as was found for the transmission coefficient, Y^^ (lower half of Fig. 10). 

 Comparison of Figures 10 and 20 suggests that the amount of wave energy found 

 at the forcing period will increase as the transmission by overtopping coeffi- 

 cient increases. 



The case of irregular waves is where the incident wave energy is distributed 

 over a range of wave frequencies (several measured incident laboratory wave 

 records and computed wave spectra are shown in Figs. 4 and 5). Tests with 

 irregular waves indicate that the shapes of the incident and reflected wave 

 spectra are approximately the same (two examples are given in Fig. 21). The 

 approximately constant spectral shape is shown by the spectral-peakedness 

 parameter, Qp, where the value for the reflected waves, Qpj), is approxi- 

 mately equal to the incident spectral peakedness, Qp^ (Fig. 22). The shape 

 of the transmitted spectrum may be approximately equal to or sharper than the 

 incident spectrum (Fig. 22) with the spectral-peakedness parameter of the trans- 

 mitted waves, Qp^, greater than or equal to Qp^ (Fig. 22). Secondary waves 

 may appear in the transmitted wave spectrum at harmonics of the period of peak 

 energy density, Tp, (Fig. 21). 



A zero up-crossing analysis (Fig. 23) was performed on the wave records to 

 allow statistical examination of individual wave heights and periods. Since 

 reflected waves contaminate the incident wave conditions, an analysis was 

 performed for the record from each gage, then results averaged to minimize the 

 influence of reflection. Cumulative height distributions were then prepared 

 for incident and transmitted waves. The cumulative curves were put into dimen- 

 sionless form by dividing by the observed rms wave height, ^rms > ^""^ ^^e 

 dimensionless heights at various probability levels, p, determined (p = 0.01, 

 0.02, 0.05, . . . 0.60). A plot of these dimensionless heights for transmitted 

 versus incident waves indicates the shape of the transmitted wave height distri- 

 bution as a function of the incident wave height distribution. For the case 

 of a breakwater with the water depth at the crest level (dg/h = 1.0 or F = 0) 

 the transmitted wave height distribution is approximately the same as the 

 incident height distribution (Fig. 24) . If the water level is below the crest 

 elevation (dg/h = 0.80, positive freeboard), the transmitted wave height distri- 

 bution is skewed toward larger waves (Fig. 25) . This means that the larger 

 transmitted waves are bigger than predicted by the transmission coefficient, 

 Ky^. For example, at the 5-percent level, transmitted waves are 30 percent 

 larger than expected from the overall transmission coefficient and at the 

 1-percent level 100 percent larger. 



The above observations are consistent with the wave transmission by over- 

 topping model given by equation (14) . At zero freeboard the transmission 

 coefficient is approximately constant, so all waves in a distribution will 

 transmit the same amount and the distribution will remain unchanged. However, 

 for subaerial breakwaters the larger waves will have smaller F/R ratios and 

 transmit more efficiently than small waves, so that the transmitted wave dis- 

 tribution is skewed toward large waves. 



The joint distributions of wave heights and periods observed in the 

 laboratory illustrate the same overall trends found in the field. Larger 

 waves have a mean period approximately equal to the period of peak energy 

 density in the spectrum, Tp (Coda, 1978), with the average wave period 

 decreasing for smaller wave heights (Fig. 26). The correlation between 



35 



