Some examples 
The Lebesgue Stieltjes Power Integral just defined in- 
cludes as special cases all of the representations of the sea 
surface in the previous chapters which were infinitely long in 
duration. From the definition of the integral, it is evident 
that the function 7(t) never attains a constant value of zero. 
Example one is another way to express equation (2.19) when 
x and y are zero and 6 equals 37/2. E(p) is given by equation 
(7.11) which shows that it is piecewise constant with a value 
of zero below 21/T and of A above 21/T. The function, (ph), 
could be given, for this example, by equation (7.12), but act- 
ually W(p) could be anything outside of a small interval 
about 27/T. 
If the limiting process defined by equation (7.7) is car- 
ried out, the value of [E(H5,45) - Bp ig JIE is zero for all 
n except for that particular n, say n = p, for which 
m ops 2n/T <M apse" The square root for this particular inter- 
val is equal to A. Also since Pon < Hops <Hon+2? the expression 
[H on47 - 21/T| can be made as small as one pleases. As the mesh 
approaches zero, F opt falls on the interval p = 2r/T +e 
and determines the phase. In the limit then, equation (7.13) 
is the result. The potential energy is given by equation (7.14), 
and since E,., equals A° the results confirm equation (7.10). 
The various functions employed in example one are shown 
in the graphs applicable to the example in figure 13. In this 
example, since E( p ) is a step function, another function de- 
fined as JE(u), the jump in E(w), can be defined and graphed. 
= 130i 
