the potential function. At z equal to zero the expressions simplify 
considerably, and possibly some interesting properties about the 
power spectra in the transition zone can be deduced by considerations 
Similar to those of the previous chapter. 
The kinetic energy integrated over depth and averaged over 
time and the y, direction is given by equation (12.47). The Itcoth 
of H 2H/e times the hyperbolic tangent of }# *n/elI( # 4H)] is equal 
to one by virtue of equation (12.11). Thus the average kinetic energy 
is equal to the volume under ec ee ils (that is, equation (12.48)) 
times pg/4. From previous considerations this is equal to the po- 
tential energy averaged over YR and t. At a fixed point, say, Xp 
and yp equal to zero, where the statement is exact these values also 
hold and the potential energy and the kinetic energy (integrated over 
depth) averaged over time are both equal to (pg E )/4. This 
RHmax 
statement can be proved by use of the results of Chapter 10. 
The flux of energy toward shore in ergs/sec per centimeter of 
length along the YR axis, is the average value of the work being 
done on the Ypoz plane determined by setting XR equal to zero. The 
wave power is then given by equation (12.49), and the results check 
with the same result in Lamb [1932] where the flux is determined 
for a pure sine wave. 
If the short crested wave system is concentrated in a narrow 
8p band width at the point of observation, and if the important 
spectral components are all traveling in nearly the same direction, 
Say On,, then it is possible to omit the cos 8, term in equation 
(12.49). Then equation (12.49), as modified, is the flux of energy 
in the Ont direction at the point of observation. It can then be 
estimated (except for the short crested effect) from [Ang (H Ie as 
61 
