and G2(@), 2) is the quadratic response function 
29(T), Tr )€ Ord dT. 
——_ 8 
G7(@1,@2) = | 
8 
Through these manipulations, Hineno obtained his expression for the probability 
distribution function as 
2 h h 2 h D) 
P(z) = exp] -—— <0 | 30 | = 3 } +— 42). 12.120) 
2h20 hyo = 6h \ h20 2ho2h20 
The probability distribution density function p(z) and the expected 1/n highest value 
Z1/n are obtained from the relations 
p(z) = “ey @2s2 1) 
dz 
| z p(@)dz eo 
Z1/n = a =n | z p(z)dz. (12.122) 
{ P(z)dz eae 
Z1/n 
If we use Eq. 12.120 for P(z) and consider that in real calculations, the smallness 
parameter € is absorbed in the computation of G2(w ,w2) and does not appear explicitly, 
fe Zi hijo _ h30 hy 1 
PU) expla nn ae pee 
2h20 hoo =2h3g = 2h20ho2 = h20 
1 h h h h h 
— | 0 | AL 2.123) 
This is the final expression of the probability distribution density function of the maxima. 
Using this expression, M. Hineno calculated the probability characteristics of the maxima 
and minima of the waves that were treated as nonlinear with the quadratic response 
function, as shown in Eq. 9.10 in Section 9.2.1, 
361 
