-5 2 

 with V = 10 ft /sec. It is readily shown that the major part of the 



experimental data presented in Appendix B satisfy this criterion. The 



experiments and the empirical formulas developed for the wave friction 



factor, f , therefore correspond to fully rough turbulent flow conditions. 



When fully rough turbulent flow conditions exist in the model a 

 Froude model will correctly reproduce the prototype conditions. A check 

 of whether or not fully rough turbulent flow conditions may be expected 

 in a particular model test may be performed in a manner similar to that 

 described above, using Jonsson's (1966, Fig. 6) wave friction factor 

 diagram. 



5. Discussion and Application of Results . 



A theoretical analysis of the reflection of water waves from rough 

 impermeable slopes was performed based on the assumptions of relatively 

 long, nonbreaking, and normally incident waves. The general solution 

 for the reflection coefficient is presented in graphical form in 

 Figure 15 which gives R as a function of the horizontal extent of the 

 slope relative to the incident wavelength, I /L, and a slope friction 

 angle, 4> . ^ 



The theoretical analysis accounts for the energy dissipation on the 

 rough slope by including a term expressing the bottom shear stress. 

 Therefore, the analysis introduces and identifies the physically 

 fundamental parameter of the problem as a wave friction factor, f^^. This 

 wave friction factor expresses the effect of slope roughness and is 

 related to the slope friction angle, (j), through the use of Lorentz' 

 principle of equivalent work. A series of experiments was performed in 

 which an accurate determination of the reflection coefficient of rough 

 impermeable slopes was used in conjunction with the theoretical analysis 

 to evaluate the magnitude of the wave friction factor, f^^. From these 

 experimental values of f^^, two empirical relationships for f^^ as a 

 function of the relative slope roughness were obtained. One of these 

 relationships (eq. 124) leads to a simple expression for the slope 

 friction angle (eq. 126) which shows the value of (}) to be independent of 

 the incident wave amplitude. Therefore, equation (126) is particularly 

 convenient for use when the incident wave is given in terms of its 

 amplitude spectrum. The other empirical relationship for f^^ (eq. 125) 

 leads to a more elaborate and accurate relationsliip for (j)(eq. 127). 

 With this relationship the reflection coefficient of rough slopes 

 decreases slightly with increasing incident wave amplitude, thus 

 reflecting the nonlinear nature of the energy dissipation on the slope. 



The resulting semiempirical procedure for the prediction of 

 reflection coefficients of rough impermeable slopes was tested against 

 a separate set of experiments and yielded excellent results for values 

 of the slope angle 1/3.0 <_ tanBg 1_ 1/2.0. For slopes steeper than 

 corresponding to tanBs = 1/2.0 the procedure overestimates the reflection 

 coefficient and a correction factor varying from 1.0 for tang = 1/2.0 



75 



