III. REFLECTION COEFFICIENTS OF ROUGH IMPERMEABLE SLOPES 



1 . Preliminary Remarks . 



In the previous section of this report an analytical solution for 

 the idealized problem of wave transmission through and reflection from 

 rectangular breakwaters was obtained. Since most breakwaters are of 

 trapezoidal, rather than rectangular cross section, a considerable amount 

 of energy may be dissipated on the seaward slope of the breakwater. 

 This external dissipation of energy is not accounted for in the analysis 

 of porous crib-style breakwaters. To account for the external dissipation 

 of energy on the seaward slope of a trapezoidal breakwater the associated 

 problem of energy dissipation on a rough impermeable slope is considered 

 both theoretically and experimentally. 



A theoretical analysis of this problem is based on the following 

 assumptions : 



(a) Relatively long normally incident waves which may be considered 

 to be adequately described by linear long wave theory. 



(b) Energy dissipation on the rough impermeable slope may be 

 represented as the energy dissipation due to bottom frictional 

 effects . 



The first assumption is identical to the assumption made in 

 Section II of this report. The second assumption presumes that the 

 effect of energy dissipation due to wave breaking is minor. This may 

 seem to be a restrictive assumption. When realizing that the seaward 

 slopes of breakwaters are generally steep, this assumption is quite 

 reasonable. At any rate, the main purpose of the theoretical analysis 

 is to produce a rational framework within which the experimental results 

 for reflection coefficients of rough impermeable slopes may be 

 analyzed. 



The essential features of the mathematical manipulations and the 

 derivation of the governing equations are presented in Appendix A to 

 enable the treatment to be relatively brief and to the point. The 

 frictional effects on the rough slope are accounted for by introducing 

 a term relating the bottom shear stress to the square of the horizontal 

 orbital velocity through the use of a wave friction factor, f^, 

 analogous to that introduced by Jonsson (1966). The bottom shear stress 

 is linearized and a theoretical solution for the reflection coefficient 

 of rough impermeable slopes is obtained in terms of a linearized slope 

 friction factor. By using Lorentz' principle of equivalent work an 

 implicit solution for the reflection coefficient of rough impermeable 

 slopes is obtained in terms of incident wave characteristics, slope 

 geometry, and the wave friction factor, f , which expresses the effect 

 of slope roughness. 



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