458 Lord Kelvin on the Front and Rear of a 
Suppose for example X=4<; we have 
4mz 
€ A =e "='043214; A;/A,;='02495; A;/A,=:03347. (39). 
Thus we see that A; is about 1/40 of A,; and A;, about =! 30. of 
As. This is a fair approach to sinusoidality ; but not quite 
near enough for our present purpose. Try next A=2z; 
we have 
Ge jy SBM; e-2"=-(01867; Ag/A;="001078 . (40). 
Thus A; is about a thousandth of A,; and A; about 1) x 10-6 
of A,. This is a quite good enough approximation for our 
present purpose: A; is imperceptible in any of our caleu- 
lations: Ag is negligible, though perceptible if included in 
our calculations (which are carried out to four significant 
figures): but it would be utterly imperceptible in our 
diagrams. Henceforth we shall occupy ourselves chiefly 
with the free surface, and take z=h, the height of O, the 
origin of coordinates above the undisturbed level of the 
w ater. 
$18. To find the water-surface at any time ¢ after being 
left free and at rest, displaced according to any periodic 
function P(#) expressed Fourier-wise as in (27) ; take first, 
for the initial motionless surface-displacement, a simple 
sinusoidal form, 
—€,=Acos(ma—c).s . . « » (aR 
Gat back to'(2), (3), and (4) above, let w(z, x, t) be the 
downwards vertical component of displacement. We thus 
have, as the differential equations of the motion, 
dw es: 
I dz” dé (=) 
dw  d?w 
dat + d-2 =() ° aah ui ° ° (43). 
These are satisfied by 
w=Ce-"* cos(mz—c) costa/gm. . . (44), 
which expresses the well-known law of two-dimensional 
periodic waves in infinitely deep water. And formula (44) 
with Ce-™=A and t¢=0, agrees with (41). Hence the 
addition of solutions (44), with jm for m; with A successively 
put equal to A, A,..., B,, B, ...; and with c=0 for the 
A’s, and =47 for the B’s, gives us, for time ¢, the vertical 
component-displacement at depth -—h below the surface, if 
at time t=0 the water was at rest with its surface displaced 
according to (27). Thus, with (38), and (24). we have P(z, ¢). 
