S20" 
(38) plit< 0) =—————— Ne dt'=)——— { e dt . 
Equation (38) may be interpreted to be the probability that some particle, 
a.) will be found at the top of a loop between the points where z = z(x) is 
vertical for some particular time of observation. Sucha particle is 
surely involved in a breaking wave. Other particles that do not satisfy 
this condition also may be involved in a breaker. 
Since the spectrum S(k) has not as yet been measured, it is 
necessary to investigate possible relationships between S(k) and spectra 
of z(x,0) and z(0,t) before estimating the probability in (38). 
Marginal distribution of z=z(x) 
The graphical construction used to illustrate the generation of the 
random function z = z(x) can be used to obtain the marginal distribution of 
z= 2(x). The joint density of z(a,0) and & m(a,0) is bivariate normal. 
The variance of z(a) is ee the variance of m(x) is bo, and the covariance 
is oT as defined in (34), (35) and (36). 
If f(z,m) is the bivariate normal distribution of z and m, we 
note that 
(39) F(z, m) = f(z, m)m 
nearly has the required properties of a probability density function since 
m is dimensionless and E(m) = 1. Also since x and a have the same 
dimensions, a given number of equally spaced points in a small interval 
of a given length over a produce m times this number of points with this 
same spacing in an interval of a different size on the x axis. 
The function F(z,m) is not a probability density function strictly 
because it is negative for m< 0, but this corresponds physically to im- 
possible configurations of z = z(x), and points on z = z(x) generated when 
m <0 will be shown to have extremely low probabilities. Actually 
even more points on z = z(x) are probably destroyed by the breaking pro- 
cess, and z = z(x) is modified in form long before m becomes zero for 
some sea conditions. 
However, as an approximation, we can truncate F(z,m) at 
m = 0 and renormalize. Thus 
