1. Monochromatic Wave Crest Elevations . 



a. Waves Traveling Over a Horizontal Bottom . Dean (1974) developed a 

 stream-function wave theory to predict a number of wave characteristics for 

 waves propagating over a flat horizontal bottom for a number of relative depths 

 and wave heights equal to 0.25, 0.50, 0.75, and 1.00 of the breaking limit. 

 An examination of the tabular values of ri£./H (Dean, 1974, Vol. II) shows that 

 relative crest height can be approximated using the empirical relation: 



^ ^ fn (3) 



H 



1.0 + tanh -A2 in. (A3 



e] 



where Lq is the deepwater wavelength given by Airy theory as 



Lo = ^ (4) 



Wave trough elevation, n^, is equal to wave height minus crest elevation. 

 Dean's tabular results are compared with equation (3) and Aj is found to be 

 1.0 for wave traveling over a flat bottom (m = 0.0). Values of A2 and A3 

 are functions of relative depth (d/L^ or d/(gT^)), as shown in Figure 8 and 

 given in Table 2. 



Laboratory data in this study show excellent agreement with Dean's stream- 

 function theory and give a value A^ = 0.992 ± 0.0393 (Table 3). Cnoidal waves 

 (Uj^ > 25) have slightly smaller values of this empirical parameter with Aj = 

 0.95 ± 0.038 (Table 3). 



b. Waves on a Slope . Slngamsettl and Wind's (1980) data are used to 

 evaluate Aj for monochromatic waves at the breaking point on various beach 

 slopes. Wave crest elevations, as indicated by A^, decrease approximately 

 linearly with beach slope, m (Table 3, Fig. 9). For example, waves at the 

 breaking point on a 1 on 5 slope (m = 0.20) have only 67 percent of the crest 

 elevations of a wave with same H/(gT^) and d/(gT^) traveling over a flat 

 bottom. 



Figure 10 illustrates the predicted Influence of beach slope on the 

 breaking crest elevation for a sample wave tank condition. In this case waves 

 in the Incident flat part of the tank have d/(gT2) = 0.019 and H/d = 0.37. 

 The figure shows the predicted crest elevation at the breaking point using the 

 method described in Appendix F for estimating ti„. In this example slopes 

 flatter than 1 on 20 have little influence on the crest elevation, while slopes 

 steeper than 1 on 20 produce lower crest elevations. 



2. Crest Elevation Prediction Aids . 



Equation (3), which is used for predicting monochromatic wave crest eleva- 

 tions, can be presented in a number of graphical forms useful for predicting 

 and understanding trends in relative crest elevation. Figure 11 shows T]^/Yi 

 versus relative depth for various wave steepnesses. Relative crest elevation. 



