for the motion can be deduced immediately by integration of the 
variable part of the pressure with respect to time. The potential 
function is then given by equation (11.7). The u, v, and w velo- 
city components can then be found immediately from the potential 
function and they are given by equations (11.8), (11.9), and (11.10). 
Note that pena Vy PMs > Pt Pyy ce QO, and the equation of 
continuity and, consequently, the potential equation are automatically 
satisfied for any functional form for [aj (u ,0)1°. 
The kinetic energy integrated over depth and averaged over y 
and t at any x is then given by equation (11.11). This is, of course, 
to be expected from Lamb [1932]. The proof will be left to the reader, 
The techniques of Chapter 9 can be employed. It can also be proved 
that the kinetic energy integrated over depth and averaged over t at 
any fixed point is also given by pg EB nax/* The proof follows by 
the application of the methods of Chapter 10. 
The u component of the velocity for a fixed x and y and for 
any depth, z, can be written as a stationary Gaussian Integral as 
a function of time as given by equation (11.12). The functions, 
[D(p» 7° and F(u ), are given by equations (11.13) and (11.14). 
The u velocities decrease in range with depth and change back and 
forth more slowly with time at greater depths. A graph of u as 
a function of time for some fixed depth, z, would look like a pres- 
sure record as the velocity shifts back and forth. However the power 
spectrum of the u velocity record would not have the same shape 
as the power spectrum of the pressure record for the same short 
crested wave system passing overhead. The interrelations are given 
by equations (11.14) and (11.4). 
