original value. Thus the larger the value of ao , the wider the 
spectrum of the disturbance, and the more rapidly the disturb- 
ance dies down in amplitude as it travels along. 
The time required for the wave group to pass the point of 
interest is simply two tenths of the time required for the wave 
group to pass the corresponding point of interest in Table l. 
Thus, the Modi eatioe of the wave group with a large value of 
0 occurs much more rapidly than it does for a small value ofc . 
In summary, the envelope travels in the positive x direction 
with a speed determined by the group velocity of waves with a 
period T. The larger the value of T, the more rapidly the group 
travels in the x direction. Its maximum amplitude decreases as 
it travels along, and it spreads out over the sea surface more 
and more the further it gets away from the origin. The larger 
the value of T and the larger the value of o , the more rapidly 
the group disperses in time. 
Apparent local period 
The rapidly varying sinusoidal term which determines the 
nature of the waves as modulated by the envelope can now be con- 
sidered, The sinusoidal term is given by S(7m ) in equation (4.15). 
The term varies between plus one and minus one as x ane t are 
varied and it is defined everywhere in the x,t plane. It is 
not periodic in t since there exists no constant T such that 
S(ym )[t] = S(m)[t + 7]. In addition, the sinusoidal term is 
not periodic in x. In fact, the entire solution is not periodic. 
Consider the term in the brackets in equation (4.15) for a 
fixed positive value of Xy° As a function of t, it is a parabola 
ee 
