-19- 
already capable of differentiating between completely unrealistic spectra 
and more realistic ones. If the space scale of x) and z, is changed for 
the same vector realization used to construct figures 1 and 2, the result 
is figure 3. Here a is triple valued for a certain range of x, and when 
z(x) is plotted the surface forms a large loop. This may be indicative of a 
very unstable state in the waves that could not be attained in nature be- 
cause breaking waves would form each time such a loop would be initiated 
in the space-time representation and the kinetic energy of the wave motion 
would be dissipated by turbulence in the whitecaps to an extent that the 
particle motion spectra would be modified to a more realistic condition. 
Conditions for a loop in z = z(x) 
A loop occurs in z = z(x) if z is triple valued as a function of x. 
This in turn occurs if x(a) is not monotonically increasing. Thus if 
m <0 where 
a 
2s — - - 
(33) Wear l Zalk cos(k a w itt c)) 
a loop will form. 
The probability that 
(34) Dawe KmCoOs (kia@i-iwert teem) eal 
nen n n n 
must therefore be investigated. 
The three moments given by 
(35) a = f S(k) dk 
(eo) 
(36) #, = fS(k) k dk 
(o) 
and 
(37) vo = T S(k) mora 
{e) 
need to be considered in this connection, and from (34) it is evident that 
b, must exist and be bounded. 
The probability that m will be less than zero is given by 
