566 Proceedings of Royal Society of Edinburgh. [sess. 
which satisfies (65) and expresses a sinusoidal wave-disturbance, 
of wave-length 27 r/m, travelling awards with velocity v. 
§ 39. To find the boundary- pressure II, which must act on the 
water-surface to get the motion represented by (66), when m, v, k 
are given, we must apply (64) to the boundary. Let 2=0 be the 
undisturbed surface ; and let d denote its depression, at ( x , o, i), 
below undisturbed level ; that is to say, 
d = £(#, o, t) = — cf>(x, z, t) z=0 = mk sin mix - vt) 
dz 
(67), 
whence by integration with respect to t, 
d = — cos mix-vt) (68). 
v 
To apply (64) to the surface, we must, in gz , put z = d; and in 
dcfi/dt we may put 2 = 0, because d, k, are infinitely small quantities 
of the first order, and their product is neglected in our problem of 
infinitesimal displacements. Hence with (66) and (68), and 
with n taken to denote surface-pressure, (64) becomes 
kmv cos mix - vt) = - k cos mix - vt) - n + gC . (69). ; 
whence, with the arbitrary constant C taken = 0 , 
II = kv(^L - cos m{x -vt) (70) ; 
and, eliminating k by (68), we have finally, 
II = (g - mv 2 ) d 
(71). 
Thus we see that if v^Jg/m, we have n = 0, and therefore we 
have a train of free sinusoidal waves having wave-length equal to 
27r/m. This is the well-known law of relation between velocity 
and length of free deep-sea waves. But if v is not equal to Jg/m , 
we have forced waves with a surface-pressure (g — mv 2 ) d which 
is directed with or against the displacement according as 
v< or >\Jg/m. 
§ 40, Let now our problem be : — given n, a sum of sinusoidal 
functions, instead of a single one, as in (70); — required d the 
resulting displacement of the water- surface. We have by (71) 
and (70), with properly altered notation, 
