can be graphed by first constructing the envelope and then by draw- 
ing in the wave crests with appropriate regard to phase. 
There will be some point X49 at which the approximations em- 
ployed in equations (5.25), (5.26) and (5.27) begin to fail because 
the modulation on the rear edge of the wave train will lap over and 
combine with the modulation of the forward edge of the wave train. 
Figure 9 shows that the modulation is only important for 6000 sec- 
onds if xX, = 439 km. Modulation which is effective for more than 
18,000 seconds would affect the rear half of the train. Therefore 
an estimate of nine times 439 km or approximately 4,000 km is better 
than the previous crude estimate of 10,000 km for the point at 
which the use of equation (5.15) directly would be required in 
evaluating the solution. 
Some objection might be raised on the: physical reality of the 
problem because of the behavior of the solution near xy equal to 
zero. The unrealistic behavior is due to the discontinuous char- 
acter of the functions employed. Wave trains in nature would not 
be so sharply delineated. To eliminate this objection, just con- 
sider the wave train as a function of time as it passes the point 
x, = 5 km. Let this value of x, be the new point of origin of the 
wave train. Then the new wave train at its starting point would 
have smoothed ends, and beyond the new reference point for all times 
the solution would be a well behaved function. 
Summary 
The behavior of the finite wave train can now be Summarized. 
If the wave train takes a given number of hours to pass a given 
point, it will take essentially.the same number of hours to pass 
- 99 = 
