the waves. John [1949] has solved the problem for a purely periodic 
wave. His solution is given in the form of a Fredholm integral 
equation (which may or may not be solvable itself). Stoker,in a re- 
cent conference, suggested more direct methods which could yield 
immediate practical results. By the principals of Chapters 9 and 
10 these results can be extended to the Gaussian case. 
Cross correlation functions 
Another important tool for the study of ocean waves is the 
cross correlation function. There are many possible cross corre- 
lation functions which can be constructed. For example, X(p), 
given by equation (11.28), gives the cross correlation between the 
height of the free surface at a fixed point and the pressure re- 
corded by a pressure recorder at some depth, z, below the surface 
at that same point. As another example, equation (11.29) gives 
the relationship between the free surface at two different values 
of y at a fixed time. 
With these cross correlation functions, many properties of the 
sea surface, the velocity component fields, and the pressure fields 
can be interrelated and studied. A detailed study of equation (11.28) 
would probably show that the deeper the pressure recorder the less 
it reflects the passage of high short "apparent" period waves over= 
head and that it is easily possible for a pressure recorder to record 
a crest when actually a trough is passing overhead (or conversely). 
The cross correlation functions must be studied by carefully keeping 
the same net and the same ¥(»,0) for each term in the net for the 
two functions being studied. Although values are Gaussian, for 
example, an accidental high crest is related to an accidental high 
22 
