II. THE FIRST MODEL (PELNARD-CONSIDERE) 



The idea of mathematically formulating shoreline evolution is attri- 

 buted by Bakker (1968a) to Bossen, but no reference to Bossen is given. 

 The first report which appears in the literature, on mathematical model- 

 ing of shoreline evolution, is by Pelnard-Considere (1956). His 

 theoretical developments were substantiated by laboratory experiments 

 made at Sogreah (Grenoble), France. The experimental results fit the 

 theoretical results very well. It is surprising that such relatively 

 simple theory has not been more frequently applied to prototype cases by 

 the profession (at least as it would appear from the open literature) , a 

 fact which may be attributed to the lack of knowledge of wave climates. 



Pelnard-Considere assumed that: 



(a) the beach profile remains similar and determined by 

 the equilibrium profile. Therefore, all contour lines are 

 parallel. This assumption permits him to consider the problem to 

 be solved for one contour line only. 



(b) The wave direction is constant and makes a small angle 

 with the shoreline (<20°). 



(c) The longshore transport, Q , is linearly related to the 

 tangent of the angle of incidence a • (Q - f (a) , f (a) = tan a) . 



(d) The beach has a fixed (ill-defined) depth, D (Fig. 1). 

 D is a factor relating erosion retreat to volume removed from 

 profile, which could be defined by the threshold velocity of 

 sand under wave action. A practical method of determination 



of D is given in Section VIII. 



Despite the crudeness of these approximations, the Pelnard-Considere 

 model can be considered as a milestone in demonstrating the feasibility 

 of mathematical modeling of long-term shoreline evolution. For this 

 reason, it is judged useful to describe in some detail his theoretical 

 development . 



Consider an axis, ox , parallel to the main coastal direction and 



an axis, oy , perpendicular seawards (Fig. 2). The angle the deepwater 



wave makes with the axis, ox , is a . The angle of the wave with the 



o 

 shoreline a at any location is assumed to be small; therefore, 



-I 3y 3y 3y ,,. 



a = a - tan ^ s a - -^ or a - a = - —- . (1) 



o 9x o 3x o Sx 



(y = f (x,t) gives the form of the shoreline as function of time t) . 

 The littoral drift Q is a function of angle incidence a and can be 

 put into a Taylor series: 



13 



