all of the time. The constants, Cis Coy and CS were determined from 
f, (8), f,(e), and f,(0). These functions of © must be positive 
everywhere or else [ay (pn ,0)1° can have negative values. Therefore 
f,(e), f,(@), and f,(8) must be zero. Therefore for values of # 
greater than p K? the power spectrum must be of the form f,(0)/u 4 
(at least) such that when multiplied by» 3, it goes to zero asp 
approaches infinity. A better way to state this requirement is 
given by equation (11.16) because fractional powers in the series 
expansion are then also possible. 
The results that have just been obtained can be interpreted in 
a very easy way by considering sea surfaces composed of purely per- 
iodic ten second waves, purely periodic five second waves, purely 
periodic two and one half second waves, and so on. If the various 
separate wave trains are all of the same height then the particle 
velocities at the surface are twice as great in the five second waves 
as in the ten second waves and four times as great in the two and 
one half second waves as in the ten second waves. If a condition 
such as the one just derived is not imposed, very strong velocities 
must result. 
The slope and curvature of the sea surface 
These power integrals can also be differentiated and integrated 
with respect to the time and space wariables. The slope of the sea 
surface in the x direction as a function of time and space variables 
is given by equation (11.17). By the methods of Chapter 10, this 
equation can be reduced to a function of time at any point. It then 
follows that the slope in the x direction is given by an integral 
of the form of equation (7.1) except that the power spectrum is 
