above empirical results are from monochromatic model tests of limited duration 

 which do not account for the natural irregularity of ocean waves nor the ef- 

 fect of variable duration of exposure. The many untested or otherwise unre- 

 solved questions about breakwater damage modes should not, however, prevent 

 designers from applying the information that is available. The need to con- 

 firm analytical predictions of breakwater stability and performance by scale 

 model testing prior to construction cannot be overemphasized. 



63. A characterization of damage as a function of incident wave height, 

 with the features of Equation 14, allows the "expected" or long-term average 

 damage to be estimated. The statistical definition of expectation for con- 

 tinuously distributed variables is . 



E{x} -f 



xf(x) dx (15) 



where f(x) is the probability density function (pdf) of x (DeGroot 1975). 

 A function of x , g(x) may be substituted for x in Equation 15 without 

 changing the definition, thus 



Eg(x) = rg(x)f(x) dx (16) 



The long-term distribution of wave heights formulated for most current design 

 exercises to represent the incident wave climate is derived as a cumulative 

 probability distribution (cpd) F(H) where 



f(H) = ^^ (17) 



dH 



The expected annual damage can then be estimated from a damage function 

 7oD(H^) , such as Equation 14, and a cpd for wave heights F(H) by using the 

 following equation: 



E ^ = X 



/»Ki^)[^>" 



where X = the Poisson parameter or average number per year of extreme 

 events represented by H values. This formulation assumes that the number 

 of storms per year is a random variable and can be represented by a mean 

 value. It assumes further that this number is independent of the H values 



41 



