because all terms in equations (5.15), (5.16) and (5.17) would have 
to be evaluated carefully over small increments in the variable 
quantities. The disturbance would die down in amplitude and spread 
out over the surface quite rapidly. In addition, the variation in 
amplitude and sign of the terms G and H would be as rapid as the 
variation of the sine and cosine terms so that the simplifying con- 
cept of a slowly varying envelope and a relatively rapidly varying 
Sinusoidal term under the envelope would not be applicable. 
Simplification of results 
For large values of n the analysis of the solution and the 
physical interpretation of the results are simpler. One could con- 
sider the problem to represent a train of waves of constant apparent 
local period which takes, say, ten hours to pass a given point, 
x = 0. Then if T equals ten seconds, n equals 1800, and nT equals 
18,000. 
Now consider the integrals in equation (5.18) and (5.19). If 
the upper limit were replaced by plus or minus ten in these inte- 
grals, the values of the integrals would still be very close to plus 
or minus one half. Thus, interest should be concentrated on times 
and places where the upper limits of integration in equations (5.16) 
and (5.17) lie between minus ten and plus ten. One way to do this 
is to study the variation of the solution at a fixed value of x as 
a function of time. Pick a fixed value of x and call it X, as in 
Chapter 4. Then the forward edge of the wave train arrives at xy 
at a time, t, determined by the group velocity of waves with a 
period equal to the apparent local period of the wave train. When 
=OOm= 
