The assumption made in the formulation essentially noes the 
form of 1 (0,t) which can be arbitrarily given in such a prob- 
lem. The possibility of very low waves traveling in the opposite 
direction out of the group is indeed a very small price to pay 
for a closed complete easily evaluated solution. 
Envelope 
The free surface given by equation (4.9) is a product of a 
slowly varying term which determines the envelope of the disturb- 
ance times a term which is the sine of a complicated function 
of x and t and which varies rapidly as a function of x and t in 
order to give the individual waves in the wave group. Consider, 
first, the envelope of the free surface given by E(9 ) in equation 
(4,12) where D is defined in equation (4.11). The minus sign 
can be considered to be part of the phase of the sinusoidal tern. 
At x = 0, the envelope becomes Ae~ ote 
Substitute some constant value for x into the equation for 
the envelope, say x = xy and keep that value. As the time varies, 
what happens to the amplitude of the disturbance? The disturbance 
is greatest when t = 4nrx,/gt which shows that the envelope moves 
in the positive x direction with the group velocity of waves of 
the period T. The maximum value of the envelope is given by 
a/(D) 1/4 and so the greatest value of the amplitude of the dis- 
turbance decreases as the wave group travels in the positive x 
direction. 
Letet = 4rx,/er Gia te + t' (Equation (4.13)), so that 
attention can be concentrated on the times near the time when 
the wave group passes the point Xz° The exponent of e in the 
ieee 
