and will never stop. Consequently this representation lacks reality 
in the same sense that equations (6.1) and (6.3) lacked reality 
in that the storm which produced the record would have to last for- 
ever at the origin. If 7 (0,t) is given by equation (7.35) then 
by virtue of the linearity of the problem, 1) (x,t) is given by 
equation (7.36). 
Equation (7.37) is another way to express equation (7.36) 
and the variation with x has been absorbed in W(p). For any 
partial sum such as equation (7.9), and for a fixed x, (say posi- 
tive), it is always possible to add an integral number of 2r's 
to Y (Honey) - Gree) ox, 7 and obtain a new value, W'(PH>5,41); 
which satisfies equation (7.38). The W'(u5,,,) will be distri- 
buted according to the same probability laws that govern the distri- 
bution of the original W CH ony)? and consequently W'(#) is 
another point set function like V(H). Consequently 7 (x,t) at 
any x is Gaussian and has the same cumulative power distribution 
that the record at the origin had. 7 (x,t) also has the same 
power spectrum. Ina statistical sense, then, the sea surface 
has the same properties at all points. 
Wave record of finite duration 
In order to generalize the model to a record at the source of 
finite duration, consider the multiplication of 7 (0,t) by F(t). 
F(t) is any function of time which varies very slowly compared to 
the individual waves in 7 (t).. F(t) should also be essentially 
zero outside of a certain range of t. F(t) operating on the inte- 
gral as in equation (7.35) is equal to the effect in the limit of 
F(t) operating on one of the partial sums which represent 7 (t). 
SiG. 6 
