EE eS 
In Chapter 5, it was shown that the forward edge and the 
rear edge of the finite wave train travel forward with the group 
velocity of the waves under the envelope and that the edges are 
modulated by appropriate combinations of the Fresnel Integrals. 
The disturbance is present at a given Xy for only 2nT seconds 
approximately. Therefore if equation (6.19) is applied to the 
solution, for any t*, and if T is allowed to approach infinity 
P.E. will become zero. The potential energy averaged over a long 
long time after any initial time at any fixed point is zero. 
But at a given X,>0; the disturbance is present for 2nT 
seconds, and the 2nT seconds could stand for ten or twenty hours. 
If, in equation (6.19), t* were a time after the train had arrived 
and if t* + T were a time before the train had passed, then PoE. 
would very nearly equal oga/4. Note that for the elementary 
cases discussed above, the average over ten cycles is only two 
per cent in error. 
The modulation of the edges has to be considered, and the 
value of P.E. is not given by oga-/4 if equation (6.19) is evalu- 
ated in the modulation zone. 
Equation (6.24) formulates the above discussion in terms of 
inequalities. Apart from the modulation effects of the edges 
expressed schematically by E(% 5 n, T) (a positive number), the 
value of P.E. will be approximately pea-/4 if the first inequa- 
lities indicated are satisfied and it will be approximately zero 
if the second two inequalities are satisfied. If none of the 
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