1903-4.] Lord Kelvin on a Free Procession of Wares. 317 
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 0, is simply given by the 
calculation, for positive values of x, of the motion due to an initial 
motionless configuration of sinusoidal furrows and ridges on the 
left side of 0. Fig. 8 represents a static initial configuration, 
which we denote by Q ( x , 0), approximately realising the con- 
dition stated in § 20. Fig. 9 represents on the same scale of 
ordinates the surface displacement at the time 25r in the sub- 
sequent motion due to that initial configuration ; which, for any 
time t, we denote by Q (x, t) defined as follows : — - 
Q(a;, t) = 7j<f>(x, t) - <j>(x + 1, t) + + 2 ,t)- ... ad. inf. (46), 
where <f> 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, t , after the beginning, is 
simply the P (x,t) of § 13 calculated according to §§ 18, 17; 
and the motion there is still merely standing waves, ideally 
resolvable into rightward and leftward processions. 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, f, becomes sensibly in- 
fluenced by its own rear. Between I and B the whole motion is 
transitional in character, from the regular sinusoidal motion P(#, t) 
of the water on the left side of I, to regular sinusoidal motion of 
half wave-height JP(a?, t), from B to 0 ; and on to F of fig. 9, the 
beginning of the front of the disturbance in the rightward proces- 
sion. Hence to separate ideally the leftward procession from the 
whole disturbance due to the initial configuration, we have only 
to subtract JP( x , t) from Q(.t, t) calculated for negative values 
of x. Thus the expression for the whole of the leftward pro- 
cession is 
t) - -|P(^, t ) for negative values of x . . . (47). 
Fig. 10 represents the free surface thus found for the leftward 
procession alone at time ^ = 25r. 
