57. The problem consists of two coupled partial differential equations 

 with appropriate boundary and initial conditions. Since the configuration is 

 symmetric with respect to the center of the river mouth (if q is constant), 

 only half of the problem domain has to be treated. The boundary conditions 

 are no sand transport through the center of the river (symmetry), and mass 

 conservation should apply between the two solution areas. Also, the beach 

 must be continuous at all times over this boundary. Furthermore, the shore- 

 line is unaffected by the river sand discharge as x approaches infinity. 

 According to Carslaw and Jaeger (1959, p. 80) the solution is 



yi (x,t) =— 



1 - 2i erfc 



2/et 



-.2 , / a + x 



- 2i erfc . 



2/et 



(55) 



for t > and < x < a . 



y (x,t) = 



2 V 



• 2 f 



i erfc 



2/et 



.2 ,, / x + a 

 - i erfc . 



2/et 



(56) 



for t > and x > a . 



58. The function ierfc is defined in Equation 23 and the superscript 

 2 denotes a double integration. An exponent n represents n integrations 

 of the complementary error function. The following recurrence relation holds 

 for n > 1 : 



„ .n . n ~2 .n-1 . 



2n l erfc x = i erfc x - 2x l erfc x 



(57) 



In Figure 26 the solution to Equations 55 and 56 is illustrated, 



42 



