Two approximations were employed in the derivation of the 
solution. The first approximation was in the formulation of 
the integral representation given by equation (8.37). This in- 
tegral representation permits some of the spectral components 
to have a component of travel in the negative x direction. By 
the arguments given in Chapter 5 for the simpler case, this ap- 
proximation is probably not too bad. The effect of the other 
approximation, namely that given in equation (8.49), is probably 
more important. The approximation is more accurate for small 
values of ©,. With ©, greater than 7/4 or less than -7/4, the 
approximation becomes poorer. A more accurate evaluation of the 
integral might show that the parallelogram form in the x,y plane 
shown in figure 18 would tend to lose the sharper corner 
and assume the shape of a rectangle with sides parallel to the 
‘ 
X and Y axes as it travels along. The approximation is adequate 
in the sense that it locates the disturbance fairly precisely 
and shows where it is not located to a great degree of accuracy. 
Additional comments 
It would be possible to take the two initial value problems 
given in this chapter and manufacture some model wave systems 
from storms at sea which have properties which would be analogous 
to those models studied in Chapter 6. Models with discrete 
spectral components which would travel in all directions within 
a 180° sector could be manufactured. They could be made to be 
infinite in duration and width, infinite in duration and finite 
in width, finite in duration and infinite in width, and finite 
= 2040 
