1903-4.] Lord Kelvin on a Free Procession of Waves. 
315 
are carried out to four significant figures) : but it would be utterly 
imperceptible in our diagrams. Henceforth we shall occupy our- 
selves chiefly with the free surface, and takez = 7i, the height of 0, 
the origin of coordinates above the undisturbed level of the water. 
§ 18. To find the water-surface at any time t after being left free 
and at rest, displaced according to any periodic function P(x) 
expressed Fourier-wise as in (27) ; take first, for the initial 
motionless surface-displacement, a simple sinusoidal form, 
- £ 0 = A o,os(mx - c) (41). 
Going back to (2), (3), and (4) above, let w ( z,x,t ) be the down- 
wards vertical component of displacement. We thus have, as the 
differential equations of the motion, 
dw d^w 
(J ~dz = dt 2 
d 2 w dho _ q 
<h? + d£~ 
(42) , 
(43) . 
These are satisfied by 
io= Ce~ mz cos(mx - c) cos tjgm .... (44), 
which expresses the well-known law of two-dimensional periodic 
waves in infinitely deep water. And formula (44) with C€ _w7l = A 
and t = 0, agrees with (41). Hence the addition of solutions (44), 
with jm for m ; with A successively put equal to A x , A 2 . . . , 
Bj , B 2 . . . ; and, with c = 0 for the A’s, and = J7r for the B’s, gives us, 
for time t, the vertical component-displacement at depth 2 — h below 
the surface, if at time t = 0 the water was at rest with its surface dis- 
placed according to (27). Thus, with (38), and (24), we have ~P(x, t). 
§ 19. Looking to (44) and (27), and putting m = 27r/X, 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 X/j and time-periods expressed by 
2tt 
J Jpa ** jg 
(45). 
The whole motion is not periodic because the periods of the 
constituent motions, being inversely as Jj, are not commensurable. 
But by taking A. = 2 h as proposed in § 17, which, according to (40), 
makes A 3 , for the free surface, only a little more than 1/1000 of 
Aj, we have so near an approach to sinusoidality that in our illus- 
