time. The quantities in the figure are dimensionless, with the sand discharge 



from the river normalized by the amplitude of the sand transport rate Q and 



2 ° 



the angular frequency of the oscillation normalized by e/L . Figure 25 



clearly shows how the superimposed sinusoidal-shaped variation damps out with 



distance from the source along the x-axis. 



Sand Discharge from a River Mouth of Finite Length 



56. If the river mouth has a finite width in comparison to the area 

 into which it is discharging sand, an approximation by a point source is no 

 longer accurate. Instead of supplying sand to the system via the boundary or 

 initial conditions, the mass conservation equation in the full form of Equa- 

 tion 3 is applied. The sand discharge from the river q is considered a 

 continuous function of x , varying along the river mouth. The river mouth is 

 assigned a length 2a , and the sand discharge is measured per unit width. 

 Mathematically, the situation is expressed as 



2 



3 y, q R 3yi 



-r + d 5 ■ it ° s * s * (50 > 



3x 



3 y 2 3y : 



2~~ = It 



e — =— = — — x > a (51) 



y 1 (x,o) = y (x,o) = (52) 



3y x 3y 2 



3x ~ 9x 



9y 



.,. - x = (53) 



y, = y 



1 J 2 



y 2 = x -»■ » (54) 



41 



