where 



Rx (2rit ~ ratio of proportions of native material to the 

 borrow material at the critical phi value (when 

 phi value is that 4> size with greatest ratio 

 of^ the proportions of native sand to borrow 

 material) , 



Ojj = (4)g - <i>-, f:\j2 Standard deviation is a measure of 



sorting. (See Section 4.2) (5-2) 



^<j> ~ '^Qk ■*■ "^le)/^ P^''" ""^^^ diameter of grain size 



distribution, (See Section 4.2) (5-3) 



-^ = subscript b refers to borrow material 



-„ = subscript n refers to natural sand on beach 



<|>8i+ = 84th percentile in phi units 



i|>jg = 16th percentile in phi units 



e = (base of natural logarithms, 2.718) 



This formula assumes that both composite native and borrow material 

 distributions are nearly lognormal. This assumption can be e^qjected to 

 be satisfied for the composite grain size distribution of most natural 

 beaches and for naturally deposited borrow material that is almost homo- 

 geneous. Pronounced skeuness or bimodality might be encountered with 

 borrow sources that contain alternating horizons of coarse and fine 

 material, such as clay-sand depositional sequences, or in borrow zones 

 that cross cut flood plain deposits associated with ancient river channels. 



The formula for R(j) arit is not applicable to all possible combina- 

 tions of grain size moments for borrow and native material. The possible 

 combinations can be subdivided into the four basic cases given in Table 

 5-1, and indicated as quadrants in Figure 5-3. Table 5-1 shows that, 

 R<f> crit is assumed to have direct application in only one of the four 

 cases. 



In Case 1, the borrow material has an average grain size finer than 

 that of the native material, but the borrow material is more poorly sorted. 

 The basic prerequisite is satisfied that there is some borrow material 

 present for each of the size classes which contain native material. There 

 will be some coarser size classes beyond which this supply will be more 

 than adequate. However, a large part of the borrow material is finer 

 than that expected to remain in the active littoral zone. The best esti- 

 mate of the overfill ratio is the critical ratio calculated by Equation 

 5-1. This should be a conservative estimate since overage quantities in 

 all size classes other than that for Ra ^r-tt would not all be expected to 

 be completely lost from the active area. 



5-11 



