times. If a certain volume of sand V is instantaneously released at a point 



x at time t , the solution can be written 

 s s 



,, -(x-x ) 2 Ae(t-t ) 



, „s V \ s/ / s . . 



y(x,t) = — e (45) 



2D/ire(t - t ) 



for t > t and -°° < x < °° . 

 s 



Equation 45 has been discussed by Le Mehaute and Brebner (1961) and by 

 Le Mehaute and Soldate (1977). Accordingly, a superposition of an infinite 

 number of such released quantities can be used to represent the sand discharge 

 from a river. According to Carslaw and Jaeger (1959, p. 262), the solution 

 for a point source with a continuous time variable sand discharge q may be 

 expressed as 



)/^J R 



-(x-x s ) 2 /4e(t-0 _ :| 



y(x,t) — / q U) e x °" s (46) 



2D/^ J /t - C 



for t > and -°° < x < 



If q„ is constant and equal to q , the solution is 

 K O 



, r , q o rr -(v x ) 2 / 4et % 



- X x-x 



-=— 2i erfc J ^ (47) 



2e 2/FE 



for t > and -°° < x < °° . 



Equation 47 is identical to the solution describing a constant flux q /2 on 

 the boundary (x = 0) for a beach of semi-infinite extent. Figure 24 illu- 

 strates the solution where L is used as a normalizing length, and the point 

 source is located at x = L . The nondimensional quantity containing the 

 shoreline position is formed as the ratio between the amplitude of the sand 

 transport rate and the sand discharge from the river. 



38 



