bracketed term. If Y is zero and W, is fairly large, for an 
x which is not too large, the integrals will have to be evalu- 
ated from a large negative value to a large positive value and 
by the same arguments employed in Chapter 5, the value of the 
bracketed term will be essentially two. If, say, ¥ = K,W./2 
then the integrals will have to be evaluated from zero to a 
large positive value, and the value of the bracketed term will 
be one half. Thus when Y = KW, /2 or Y¥ = -K,W./2, the potential 
energy, averaged over a relatively short interval of time, of 
the waves at that point under the envelope will be one fourth 
of its value near the center of the disturbance. Similarly in 
the second bracketed expression if (2 p4x/eK,) -t- (D/2) =—0), 
or if (2m 4x/eK,) -t +(D,/2) = 0, the average potential energy 
will be one fourth the value at the center of the disturbance. 
If the four equations treated above are put back into their 
original form as a function of x,y and t and 85 by the use of 
(8.51), (8.52), (8.53), and (8.41), then y and t can be found as 
a function of x and the other parameters of the solution. The 
result is equations (8.63), (8.64), (8.65) and (8.66). Fora 
fixed value of x, O55 WW, Dis and p 1? these equations are equa- 
tions of four straight lines in the y,t plane. Segments of these 
straight lines are graphed for x = xy in the upper right of the 
y,t plane shown in figure 18. Their intersection determines a 
rectangle in the y,t plane. Inside the rectangle, the disturb- 
ance is at essentially full amplitude, and at the heavy bound- 
aries as indicated on the figure, the average potential energy 
if one fourth of what it is in the center. 
- 200 - 
