is pushed shoreward by the waves. To reproduce the onshore-offshore 

 movement in a mathematical model, it is necessary to schematize the 

 coast by two or more contour lines instead of one. 



Bakker's [1968b} two-contour-line model is not easily applied to 

 practical engineering problems encountered by designers, due to lack of 

 knowledge about onshore-offshore transport. However, his contribution 

 toward establishing a realistic mathematical model of shoreline evolu- 

 tion is of sufficient importance to deserve detailed review. 



Bakker (1968b) assumes that the profile is divided into two parts 

 (Fig. 13). The upper parts extending to a depth, D , are affected by 



the groin, the part below D extends offshore to a depth of D + D 



which is the assumed practical seaward limit of material movement. 



The "equilibrium distance", w , is defined by a distance (y" - y, ) 



corresponding to an equilibrium profile under normal conditions; i.e., 

 far away from the groins . 



The onshore-offshore transport is defined by: 



Q = q 

 y y 



(y^ - y2 + w) 



(25) 



where q is a proportionality constant (dimension LT ) . When 



(y-i - y^ + w) is positive, the transport is seaward; when negative, it 



is shoreward. q has been found by Bakker for a part of the Dutch 

 coast equal to 1 to 10 meters per year for a depth D = 3 meters. 



Letting y2 = y; - w > then, Qy = q^ (y^ " ¥2-) . 



Now, following Pelnard-Considere; i.e., developing the expression 

 for the longshore transport rate Q in a Taylor series in terms of a 



(a - a ) 



(26) 



which gives in linear approximations: 



Q = Q. 



dQ 



da 



(27) 



33 



