^ J C2k £ /T^ifT) + ^— J, (2k ii, A-if, ) 



0^ O S b r-i r-j— 1^ o s b 



/ 1-lf, 

 b 



and the complex runup amplitude 



ik i 



_X e ° ^ . (95) 



2a. 



^ J (2k £ /l-i£, ) + i— J, (2k a /1-if, ) 



o^ o s b-^ n — rf- 1 s h-' 



b 



It can be seen from equation (94) that [a^,], which is the physical 

 amplitude of the reflected wave, is equal, to a^ for f-^ = 0. Since 

 f|2 = expresses the condition that the energy dissipation on the slope 

 is zero, this is to be expected. 



Equations (94) and (95) show that the important parameters in 

 determining the reflected wave amplitude and the runup amplitude are 

 the length of the slope relative to the length of the incident waves 

 in front of the slope, £.3/1, and the friction factor, f^, arising from 

 the linearization of the bottom frict ion t erm. Since the linearized 

 friction factor appears in the form vlTTf^ it is expedient to introduce 

 the friction angle 4) defined by 



tan2(i) = f^ ; <_2<^ <_j , (96) 



/rif^ = (1 + tan^2*)^/'^ e"^* . (97) 



In terms of the relative slope length, l^/L, and the friction angle, 

 (\> , the reflection coefficient, R = |a |/a., may be determined from 

 equation (94) . This solution is presented in graphical form in 

 Figure 15. 



Similarly the nondimensional runup amplitude. 



2a. 



(98) 



52 



