predicted by Miche (1951) (eq. 4) for incipient breaking. For a given 

 slope, however, maximum relative runup for successively smaller values 

 of d s /gT 2 occurs at correspondingly smaller values of H^/gT 2 . This 

 relationship is shown in Figure 3 which is a set of runup data curves 

 for a smooth 1 on 2.25 slope fronted by a horizontal bottom. Each line 

 represents a different d s /gT 2 value, and it shows that the maximum 

 R/H^ value occurs for a range of H^/gT 2 values as dg/gT 2 varies. 



Comparison of data for different slopes indicates that, when H and 

 H^ are considered approximately equal, equation (5) gives roughly the 

 maximum wave steepness for nonbreaking waves. It does not, however, 

 preclude breaking waves for lower values of H^/gT 2 and dg/gT . 



Miche (1944) developed the following theoretical equation for non- 

 breaking wave runup for structures in deep water: 



H * \28 ' 



(6) 



where 9 is the structure slope measured in radians. This equation is 

 applicable only to waves which are in deep water at the structure toe, 

 and to steeper structure slopes. 



Hunt (1959) gave an empirical equation for runup from waves breaking 

 on a structure slope, using equation (5) as a limiting condition, as 



£ = 0.405 tan 9 6 i;9 for -S5- > 0.031 tan 2 9 . (7) 

 H (H/gT 2 ) U2 gT 2 



Hunt's equation was developed from the observation that, for the steeper 

 waves which break on the structure slope, relative depth loses its sig- 

 nificance in determining runup. 



Since a wave may break on a slope for differing wave steepnesses as 

 relative depth, dg/gT 2 , varies, Figure 4 was developed from smooth- 

 slope runup data to show the variations. The lines in the figure are 

 based on estimates of the wave steepness Values for which a curve of 

 constant dg/gT 2 becomes tangent to the "line of complete breaking" 

 which is determined empirically for each structure slope from data plots 

 (see example in Fig. 3). The lines in Figure 4 give estimates of the 

 minimum wave steepnesses necessary for incident waves to break on a 

 given slope for the particular relative depths, dg/gT 2 . From the 

 empirical data, an equation similar to equation (7) but developed for 

 the deepwater wave height is 



— = (cot 0)- 1 - 04 (4.23) (lO) 2 ^" 1 ) -2_ for cot > 2.0 . (8) 

 HI \gT 2 / 



20 



