When x is not zero, as a function of y, the profile of 
-FF(x,y,t) where it exists looks at an edge like the dashed curve 
in figure 9. Similarly when x is not zero, as a function of t 
the same profile would be found. The product of the two profiles 
is rather complex at the corners of the rectangle. 
As X approaches zero in equation (8.62) the radicals in 
the integrand become infinite. The quarter power lines move 
to the position indicated at the origin of the y,t plane in figure 
20. This shows that the solution reduces to the initial values 
given in the formulation of the problem despite the approximations 
employed in evaluating the integral. 
N ow the function will be studied for fixed values of t as 
a function of.x and ye A second set of coordinate axes defined 
by X and ¥ as given in equations (8.52) and (8.53) are also use- 
ful. The bottom graph in figure 18 shows the two coordinate sys- 
tems. The quarter power points in the Y direction are given simply 
by equations (8.67) and (8.68). In the x direction, they are 
given by equations (8.69) and (8.70). The area in the x,y plane 
occupied by the waves is consequently a parallelogram with sides 
parallel to the Y 
axis and the x axis. The individual wave crest 
segments are parallel to the Y axis and travel in the positive 
X direction. On the X axis, the value of X for the forward edge 
of the disturbance is given by X = x/cos0, = g(t + D,/2)/2y, 
which shows that the forward edge of the disturbance travels in 
the positive X direction with the group velocity of waves with 
a period }# j- 
= 20Oe) = 
