when x' is zero. The term Q? g °/2r-D is small compared to one 
for the values of T and o0 employed in the evaluation. For a 
fixed X,, as x' increases positively, the apparent local wave 
length increases. As x' decreases the apparent local wave 
length decreases. Thus for a fixed t as a function of x, the 
longer waves are in the front of the group. 
One final point needs to be made. It was shown that a posi- 
tive increase in t by the amount T* caused a decrease in f(x,t) 
by the amount 27. In equation (4.27), it was assumed that a 
positive increase in xX by the amount L* caused a positive in- 
crease in f(x,t) by the amount 27. Equation (4.31) then gave 
a positive value for L* over the range of x and t where the maxi- 
mum value of the envelope occurs. The derivation therefore shows 
that those wave crests which are under the maximum value of the 
envelope as it travels along are moving forward in the positive 
x direction. 
Apparent local speed 
The total change in f(x,t) at a wave crest should be zero 
if the observer moves with the speed of the crest. Equation 
(4.32) imposes this condition and yields the result that the 
wave crests advance with an apparent local speed given by C* 
in equation (4.33). In equation (4.33) it should be understood 
that L* and T* are given by equation (4.26) and (4.29). For 
values of x and t which give a maximum value for the envelope, 
it then follows that the wave crests are moving forward with a 
speed twice that of the envelope. 
a) GR} a 
