These questions can be answered. in a statistical sense, 
put first some general properties will be obtained. The hope, 
of course, is that an actual wave record can be thought of best 
as one of the possible functions from the statistical class 
described above. 
If equations (7.27) and (7.28) hold, then the limit of 
equation (7.7) exists, and the integral given by (7.29) has 
the value (1/2)E,,,. In addition, if the derivative of E(p ) 
is continuous or piecewise continuous, it must be everywhere 
positive. Under these conditions, the derivative can be written 
as the square of some function A(m) as in equation (7.30). 
With this new representation for dE(pu ), equation (7.1) can 
be rewritten as equation (7.31) which is no longer a Stieltjes 
Integral since there are no jumps in E(#). It might be termed 
a Lebesgue Power Integral since the point set function W (#) 
‘is still involved. 
The function, Ch Cra) =, is the power spectrum of 7 (t). It 
has the dimensions [L°r], and since dy has the dimensions of 
Geel the dimension of wh RCL eas! is [L]. The power spectrum 
is easily measured in a statistical sense, and the methods for 
such measurements have been presented by Tukey and Hamming [1949]. 
It will be assumed that (AC ))? can be determined with a known 
degree of statistical reliability. The procedures for so doing 
will be described in a later chapter when short crested waves 
will be considered. 
The random walk 
If the same net that was given in equation (7.6) is applied 
= 137 - 
