1903-4.] Lord Kelvin on Two-dimensional Waves. 
185 
On Deep-water Two-dimensional Waves produced by 
any given Initiating Disturbance. By Lord Kelvin. 
(Read February 1, 1904. MS. received February 13, 1904.) 
§ 1 . Consider frictionless water in a straight canal, infinitely long 
and infinitely deep, with vertical sides. Let it be disturbed from 
rest by any change of pressure on the surface, uniform in every 
line perpendicular to the plane sides, and left to itself under 
constant air pressure. It is required to find the displacement and 
velocity of every particle of the water at any future time. Our 
initial condition will be fully specified by a given normal com- 
ponent velocity, and a normal component displacement, at every 
part of the surface. 
§ 2. Taking 0, any point at a distance h above the undisturbed 
water level, draw 0 X parallel to the length of the canal, and 0 Z 
vertically downwards. Let £, £ be the displacement -components 
of any particle of the water whose undisturbed position is ( x , z). 
We suppose the disturbance infinitesimal ; by which we mean 
that the change of distance between any two particles of water is 
infinitely small in comparison with their undisturbed distance ; 
and the line joining them experiences changes of direction which 
are infinitely small in comparison with the radian. Water being 
assumed frictionless, its motion, started primarily from rest by 
pressure applied to the free surface, is essentially irrotational. 
Hence we have 
£=j4; t - • fl)j 
where <£(&, z, t), or <£, as we may write it for brevity when con- 
venient, is a function of the variables which may be called the 
displacement-potential ; and <£(&', z , t) is what is commonly called 
the velocity-potential. Thus a knowledge of the function </>, 
for all values of x , z , t, completely defines the displacement 
and the velocity of the fluid. And, by the fundamentals of 
hydrokinetics, a knowledge of <f> for every point of the free 
