regular train of wave groups.) P.E. can then be given by 
equation (6.22). 
The process of squaring a sum of terms is carried out in 
the second expression in equation (6.22). The sum of the squared 
terms occurs in the first integral, and a sum of cross product 
terms involving two different wave lengths and two different 
periods ( T/m and T/q) occurs in the second integral. In the 
second integral, the product of two sines with two different 
arguments is given by half the cosine of the difference of the 
arguments minus half the cosine of the sum of the arguments. 
Thus each term in the second integral is sinusoidal, with T as 
the greatest possible period, and the average of a sinusoidal 
term is zero. | 
The first integral in the second expression for ou slen 
(6.22) is a sum of integrals like those treated in equation (6.20). 
Each term can be treated like equation (6.20) was treated. Since 
each term of the first integral yields an average if Teal large, 
the limit for large T is given by equation (6.23). More refined 
investigation would show that if T were of the order of ten or 
twenty times tT , the averaged value of the potential energy 
would be within a few per cent of the limiting value. 
The results show that the potential energy of the sea sur- 
face averaged over a fairly long time for the infinite periodic 
train of wave groups is the same everywhere at any time. The 
disturbance studied never started and it will never stop. It 
covers the whole xy plane. 
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