316 Proceedings of Royal Society of Edinburgh. [ses>*. 
trations we may regard the motion as being periodic, with period 
(45) for/ = 1. This makes t= J-n- when, as in § 5, we, without loss 
of generality (§ 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 /\At. 
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 0 Y', that is the left side of 0 the origin of 
coordinates ; and leave the water plane and motionless on the right 
side to begin. At all great distances on the left side of 0, 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/Jz r. The smooth water on the right 
of 0 is obviously invaded by the rightw r ard 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 0 R on the left side of 0, broadening with time, 
nor anywhere on the right of 0, perceptibly disturb the rightward 
procession. 
§ 22. Our investigation also proves that the surface at 0 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 0 F (fig. 9) to the right 
of 0, broadening with time ; and that, at any particular distance 
rightwards from 0, this approximation becomes more and more 
nearly perfect as time advances. What I call the front of the 
rightward procession, is the wave disturbance beyond the point F, 
at a not strictly defined distance rightwards from 0, where the 
approximation to sinusoidality of shape, and simple harmonic 
quality of motion, is only just perceptibly at fault. We shall find 
that beyond F the waves are, as shown in fig. 9, less and less high, 
and longer and longer, at greater and greater distances from 0, 
at one and the same time ; but that the wave-height does not at 
