460 Lord Kelvin on the Front and Rear of a 
advances. What I call the front of the rightward procession, 
is the wave disturbance beyond the point I, at a not strictly 
defined distance rightwards from O, 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 
O, at one and the same time ; but tnat the wave-height does 
not at any time or place come abruptly to nothing. The 
propagational velocity of the beginning of the disturbance is 
in reality infinite, because we regard the water as infinitely 
incompressible. 
§ 23. Thus we see that the front of the rightward procession, 
with sinusoidal waves following it from O, is simply given 
by the calculation, for positive values of 2, of the motion due 
to an initial motionless configuration of sinusoidal furrows 
and ridges on the left side of O. Fig. 8 represents a static 
initial configuration, which we denote by Q («, 0), approxi- 
mately realising the condition stated in § 20. Fig. 
represents on the same scale of ordinates the surface displace- 
ment at the time 257 in the subsequent motion due to that 
initial configuration ; which, for any time ¢, we denote by 
Q (a, t) defined as follows :-— 
Q(2, t)=46(«, )—b(@ +1, t)+$(@+2,t)—.-- ad.inf. (46), 
where ¢ is the function defined by (17), with z=1 and g=4. 
§ 24. The wave-height, at all distances so far leftward from 
O that the influence of the rear of the leftward procession has 
not yet reached them at any particular time, ¢, after the 
beginning, is simply the P(, ¢) of § 16 calculated according 
to §§ 18, 17; and the motion there is still merely standing 
waves, ideally resolvable into rightward and leftward pro- 
cessions. Let I, beyond the leftward range of fig. 10, be the 
point of the ideally extended diagram, not precisely defined, 
where the leftward procession at any particular time, ¢, 
becomes sensibly influenced by its own rear. Between [| 
and R the whole motion is transitional in character, from the 
reoular sinusoidal motion P(#, t) of the water on the left side 
of I, to regular sinusoidal motion of half wave-height 4 P (a, ¢), 
from RtoO; and onto F of fig. 9, the beginning of the front 
of the disturbance in the rightward procession. Hence to 
separate ideally the leftward procession from the whole dis- 
turbance due to the initial configuration, we have only to 
subtract £P(a, ¢) from Q(w, t) calculated for negative values 
of w. Thus the expression for the whole of the leftward 
procession is 
Q(v, t) —4P(a, t) for negative values of z . (47). 
ss 
