unaltered. For slightly larger values. of X49 the effect of G* 
and H* is to produce strangely distorted wave crests shortly after 
t' = 0 and crests with many different apparent local periods short- 
ly before t' = 0. For moderate values of x,, the effect of G* and 
H* is to cause the individual wave crests to be modulated in ampli- 
tude and phase and to have an apparent local period equal to T for 
all values of t'. Finally for values of xX, near ten thousand kilo- 
meters, G* and H* are no longer appropriate and the original form 
of the solution must be studied. 
Figure 8 shows the forward edge of the finite wave train as 
a function of t' for various values of x, in order to illustrate 
the above remarks. T is ten seconds, n is 1800, and xX, has been 
chosen to be 1/2 centimeter, one twelfth of a wave length, five 
wave lengths and one hundred wave lengths. The time scales for 
the graphs on the left are different in order to show the fine 
details, and the times scales for the graphs on the right are the 
Same in order to show the overall behavior. For Xy equal to 1/2 
centimeter, the effect of G* and H* is to put a few high frequency 
ripples on the very first wave crest as in the graph at the upper 
left. The graph on the upper right shows that the major portion 
of the train is essentially unaffected. For x = L/12 and x = 5L 
a few crests at the forward edge are affected but the major portion 
of the train is still unaffected. For X = 100L, about ten crests 
are found of substantial amplitude in advance of t' = 0, and about 
twenty crests are modulated in amplitude behind t' = 0. The transi- 
tion from constant amplitude to small amplitude is quite gradual. 
For values of x, which are large enough, G* and H* are slowly 
~- %- _ 
