the other hand p is rather large, there is a good chance that the condi- 

 tions around the spot on the bed where the particle is coming to rest 

 after completing a step of length A'^D are not favorable to deposition; 

 the result is that the particle will be forced to take an additional step 

 of length A'lD. This can be repeated a number of times until the particle 

 finally finds a point on the bed where it is permitted to rest. This model 

 suggests that the actual distance travelled will be proportional to the 

 length of each step (A'lD) times the number of consecutive steps made in 

 each realization. This number of steps may be thought of as a discrete 

 random variable having a binomial distribution with parameter p. The 

 probability that the particle will travel a distance A'j^D is (1-p) which 

 is the probability of failure in the first trial. The probability of 

 covering a distance 2A'lD is p(l-p), the probability that failure will 

 follow one success. In general the probability of covering a distance 

 (n + 1) A'lD is p n (1-p) which is the probability of the first failure 

 occurring after n consecutive successes. The expected value of the dis- 

 tance covered by the particle in a single realization can be expressed 

 then by AlD where 



A L D = 2 p n (1-p) (1+n) A' D = - 

 n=o L 1 



A' L D 



(4-22) 



We introduce now this expression for A T D in equation (4-20) to obtain 



A „ A \ 

 _ _2 L _p_ 



3 B A A 1-p 



Y D 2 / g^Pf ) 



DP, 



(4-23) 



Solving (4-23) for p we get 



A* $ 

 1 + A„§ 



(4-24) 



where 



A, = 



A 1 A 3 



A A ' 

 A 2 A L 



(4-25) 



and 



9£ 



3/2 



-Pr) 



(4-26) 



We equate finally the right hand sides of equations (4-13a) and (4-24) to 

 obtain the very important relationship between flow intensity and bed-load 

 rate in the form of equation (4-27) 



tt/2 



2 



t_ 



o <- 



e 



TrV^rT 



B **- To 



dzdUJt 



A *$ 



(4-27) 



1+A *§ 



18 



