In the second case 2a must be positive. In putting it equal to h, and in substituting -ŋ' for ŋ, we have from (16)

This is the equation of a negative solitary wave, and we are able now to draw the conclusion that whenever σ is negative;

that is whenever the depth of the liquid is less than T

др

the stationary wave is a negative one. For water at 20°C. this limiting depth is equal to 0.47 cm. (T=72, g=981, p=0.998 B.A.U.).

Now, for a further discussion of equation (15), we drop the assumption that the fluid is undisturbed at infinity. If then I be taken equal to the smallest depth of the liquid, we must

δη

have =0 for n=0, and therefore in virtue of (15) c=0.

дх

On supposing then σ positive *, c1 must be negative in order

δη

that may be real for small positive values of n, but then the equation

has a positive root h and a negative —k, and we may get from (15)

δη

1

=±√ √ = n(k−n) (k+n).

By substitution in this equation (19) of n=h cos2 x and by integration, we find

*When σ negative, let then be equal to the greatest depth. On substituting σ=—σ', n=—n' we have again c1 negative,

where h and —k are the roots of n'2-2an+6c1 =0.

which is the equation of a train of periodic waves whose wave-length increases when k decreases.

For k=0 this length becomes infinite, and the equation may be shown to coincide with (17).

The following figure (fig. 1) represents such a train of stationary waves for the case in which k=h, M=0.8.

III. Stationary Periodic Waves (Cnoidal Waves).

Proceeding now to a further investigation of the waves determined by equation (20), we calculate from (10) and (11) the value of y. From these equations we get

where the constant of integration is rejected because its retention would only have had the effect of augmenting in equation (9) the value of the arbitrary constant a.

On substituting, then, ƒ from (9) in (1) and (2), observing that in virtue of (14)

1

+ } kh−3n2] } + {/√√/Z { (h−k)n+3hk—}y'}y2+...... (21)

20

gn(h―n) (k+n)

v=

.y.

Ισ

(22)

When k=0 they determine the motion of the fluid for a solitary wave.

In the first place we now will endeavour to calculate the velocity of propagation. For the solitary wave this is simple enough. If we consider that the liquid at infinity is brought to rest when a uniform motion with a horizontal velocity

is added to the motion expressed by (21) and (22), it is clear that this velocity, with reversed sign, must be taken for the velocity of propagation of the solitary wave.

*

But for a train of oscillatory waves Sir G. Stokes has shown that various definitions of this velocity may be given, leading at the higher order of approximation to different values. It seemed to us most rational to define it as the velocity of propagation of the wave-form when the horizontal momentum of the liquid has been reduced to zero by the addition of a uniform motion. This definition corresponds to the second one of Sir G. Stokes. According to it, we have to solve the equation

dx (u−q)dy=0,

where q denotes the velocity of propagation, and where

(24)

σ

kK

K}

V = S^ndx= 4√ √ + { (h + k) E (K) –£K }

h+k

=λ {(4+ k) B(K) − k }

πλ

(26)

denote the volume of a single wave reckoned from above its lowest point, we get from (24), retaining only such terms as are of the first order compared with ŋ, h, and k:

* Math. and Phys. Papers, vol. i. p. 202.

the first is due to a true change of form of the wave, whilst the second may be attributed to a small ing motion of the wave, which is left after the addition uniform motion with velocity go=gl. To this effect ve still at our disposal the quantity a, whose close conwith the uniform motion, which we have added in to make the wave nearly stationary, has been indicated

of the best ways to obtain the desired separation is ly to make stationary the highest point of the wave, s is effected by fulfilling the condition

liscussing this equation (33), we see at once that a wave (31) is stationary when h4op2; and this is in ince with the equation (17) of the stationary solitary which we have obtained above. When h>4op2, the orm of the wave, calculated from (33), is shown line in fig. 2.

fore we have dx=(u'-g)dt, or to a first approximation

Of course, as all fluid particles with the same y describe congruent paths, these formulæ may be simplified by supposing x=0.

IV. Deformation of Non-Stationary Waves.

In order to study the deformation of non-stationary waves, we will now apply our formula (12) to various types of waves. Solitary Waves.-As a first example we choose a solitary wave whose surface is given by

According to (12), the deformation of this wave is expressed

But before we are able to draw any conclusion from this expression, it is necessary to separate the two parts of of

dn dt' * Z(u)=u (1– E(K)) – M2 sn2 u. du. Compare, for instance, Cayley, 'An Elementary Treatise on Elliptic Functions,' 1876, ch. vi. § 187.