The roughness value, k r , is used in describing roughness elements on 

 a slope. For stone, k r is the equivalent spherical diameter, based 

 on the weight and density of the armor unit; for a concrete armor unit, 

 kp is defined specifically as a characteristic dimension of that armor 

 unit. Because effects of porosity and roughness are difficult to differ- 

 entiate, various structure types and cross sections are analyzed indepen- 

 dently, with notation describing the structure characteristics (e.g., 

 filter layers, if any; thickness of armor layer; height of core). 



One of the above dimensionless variables is reformulated and, 

 together with the other dimensionless variables, gives the following 

 principal variables used: 



o /H' d H' - h \ 



% W % V L' dj 



where R e is the depth Reynolds number (discussed in Sec. VI, 2). The 

 term Z/L is used, rather than Z/gY 2 -, because it was assumed that if 

 the wavelength in the flat part of the tank is L < 2Z, the relative 

 runup would be a function of a wave substantially influenced by the 

 beach slope, and the relative beach-slope length, Z/L, could be 

 neglected. Some experiments had wavelengths much longer than the slope 

 length (up to L « St). For such conditions, in which L > 2Z, relative 

 runup is expected to be a function, in part, of Z/L. This beach-slope 

 effect is discussed further in Section IV, 3. 



The term dg/H^, (relative depth) is used for consistency with the 

 SPM. However, it is useful in that for each value of dg/H^, the 

 relative roughness term, H^/k r , also has a constant value for a given 

 absolute armor unit dimension and depth. An alternate form of relative 

 depth, d s /gT 2 , is used occasionally, but principally as a means of 

 deriving dg/H^ (see Sec. IV) . 



III. THEORETICAL AND EMPIRICAL EQUATIONS 



1. General . 



Theories dealing with wave runup at the shoreline are applicable to 

 either breaking or nonbreaking waves, but usually not both types. In 

 this classification, waves break because of instability caused by 

 decreasing depths instead of instability related to waves of maximum 

 steepness in a uniform water depth. Various breaking criteria have 

 been developed; a detailed discussion is given in Technical Advisory 

 Committee on Protection Against Inundation (1974). Most nonbreaking 

 wave theories are derived for rather long waves on very gentle, uniform 

 slopes extending to an "infinite" depth. Breaking wave theories gener- 

 ally are concerned with a bore-type propagation on gentle slopes, rather 

 than the plunging or spilling types commonly encountered on structures 

 or steep beaches. Breaking waves are discussed here as related to 

 structures in the coastal zone. 



