y = Be cos 



^ 



c^ - ^J 



(18) 



which indicates that shoreline undulations tend to decay exponentially 

 and disappear with time. B defines the beach undulation amplitude at 

 time, t = o , and A is related to the wavelength, L , of this undula- 

 tion through the relationship: 



A = 



(^y 



(19) 



Shoreline evolution due to the sudden dumping of material at a given 

 point may be represented by: 



y = K 



^-x /4Kt 



(20) 



Equation (20) gives the spreading of the sand along the shoreline since 



/-co 



the integration / ydx, which expresses the conservation of sedi- 



I 



ment in the system, is a constant (see Fig. 8). This solution was also 

 mentioned by Pelnard-Considere . 



It is interesting that much later, Noda (personal communication, 

 1974) investigated the same problem by taking an initial condition for 

 sand dumping. 



y = f(x,o) 



Y = constant when U <X 



when |x| >X 



as shown on Fig. 9. Using the functional relationship now commonly 

 accepted, f(a) = sin 2a , Noda found that the solution to the diffusion 

 equation to be: 



y = 



erf 



(X - X) 

 2 VkF 



erf 



(-X + X) 

 2 Vl(t 



(21) 



24 



