predecessors or successors with a label “favorite integer plus 
two." The sea surface is irregular, it does appear that the 
waves sometimes come in groups, but the groups do not persist, 
they do not have a mean time of separation, and they do not 
contain the same number of waves. 
Figure 12 shows some wave records. They are on a greatly 
condensed time scale such that the crests are all crowded to- 
gether. Note the basic features of these wave records. Iso- 
lated high waves frequently occur as at the points marked A. 
Sometimes groups appear as in the intervals marked B. At times 
the trend in the amplitudes is quite high as in the intervals 
marked C. And at other times the trend in the amplitudes is 
quite low as in the intervals marked D. The basic feature of 
the records is their irregularity, which is a type of irregu- 
larity which would almost appear to defy an adequate mathe- 
matical representation. Obviously any of the mathematical 
models employed in the past chapters do not represent such a 
wave record. Consequently, better models must be found. 
The next step then in increasing complexity is to find 
some way to represent the sea surface which is general enough 
to include this very irregular pattern. It will be found that 
Fourier Integral Theory is not enough and that an extension 
to a type of Lebesgue Stieltjes Integral is needed.* The ex- 
tension to this type of integral and the inclusion of some very 
interesting statistical methods simplify the problem once the 
basic concepts are understood and permits a tremendous stride 
*Phe Lebesgue-Stieltjes Integral is defined in James and James 
[1949] for example. 
= 323) 3 
