Figure 20 shows, clear differences in the data variation near the 

 pattern minimums for the two piles. At a circular pile, singularities 

 in the second angular derivative can occur near the water level minimums 

 (Hallermeier, 1976); the minimums then indicate the break in water sur- 

 face slope associated with flow separation (Longuet-Higgins, 1973). The 

 W(B) mimimums indicate maximum sheltering of the channel from incident 

 wave action, and must be related to channel geometry. 



The calculated velocity head of the incident crest, U^/2g, gives a 

 fairly close upper bound to the front runup of a single crest at a cir- 

 cular pile, [W(a = Q°)-W], for a subset of this study's data considered by 

 Hallermeier (1976). He reported that velocity calculations are somewhat 

 problematic for th_ese tests. For the most part, the wave conditions do 

 not match those included in the stream-function tables of Dean (1974) , 

 and the numerical stream- function theory is the only available theory 

 accurate throughout the range of d/L in these tests. Also, the test 

 waves generally have complicated profiles (see App. B) . Furthermore, 

 the runup need not be exactly u2/2g, since various regimes of flow 

 stagnation are possible (see Sec. 11,4), and the variation of horizontal 

 flow velocity with depth causes downward secondary flow at the front of 

 the pile, which might decrease runup. For these reasons, velocity head 

 calculations have not been made for the complete data. 



However, Figures 21, 22, and 23 exhibit trends consistent with crest 

 stagnation causing the front runup; i.e., the peak water level at a = 0° 

 is approximately the_incident crest height plus the velocity head. Fig- 

 ure 21 shows three [W(a)/W] patterns for the 3-inch circular pile in 

 waves of three heights at the same period. Figure 22 shows three [W(a)/W] 

 patterns at the same pil_e for^ waves at three periods with the same height. 

 Figure 23 shows three [W(a)/W] patterns for the 6-inch circular pile in 

 waves of three heights at the same period. 



Linear wave theory provides a first approximation for the peak hori- 

 zontal velocity at the wave crest: 



U = (2TrW/T) [cosh k(W+d)]/sinh kd , (1) 



where k = (2Tr/L) is the wave number and W = H/2 for the sine wave consid- 

 ered by linear wave theory. The numerical stream-function theory provides 

 a more accurate value of U for waves of finite amplitude (Dean, 1974). 

 Table 5 presents the normalized measured and calculated front runup for 

 the_ data in Figures 21, 22, and 23. The measured runup is given as 

 {[W(a = 0°)/W]-l}, and the calculated runup as (U^/2gV), where U has 

 been obtained by interpolation in the stream- function tables (Dean, 1974, 

 Vol. 1, pp. 86-91). The measured values are larger than the calculated. 

 Identical trends occur in measurements and calculations: at a given 

 wave period, normalized runup increases with increasing wave height; at 

 a given wave height, normalized runup decreases with increasing wave 

 period. These trends agree with the velocity given by equation (1). 



Minimums of [W(a)/W] generally change location and dimension with 

 changing wave conditions. However, this behavior does not seem to be 



41 



