each subsequent point reached by the train in its forward travel. 
There will be a few extra low waves in advance of the train and 
a few more lagging behind the train, but there will be only a small 
percentage of extra crests produced which are of any appreciable 
amplitude. The forward end of the train will advance with the 
group velocity of the apparent local period of the waves in the 
train, and the trailing end will follow with that same group velo- 
city. After a given distance of travel, the ends of the train will 
be modulated by a Fresnel interference pattern. 
Eventually when x becomes very, very large, the wave train 
will have a much lower amplitude. As x approaches infinity, for 
n finite, the amplitude of the wave train approaches zero everywhere. 
All disturbances of initially finite duration and amplitude must 
eventually approach zero amplitudes because of dispersion. 
The individual waves will be essentially constant in ampli- 
tude and period over the central part of the train. The crests 
will travel forward with a speed appropriate to the apparent local 
period of the waves in the train. Thus the individual crests are 
traveling with twice the speed of ane train. Therefore they must 
form in the rear of the train, grow in amplitude, travel through 
the train, and die out again at the front of the train. At the ends 
of the train, a particular crest will not have a speed exactly equal 
to the speed in the center of the train, because of the effect of 
the phase shifts shown in equation (5.28). Wave crests are created 
and destroyed. 
Agreement with classical theory 
Finally, it should be pointed out that many of the abstract 
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