varying functions of time such that they change slowly over several 
hundred seconds. Under these conditions, 7) (x,,t') can be approxi- 
mated by equation (2.8). The wave train now has an envelope given 
by (G*)° + (Ht)? l/2, Also since the most rapidly varying term 
is the sinusoidal term, the apparent local period is everywhere 
equal to T. However H* and G* still vary slowly with time so that 
the wave crests under the envelope will not all be in phase. The 
eradual phase shift with time is given by tan™~+ H*/G*. 
It is also possible to compute the potential energy at each 
point by the use of equation (5.29). The potential energy can be 
averaged over one cycle; and by suitable approximations the aver- 
aged potential energy can be given as a slowly varying function of 
time by (5.30). 
Figure 9 is a nomogram which permits the determination of the 
amplitude of the envelope of the wave train as a function of t' 
for fixed x, (and consequently tee The straight lines of various 
Slopes in the bottom part of the figure are graphs of t' = (4nx,/e)/?x 
for various xy where K is the numerical value of the upper limit of 
integration in equations (5.26) and (5.27). When t' equals 2,000 
seconds as in the example, and Xy equals 439 kilometers, K equals 
2e7- The envelope is graphed as a function of K in the upper graph. 
Thus, 2,000 seconds after t. equals 10.63 hours which corresponds 
to xy equal to 439 km, the envelope is .92 times the amplitude of 
the waves in the original train. Note that the forward edge of the 
wave train passes the point x = 0 five hours before t = 0, and the 
forward edge actually takes 15.63 hours to travel the 439 kilometers. 
For this particular value of Xy the waves build up from an amplitude 
- 9 - 
