Two approaches to mathematlcaly describe irregular waves in shallow 

 water exist. Ijima, Matsuo and Koga (1972)^^ used an equilibrium linear 

 spectral model (frequency domain) related to the deepwater spectrum. 

 McReynolds (1977) used a two-component frequency spectrum to simulate a nar- 

 row spectrum, Battjes (1974a) argued that this approach gives an upper 

 saturation limit on the spectral density values which depends on the width 

 of the spectrum — this is not realistic. Also, wave breaking is highly non- 

 linear and occurs to individual waves in physical space (space domain) and 

 not to individual spectral components. For these reasons, Collins (1972) 

 and Battjes (1974a) both adopted irregular models based on a wave-by-wave 

 height theoretical and empirical probability distribution for individual 

 waves in the space-time domain. In this model, it is the integral of the 

 spectrum which has an upper bound in shallow water and not the spectral 

 density. No rigor is claimed, only a rational approach where nonlinear pro- 

 cesses are important. 



Collins' (1972) approach was to consider the irregular sea as the en- 

 semble average of periodic components, each with its own H^, Lq, and a^ in 

 deep water. The energy, energy flux, momentum flux, radiation stresses, and 

 longshore current velocities are assumed expressible in terms of H , Lq and 

 Dtp for regular waves. The mathematical expectation of these quantities is 

 then calculated assuming that the joint probability density function of the 

 stochastic ensembles of H^, L^, and a^ is known. Suitable functions for 

 the joint probability density function are determined by empirical means from 

 field data. Implied in this approach is the assumption that various non- 

 linear interactions between the waves and the mean motion (e.g., wave setup 

 changes, wave characteristics) are properly represented by ensemble averages 

 of Individual waves. Battjes (1974a) presents arguments to show that this 

 assumption is incorrect. In the highly nonlinear surf zone, the contribution 

 of a wave with certain characteristics to wave setup, longshore current velo- 

 city, etc. is affected by the presence of waves of different characteristics. 

 Therefore, Battjes took a different approach. 



For irregular waves on gentle slopes with spilling breakers, Battjes 

 made the basic assumption that 



"...at each depth a limiting wave height H^ can be defined 

 (which may also depend on the wave period), which cannot be 

 exceeded by the individual waves of the random wave field, 

 and that those wave heights which in the absence of breaking 

 would exceed B.^ are reduced by breaking to the value H^^ . " 

 (Battjes, 1974a, p. 125). 



The energy variation in the surf zone thus results from clipping a fictitious 

 wave height distribution which is present (theoretically) if breaking did 

 not occur. This upper bound is found from the regular wave breaking ratio Y. 

 In this way, energy varies gradually due to shoaling, refraction, bottom 

 friction, and because of the increasing number of breaking waves in shallow 

 water. 



'*3lJIMA, T. , MATSUO, T. , and KOGA, K. , "Equilibrium Range Spectra in Shoaling 

 iWater, " ProoeedingSj l2th Coastal Engineering Conference, Vol. I, Washington,' 

 D.C., 1972, pp. 137-149 (not in bibliography). I 



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