m2x2 
(x0) = SRAVE @ 7 Eemikor (2.32) 
Equation (2.32) represents a wave as a function of x on 
the free surface with a wave length, L, = 2n/K. » which is modu- 
lated by a probability curve envelope. In order to evaluate 
(2.31) as a function of time also, n was expanded as a function 
of K in a Taylor series about the point aS “Kez . Only the 
first two terms of the expansion were used in the integration. 
The solution thus obtained was an approximation because 
of the series approximation of n. It showed that the probabi- 
lity curve envelope advanced with the group velocity appropriate 
to waves with a wave length, L, = 2r/K, that the envelope flat- 
tened out with time and decreased in maximum amplitude, and that 
there was a gradual phase shift of the individual waves under 
the envelope. 
The Gaussian wave packet is a far more realistic problem 
than the Cauchy-Poisson problem because the condition that the 
height of the waves be small is satisfied everywhere if it is 
satisfied at the time t = 0. As it stands, however, it is vroba- 
bly not applicable for moderately large values of o and for 
large values of time or displacement in the x direction because 
the effect of dispersion is partially neglected in the series 
approximation of n. 
-It should be noted that none of the classical solutions 
have considered the possible variation in the direction of pro- 
pagation of the wave crests as indicated in eouation (2.29). 
eho os 
