

This very important equation, to which we shall have frequently to revert in the course of this paper, indicates the deformation of a system of waves of arbitrary shape, but moving in one direction only. Before applying it, we may point out the close connexion between the constant a, which may still be chosen arbitrarily, and the uniform velocity given to the fluid. Indeed it is easy to see from (1) and (9) how a variation da of the constant a corresponds to a change

* The terms for instance with 7.3 and (?)"

3

are rejected in com

which is retained in the equations, those with

dy and

&q=-4da in this velocity, but, on taking the variation of

δα

and by multiplication with 6 dŋ and further integration,

(14)

(15)

If now the fluid be undisturbed at infinity, and if I be taken equal to the depth which it has there, then equations (14)

Therefore, in this case c1 and c2 are equal to zero, and equation (15) leads to

Here, before we can proceed, we have to discriminate between ☛ positive and σ negative. In the first case 2a is necessarily

an negative because must be real for small values of n. If, дх

then, we put it equal to -h, we have

from which, supposing x to be zero for n=h, we easily obtain the well-known equation of the positive solitary wave, viz. :