The fictitious wave heights H£ are assumed to follow the Rayleigh prob- 

 ability distribution. Battjes (1974a) presented additional arguments and 

 empirical evidence to support this choice even where waves are definitely 

 nonlinear and do not possess a narrow spectrum. Clipping this fictitious 

 wave height distribution at H = H^ gave the following approximation to the 

 true height distribution, F(H) 



H < 



F(H) = P{H < H} = {l-exp(-H2/H2) < H < H (119) 



— r m 



1 H > H 



m 



where H = the stochastic wave height 



H = the wave height of interest 



H^ = the mean square value of the fictitious wave height 



H = the maximum possible wave height in the surf zone, i.e., R,. 

 m D 



In the shallow-water surf zone, it is further assumed that the effects of 

 variability of wave period and wave direction on breaker heights are neglig- 

 ible. These factors are important, however, in calculations of the ficti- 

 tious wave heights from the complete two-dimensional spectrum. The mean 

 energy per unit area at a fixed location, taking breaking into account is 

 then calculated from linear theory as 



1 = ipgH2 (120) 



where 



12 ^ 



H2dF(H) (121) 



The radiation stresses which can be found (to second order) as the weighted 

 integral of the two-dimensional spectral density are of interest. But again, 

 in the shallow-water surf zone, the only frequency-dependent weighting factor 

 is n which is approximately unity. Consequently, Battjes (1974a) assumed that 

 the radiation stresses are reduced by breaking in the same proportion as the 

 total energy 



S.. = f- S.. = {l-exp(-H2/H^)} S. . (122) 



where S.. = components of the radiation stress tensor 



S"!"^ = components of the fictitious radiation stress tensor 

 -•f without breaking 



such that 



135 



