(a' = 0), and the side spectral components oscillate more and more 
rapidly and tend to cancel. For large values of n, the same re- 
marks about the spectral components which travel in the negative 
x direction are applicable which were made in the comments about 
the spectrum of this finite wave group. 
As n approaches infinity, for any finite values of t and x, 
equation (5.15) reduces ton (x,t) =A sin(4r°x/gT2= 29t/T) which 
is a simple sine wave traveling toward the right. Only when the 
wave train is infinitely long and lasts for an infinite length of 
time is it possible to apply the usual formula for wave speed and 
wave length without qualification. Also equation (4.9) reduces 
to the above form aS o approaches zero. 
An interesting question is, "Why are the two continuous spectra 
so different in their limiting forms?" The explanation lies in 
the way that the free surface at x equal to zero approaches its 
limiting form. The free surface studied in Chapter 4 deforms con- 
tinuously into its limiting form, namely 7 (0,t) = A sin 2rt/T, as 
0 approaches zero. The free surface studied in this chapter does 
not deform continuously into its limiting form. The sharp discon- 
tinuity from full amplitude to zero amplitude is always present, 
and an increase in the value of n just displaces the discontinuity 
in time. This difference in behavior thus explains the differences 
in the type of continuous spectra obtained. 
The solution to the problem studied in this chapter is valid 
for all integer values of n. For n egual to one, there would be 
two complete wave crests under the envelope. The evaluation of the 
solution for other values of x and t would then be somewhat difficult 
Laos 
