Equation (6.6a) is a specific example of what equation (6.6) 
might look like after different functions had been picked for 
an(#) and b,(y) and after integration overp . It could be 
graphed as a function of t but the phase shifts indicated from 
group to group would require more precision than is warranted 
for the purposes of illustration. An easier function for pur- 
poses of illustration can be found by setting ante ) equal to 
zero and the on equal to zero. The "waves" under the envelope 
then factor out and the term in the bracket represents the over=- 
all envelope in equation (6.6b). Equation (6.6b) is the last 
graph in figure 10. The number of different functions which could 
be constructed according to equation (6.6) is limited only by the 
imagination. It will be left as a problem for the reader to solve 
to find out what the functions a,(y) and b,(y) are which yield 
equations (6.6a) and (6.6b). They are all smooth piecewise con- 
tinuous and piecewise differentiable functions. 
In this chapter, it will be assumed that the wave crests are 
infinitely long in the y direction. The results will then be in- 
dependent of y, and a y could be substituted for the second zero 
in all of the equations of Plate XIV. It will therefore be omitted 
and the free surface will be treated as a function of x and t. 
Equation (6.1) is trivial. If the wave is traveling in the 
positive x direction, the only possible motion is given by equa- 
tion (2.19) where @ = 0, 5 = 3r/2, and for infinitely deep water, 
hs et /2r. The comments made on equation (2.19) still apply. 
If the observation were to represent a storm at sea, the storm 
would have started before the start of time, and it would never 
- 105 - 
