APPENDIX A 



CHOICE OF LENGTH SCALE IN SEDIMENT ENTRAINMENT PARAMETER 



Published results for oscillatory flow over horizontal sediment beds 

 indicate grain diameter, D, has a weak influence on the transition from 

 a gently agitated, rippled bed to an intensely agitated bed with lesser 

 bed forms. Mogridge and Karaphuis (1972) reported wave tank and water 

 tunnel tests showing that bed- form height reaches a maximum and then 

 steadily decreases with increasing A/D, where A is near-bottom horizontal 

 fluid orbit diameter; the maximum bed- form height depends primarily on 

 A and only slightly on D. Chan, Baird, and Round (1972) reported tests 

 in a small-diameter pipe showing a well-defined transition from a rippled 

 bed to a regime of intense bed agitation with surface particles in motion 

 throughout the oscillation cycle. The empirical expression for this 

 transition is 



A oj 



1.2 



,0.5 ^0.1 .0.2 



= 6.6, (Al) 



where (i) is oscillation frequency, v is kinematic fluid viscosity, and 

 D occurs as a relatively weak factor. 



The sediment mobility parameter (eq. 1) is a function of wave height 

 and period, water depth, and grain diameter. In writing the sediment 

 entrainment parameter (eq. 2) , replacing D by ed constitutes a direct 

 approach to clarifying the algebraic dependence of the limit depth, dg , 



on wave hei^t and period, combined in CHo/L^) . Because definitive data 

 on sediment entrainment are lacking, any choice for the length scale in 

 the denominator of this Froude number must be tentative on physical 

 grounds. Four lengths naturally occur in a shoaling wave: H, d, L, 

 and A. For the four possible expressions, Table A-1 shows the resulting 

 functional dependence of limit depth on wave steepness (corresponding to 

 eq. 6), and the value of each expression at C = 0.9 divided by its val- 

 ue at ? = 0.3, the important range of limit depth. .Using ed for the 

 length scale gives calculated results in agreement with experimental ev- 

 idence over a wide range of ? or H^/L^ (Fig- 3 and App. B) , so the other 

 possible choices cannot, because the other values in the last column 

 of Table A-1 differ greatly from the first. 



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