tan2c 



= f 



M 



w h tang 

 o s 



(109) 



in which F is a slope friction constant given by 



1 J^{2^Y^^^)\ ^ 



S 3t7 



>? y 



172 



1 



y 







1/2 



•^ y 



1/2 



dy 



(110) 



dy 



With "V given by equation (105) it is seen that the slope friction 

 constant, as given by equation (110) is a function of the relative slope 

 length, i^g/L, and of the friction angle, <}) . The evaluation of the 

 integrals and hence F = F (£ /L, (()) is performed numerically using the 

 IBM System 370 computer library routines previously mentioned and F is 

 presented in graphical form in Figure 17. 



b. Methodology for the Determination of the Reflection Coefficient 

 of Rough Slopes . It is clear from Figures 16 and 17 that 

 equation (109) is an implicit relationship for the friction angle, i), since 

 |a| as well as Fg are functions of (j) . If it is assumed that the wave 

 friction factor, f^^, in equation (109) is known for given incident wave 

 (aj^, L, hg) and slope (£3) characteristics, equation (109) may be solved 

 in an iterative manner. For an assumed value of <^ and knowing H^/L, 

 Figures 16 and 17 may be used to obtain values of |a| = R^ 2aj^ and F3. 

 With these values introduced in equation (109) a new value of ^ is 

 obtained and the procedure is continued until convergence is achieved. 

 Once the value of the friction angle is determined, the reflection 

 coefficient, R, is readily obtained from Figure 15. 



The preceding methodology for obtaining the reflection coefficient 

 of rough slopes is straightforward. However, it does rest on one very 

 important assumption--that the value of the wave friction factor, f , 

 is known. Although similar to Jonsson's (1966) wave friction factor 

 his expressions for f^^ are not expected to hold in the present context 

 which justifies asking: What has been gained by the theoretical 

 development presented in the previous sections? 



To answer this question imagine that the problem had been approached 

 on a purely empirical basis. Then, the effects of slope geometry and 

 incident wave characterisitcs in addition to the slope roughness would 

 have had to have been considered. The present theoretical development 

 has circumvented such an extensive experimental investigation by 

 establishing an analytical model, which essentially accounts for the 



57 



