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which the first is due to a true change of form of the wavesurface, whilst the second may be attributed to a small advancing motion of the wave, which is left after the addition of the uniform motion with velocity gogl. To this effect we have still at our disposal the quantity a, whose close connexion with the uniform motion, which we have added in order to make the wave nearly stationary, has been indicated above.

One of the best ways to obtain the desired separation is certainly to make stationary the highest point of the wave, and this is effected by fulfilling the condition


2(a+2op2)=3(4σp2 —h),

a=4op2—3h ;

for in that case equation (32) is simplified to



and then, for a=0,

3qoph (4σp2 — h) sech2 px. tanh3 pæ ; · (33)

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In discussing this equation (33), we see at once that a solitary wave (31) is stationary when h≈4op2; and this is in accordance with the equation (17) of the stationary solitary wave which we have obtained above. When h>4op2, the change of form of the wave, calculated from (33), is shown by the dotted line in fig. 2.

Fig. 2.


Here the wave becomes steeper in front, whilst for h<4op2 the figure would show the opposite change of form, when, contrary to the opinion expressed by Airy and others, the wave becomes less steep in front and steeper behind.

* The left side of the figure is the front side of the wave, because the wave has been made stationary by the application of a positive velocity (i. e. from left to right) to the fluid.

If, now, we take account of the fact that, as may easily be inferred from (31), the wave-surface becomes steeper in proportion as p is increased, we are then justified in saying that a solitary wave which is steeper than the stationary one, corresponding to the same height, becomes less steep in front and steeper behind, but that its behaviour is exactly opposite when it is less steep than the stationary one.

Cnoidal Waves.-Applying formula (12) to the cnoidal

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dn 3qoph (4σM2p2 —h) sn3 px. en på. dn på. (36)


Here fig. 3 shows the change of form calculated for the case h-40 M2p2>0.

Fig. 3.

When h-40M2p2=0, the waves are stationary in accordance with (20), whilst for h-40M2p2<0 they become steeper behind; and this last result, since p is inversely proportional to the wave-length, may be stated by saying that cnoidal waves become less steep in front and steeper behind when, for a given modulus and a given height, their length is smaller than the one required for the stationary wave of this modulus and height.

In proportion as M is taken smaller the cnoidal waves more and more resemble sinusoidal waves. They would take the sinusoidal form for M=0, but then an infinitely small wavelength would be required for the stationary case. For this reason sinusoidal waves may always be considered as cnoidal waves whose length is too large to be stationary, that is, they are always becoming steeper in front.

Sinusoidal Waves.-This last result is easily verified by direct application of (12) to the equation of a train of sinusoidal waves:

n=A sin



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and from this the change of form indicated in fig. 4 is easily calculated.

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More complicated Cases.-For the sake of curiosity, we represent by means of the following figures the change of form for some more complicated cases.

Fig. 5.

Fig. 6.

Fig. 7.


Phil. Mag. S. 5. Vol. 39. No. 240. May 1895.

2 G

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In fig. 54 is supposed to be small compared with

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is the case with waves of extremely small height. In fig. 6 Generally for

() to be small in regard to 41.


we suppose more complicated forms of wave these two cases have to be discriminated. When there is a moderate proportionality between the two fractions the result is still more complicated. Finally, fig. 7 refers to the equation

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It is worthy of remark that all these waves grow steeper

in front.

V. Calculation of the Fluid Motion for Stationary Waves to the Higher Order of Approximation.

In order to remove every doubt as to the existence of absolutely stationary waves, we will show how by development in rapidly convergent series the state of motion of the fluid belonging to such a wave-motion may be calculated.

Expressing again the horizontal and vertical velocity of a particle by means of the series (1) and (2) which fulfil all the conditions for the interior of the fluid, we have only, neglecting capillarity, to satisfy the surface-conditions,


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For the case of enoidal waves, which is the general one, we have found as a first approximation,

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But now, to obtain higher approximations, we assume, indicating by accents differentiation with respect to x,


n'2=an(h—n)(k+n)(1+bn + cn2 + . . .),

f=g+rn+sn2 + tn3 + un' + ...

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On writing out (39), neglecting such terms as are of a higher order than the fourth compared with ŋ, h, and k, which latter quantities are of the same order, we obtain

n'2=ahkn+ {a(h−k) +abhk}ŋ2 + {-a+ab (h−k)} n3-abn1; . (41) and by differentiation,

n"={ahk+{a(h−k)+abhk}n+{−ža+gab(h−k)}n2—2abn3. (42) From (40), by successive differentiations and substitutions, retaining all terms up to the third and the 34th order, we deduce :


f"=arhk+{ar(h−k) +abrhk+3ashk}n


f=[ar (hk)+abrhk+3ashk


fiva2rhk(hk) + {a2r (h−k)2 — fa2rhk}n

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ƒˇ= [a2r(h—k)2 — §a2rhk-15a2r (h-k)n +45 a2rn2]n' ;

where n is a quantity of the order 3.

Substituting these values in equation (1), where y=l+n, we have, retaining terms of the third order :

u1 =ƒ→ } l2 ƒ" — lnf" + 24l'fiv=q-farl2hk+4a2rl1hk(h-k)

+24a2rl' (h−k)2 — 3 a2rl+hk} n

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We find in the same way, including terms of the 31th order :

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=n'[—rl+farl3 (h−k)+fabrľ3hk+sasl3hk—126a2rl3 (h—-k) 2

+33⁄4a2rl3hk+{−2sl—r—§arl3+şabrl3 (h—k)

+fast3 (h-k)+arl2 (h−k) +}a2rl3 (h—k) } n

+{−3tl—2s—abrl3—§asl3 — farl2 — y&a2rl3}n2]. . (44)

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