For additional comment on this point, see Chapter Seven. 
Continuous spectrum 
A third possible way to analyze the actual wave record 
would be to pick out the well defined wave groups and analyze 
them by means of the Fourier Integral Theorem. For instance, 
the wave group centered at t = 0 could be defined to be identi- 
cally zero beyond the arrows which bracket it. The function 
f(t) could be given by the wave group between the arrows and 
by the zero outside of the arrows. Then equation (3.7) could 
be applied to the function and finally, f(t) could be represent- 
ed by equation (3.8). For conditions on f(t) and for defini- 
tions of the symbols used, see Sommerfeld [1949]. 
Similar analyses of f,(t - t,), fo(t - to), and f(t ~ t3) 
in (3.6) could be carried out. Hach analysis would yield a con- 
tinuous spectrum of wave frequencies given by the appropriate 
form of equation (3.7) and the relative importance of various 
parts of the spectrum would be given by equations of the form of 
(3.9). There is no known precise procedure with which one could 
start with the wave record and find the appropriate a; Cu ) and 
b,(# ), but such a procedure is theoretically possible. Finally, 
1 3(t) can represent the wave record exactly over any length of 
time chosen for analysis. 
If eguation (3.10) were applied to the actual wave record, 
the two sides of the equation would be exactly equal. Thus this 
method of analysis represents exactly the potential energy aver- 
aged over the wave record as a function of time at a fixed point. 
Such an analysis would make it possible to represent a wave 
8D e 
