four inequalities are satisfied, P.E. is of some value between 
ogA@/4 and zero. 
The average potential energy at the modulated edge of the 
train was discussed in Chapter 5. Figure 9 shows that at least 
to the eye the area under the dashed curve is equal to the area 
under a jump function given by f(t) = 0 for tbe andi byt) 
for t>t,- 
To a good degree of approximation, then, the potential energy 
averaged over a time short compared to the total duration of the 
train but long compared to a cycle is constant when the train is 
present at the point of observation. Since the train, if it has 
not traveled too far, takes 2nT seconds to pass, the total amount 
of energy present is the same as at the origin at each point of 
observation. 
For great distances of travel, dispersion modifies the re- 
sults, and P.E. decreases. No energy is lost; it is just spread 
out over a greater time interval. 
Energy balance for the finite regular train of wave groups 
The finite regular train of wave groups was broken up into 
finite wave trains of different periods. At a given point of ob- 
servation, X;, Some trains will be present, some will have passed, 
and some will not have arrived as shown by figure 11. If the 
train for m equal to K is present, and if the train for m equal 
to K + li is present, then the trains for m for values in between 
will be present. Equation (6.25) expresses this formally. Over 
a long enough time, all of the energy in the original record is 
accounted for at each point of observation. 
- 120 - 
