Free Procession of Waves in Deep Water. 459 
§ 19. Looking to (44) and (27), and putting m=27/d, we 
see that the component motion due to any one of the A’s or 
B’s in the initial displacement is an endless infinite row of 
standing waves, having wave-lengths equal to A/j and time- 
periods expressed by | 
2Qar 27rr 
= gn Nig 
The whole motion is not periodic because the periods of the 
constituent motions, being inversely as ,/j, are not commen- 
surable. But by taking X=2h as proposed in § 17, which, 
according to (40), makes As, for the free surface, only a little 
more than 1/1000 of A,, we have so near an approach to 
sinusoidality that in our illustrations we may regard the 
motion as being periodic, with period (45) for j=1. This 
makes T=,/7 when, as in § 5, we, without loss of generality 
(3 10), simplify our numerical statements by taking g=4; 
and h=1, which makes the wave-length=2. 
§ 20. Toward our problem of “front and rear,” remark 
now that the infinite number of parallel straight standing 
sinusoidal waves which we have started everywhere over an 
infinite plane of originally undisturbed water, may be ideally 
resolved into two processions of sinusoidal waves of half their 
height travelling in contrary directions with equal velocities 
2 Tr. 
Instead now of covering the whole water with standing 
waves, cover it only on the negative side of the line (not 
shown in our diagrams) Y O Y’, that is the left side of O the 
origin of coordinates ; and leave the water plane and motion- 
less on the right side to begin. At all great distances on the 
left side of O, there will be in the beginning, standing waves 
equivalent to two trains of progressive waves, of wave-length 2, 
travelling rightwards and leftwards with velocity 2/./7. The 
smooth water on the right of O is obviously invaded by the 
rightward procession. 
§ 21. Our investigation proves that the extreme perceptible 
rear of the leftward procession (marked R in fig. 10 below) 
does not, through the space OR on the left side of O, 
broadening with time, nor anywhere on the right of O, per- 
ceptibly disturb the rightward procession. 
§ 22. Our investigation also proves that the surface at O 
has simple harmonic motion through all time. It farther 
shows that the rightward procession is very approximately 
sinusoidal, with simple harmonic motion, through a space 
OF (fig. 9) to the right of O, broadening with time ; and 
that, at any particular distance rightwards from QO, this 
approximation becomes more and more nearly perfect as time 
(45). 
3 
